@@baranxlr You know, the first time I read this, the "n-word" bit went over my head, but it was still funny. Imagining the posts in the lobby being so inane and meaningless that they could only be described as "a finite number of finite word sequences".
Then you are being fooled. This paradox can be easily translated to formal language. The correct resolution is Tarsky's undefinability of truth, and there is no other resolution.
Q: How to irritate a mathematician? A: Create a mathematical paradox using natural language Q: How to really irritate a mathematician? A: Change the rules of the paradox when it is inevitably resolved
My solution to this paradox is that that sentence simply doesn't describe a natural number. Just like "set of all sets that don't contain themselves" doesn't actually describe set
Let’s try to formalise the argument a bit. Let Words be a finite set that contains some English words Words = {“smallest”, “the”, “number”, “natural”, “cannot”, “that”, “in”, “described”, “be”, …} Then consider the set Sentences that consists of sequences of up to 20 words. More specifically, Sentences = Words ∪ Words² ∪ … ∪ Words²⁰ Now, some of them will “describe a number”. We will call this subset Descriptions. Descriptions ⊂ Sentences This subset of sentences was chosen in such a way that every description corresponds to a single natural number. Let’s call this correspondence Means. Means: Descriptions → ℕ e.g., Means((“the”, “smallest”, “prime”)) = 2 The function maps short descriptions to natural numbers. Let’s call its range DescribableNumbers. Means[Descriptions] = DescribableNumbers, i.e. if we take all descriptions and evaluate what they represent, we will get a set of numbers that can be described in 20 words or less. Consider the sentence Paradox. Paradox = (“the”, “smallest”, “natural”, “number”, “that”, “cannot”, “be”, “described”, “in”, “twenty”, “words”, “or”, “less”) We claim that Paradox ∈ Descriptions. This is already not guaranteed by anything we said before! But let’s move on. Then we have the number it describes, ParadoxNumber. ParadoxNumber = Means(Paradox) We claim that this number is in DescribableNumbers by definition. (this makes total sense) ParadoxNumber ∈ DescribableNumbers We also claim that this number cannot be described with 20 words or fewer and so it’s not in DescribableNumbers. ParadoxNumber ∉ DescribableNumbers Here we are making an assumption than Means functions similar to “common sense”, but this is not guaranteed by any of our definitions! But let’s move on. We now have a contradiction. So let’s backtrack and see what assumptions might have caused it. 1. It is certainly true that Paradox ∈ Sentences, since it consists of the words we chose and it’s below 20 words in length. 2. It might not be the case that Paradox ∈ Descriptions. Our definition of “sentences that describe a single number” was a bit vague. 3. Assuming no other problems, it is certainly true that ParadoxNumber ∈ DescribableNumbers. This is because Paradox is a valid description and ParadoxNumber is obtained by applying the Means function. 4. It is not necessarily the case that ParadoxNumber ∉ DescribableNumbers. We didn’t really define how Means works internally. We simply assumed that it “makes sense”. It is entirely possible for it to not correspond to our intuition. For example, we didn’t exclude the possibility of Means((“one”, “plus”, “one”)) = 42. In conclusion, this video claims that the problem is that descriptions can correspond to multiple numbers. But even if we fix this problem, the issue does not go away. There are 2 real resolutions to the paradox. 1. The sentence we constructed does not describe a unique natural number; or 2. There is no “common sense” function that evaluates English text in the same way as we intuitively do.
Riddle me this Batman: 1) Take the words listed in the Oxford dictionary, set O, which are finite. 2) Set T contains every possible set of words from O with less than 20 words, which is finite. 3) Set Tn is the subset of T that contains every set of words that uniquely describes a natural number, which is finite. 4) There is a unique smallest natural number outside of Tn. 5) "Smallest natural number that cannot be uniquely described with less than 20 words from Oxford dictionary" simultaneously belongs and does not belong to Tn.
I'm reading "uniquely describes a natural number" as "there is exactly one number that this description can refer to" "Smallest natural number that cannot be uniquely described with less than 20 words " cannot refer to any single number, because , therefore the sentence does not belong to the set Tn, because there is no single number that it can refer to.
@kitlith Exactly! That's the conclusion I reached and I solve Russell's paradox in the same way. "The set of all sets that do not contain themselves" does not exist, only approximations exist.
@@Tata-ps4gy with Russell's paradox the problem is different. We assume (in an axiom) that for any set A, a set containing only elements of A that fulfill a certain requirement is also a valid set. So we can't just say that a set that contains every set that does not contain itself isn't a valid set. The existence of such a set is a direct consequence of the set of all sets existing. The only logical conclusion we can arrive at here is that the set of all sets doesn't exist
This doesn't seem like a good response to the paradox. It relies on an arbitrary feature of the English language, namely the fact that it requires fewer English words to describe a succession relationship than to refer to the smallest number that can't be described in less than __ words. We can easily imagine a language where this is not the case, in which the paradox would persist. There has to be a more profound and definitive way of dispelling the paradox
The way I see it, the problem is that we are mixing natural languages that are too vague with math that requires to be precise to solve problems in the first place. What does it really mean "describe"? For instance you can write an arbitrarily large number on a board and then you could say. "The number I wrote". That's certainly less than 20 words. Of course you could argue that with our finite ressources we can't possibly ever write a number that's arbitrarily large, but my point here is that natural language and math just don't always mix well.
It is the correct response to the paradox. As pointed out by someone else already: English is not formal logic, and mathematics cannot be formulated in English, or in any other so-called "natural language."
"the least natural number that this sentence does not describe", with extra steps. ...or if you wanna get abstract about it, it's "this sentence is false" with _several_ extra steps.
@@fullfungoIt’s actually any subset that can be defined using a property expressible in first-order logic. Until you define terms like “description” and “words” in first-order logic, you cannot claim that sets defined using those terms exist. Now such sets could exist, but you need to define the words “described” and “words” in FOL before you can prove it.
The way this is described is dependent upon what one means by a word and, as Sean pointed out early on, by a description. There’s a reformulation of Berry’s Paradox in formal languages that makes it clear that your objection doesn’t hold water.
@@ethannguyen2754 I think that you mean “first order property”, because “General first order property” comes with its own set of issues department from Berry’ paradox.
@@fullfungo the comment from @ethanguyen2754 is telling you that you were "completely wrong" (whatever that means). But @ethos8863 missed a beat too, which was nicely summarized by @ethanguyen2754 - Berry's Paradox _would be_ a case use of unrestricted comprehension _if_ "described by" was a condition in the formal language, but it is not, so we never even make it to unrestricted comprehension. So T is not a valid set. A logical notion one could use to define a set would be "computable with this Turing machine" (give the Turing machine) - but with this notion there is no way to get a paradox.
Your argument about the fact that a sentence can describe several number doesn't really "solve the paradox" : The implicit assumption is that a sentence describes a number if there is a unique number that satisfies the sentence. If you prefer, it is easy to rephrase the paradox to "The smallest number which can't be described by a sentence of fewer than thirty words which describe a single number". To solution to the paradox is that (and the paradox is in fact the proof of this fact) no language can do the following three things at the same time: (1) talk about itself, (2) is rich enough to formulate a sentence like the one in Berry's paradox, and (3) is complete, that is for every sentence you can always objectively decide if the sentence is true,false or meaningless. You need this to make sense of what it means that any sentence either "describe a number or don't" "English" (whatever that means) does (1) and (2), but not (3): there are sentences for which it can't be objectively decided if they describe a number or not, or which number they describe, and so you can build the set "T". But the interesting point is that this observation is Godel's incompleteness theorem. Godel's theorem is more formal of course, but that is essentially what it says: Any language which his "rich enough" (allow to basic formulate arithmetic) and "can talk about itself" (for e.g. formulate arithmetic and is recursively enumerable) must be incomplete.
2:20 Ive heard this paradox before, but this has never occurred to me until now: why would we assume that T is a finite set? Here's an english language description of a natural number: "A number less than or equal to infinity". Twelve words. Therefore, all natural numbers are describable in less than 20 words, and T is NOT a finite set
we have a dictionary with all the worlds which are finite (assumption), lets assume the number of available words are x, we can only arrange words in a string of 20 therefore you have x^20 posible arrangements of words incluiding the nonsensical, the thing is that the description should describe a number and only that number, theres a set of numbers that fit your description
@@gabrielgauchez9435 I mean, the video went on to basically make the same argument I was getting at but, why do we assume that a finite string of 20 words would describe one-and-only-one number? I mean, the paradox itself kind of illustrates this: "the smallest number not describable in under 20 words" does not describe a fixed specific number
@@rzeqdw some would describe a number, some would describe a certain a mount of number some would be griberish the string of 20 words is just puting an upper limit at the amount of number that can be uniquely described, you wont be able to describe more numbers uniquely than the amount of descriptions you can make, "the samllest number you can describe with 20 words" is something that you would think it uniquely describes a number because if it describes a bunch of numbers the smallest part would make you pick the one thats smallest, thats the paradox the point they make in the video is a way to solve it thats another topic
The problem is that we're letting the set be self referential. It's easy to construct "The first natural number not in this set" or "2 if all numbers in the set are multiplied together to make an odd number, otherwise 1".
If the statement in question defines a number then it leads to a contradiction - the result would be both in T and not in T. Therefore the statement does not define a number, any more than does the statement "One more than the number defined by this statement".
Dictionary mapping: 0 = {} 1 = {0} 2 = {0, 1} ... T is set of natural numbers whose description (number of elements in its representation) is of length n, where 0
A description that refers tk a previous steo isnt a standalone description; we wpuld need tp include the wording of the previous step to resolve it, and so that should count as part of our alloted word count.
But the central paradoxical description 'there is a least number which can be described...' cannot stand alone either. By stating the arbitrary rule that the description must standalone, that only further emphasizes the fact that 'there is a least number which can be described...' cannot be fully understood alone.
Pretty good. But I've always preferred the resolution that points out to "describe" or "name" a number is not a well-defined procedure. A sharper Berry Paradox is to consider the least algorithm expressed in a given formal language that can compute something. It is then impossible to generate a paradox. So the original Berry Paradox is not about mathematics, it is about informal language. Informal languages admit all sorts of paradoxes, because the meanings are not sharp. There is not sharp meaning of "described". There is a sharp meaning of "computed" but no paradox possible in that case.
Some things come to mind. We cannot ban the explicit reference to whether and which other entries are the set, because the paradoxical entry is formulated as such. We can, however, clean up the relation of word sequences to numbers. Note that we are allowing anywhere from 0 to 19 words in a sequence in the set of word sequences. We want each element to be associated with at most one natural number, and we want to be clear and rigorous to find the spot where the paradox happens, or an assumption is contradicted. We are given a dictionary D, a set of words. This is a finite set of size n. We generate the set W, being a set of all possible ordered lists of words that contain fewer than 20 words. This set is finite with size (n^20 - 1)/(n - 1). Logically speaking, for each element of W, it either describes one or numbers, or it does not. We create the relation t(x) from W -> Z+, relating each word-sequence to a natural number with the following rule: If x cannot describe a number, output 0. If x can describe one or more numbers, output the least natural number that it can describe. Let T be the set of all outputs of t(x) over all x in W; that is, T is now the set of all natural numbers that our dictionary can describe in less than 20 words, and the element 0. Since no element of W can relate to more than one natural number, T is finite. Consider the word sequence k = "The least natural number that cannot be described in less than twenty words." This 13 word sequence is in W. This generates the paradox as described in the video without ambiguity of a word-list having multiple number outputs. There are two red flags, and they both relate to the process of 'describing'. There are several lists of words that seem to describe a number, but are problematic. For example, g = "The greatest number that can be described in nineteen words or less plus one." This is a 14 word string, so must be in T, but it describes the largest element of T plus 1. That is, t(g) > t(x) for all x in W. Meaning t(g) > t(g), which is impossible. The problem here is that the 'paradox' tries to claim that when a phrase that seems to describe describe a number generates an impossible result, we cannot determine its result, but we can. When we generate W, the phrase "The least natural number that cannot be described in less than twenty words." is in there. It seems like it describes a number, but when we examine it, it becomes clear that it does not actually do so; If we assume k describes any number y, we conclude it doesn't describe the number y. Contradiction. This contradiction invalidates the case - that is, "The least natural number that cannot be described in less than twenty words." *does not describe one or more natural numbers*. It looks like it should, but it actually doesn't, since no natural number actually is described by it. Therefore it outputs 0. The same would be true for the 'g' statement above; since it any natural number we want to claim it describes ends up being a number it doesn't actually describe, it cannot describe any number, and so outputs to 0. (We don't actually need to output to 0, but it's a little cleaner if every possible input has an output). As such, the system described remains consistent, without these paradoxes. When a statement that seems to describe a number actually fails to describe any number at all, then that statement doesn't actually describe a number, so outputs 0. If the statement actually did describe one or more numbers, then it would output the smallest of those numbers to include in T, which remains a finite set. TL;DR - The paradox does not work, because the phrase "The least natural number that cannot be described in less than twenty words." is a phrase that does not actually describe any number (even though it *looks* like it should), and so doesn't add anything to the set. It's as meaningless to the set contents as "purple banana Eifel Tower monkey spleen".
You have created a nice description not of a given number but of the successor operation. And now hat the successor operation is properly defined, it can be used as 1 word. Then we can define all natural numbers in less than 20 words as: The Number is the successor of the number . (10 words) with a special definition for 0 to avoid iterating out of the domain of the successor operation.
I think that if you formulate the paradox in a formal language, your approach to explaining that it “has a flawed premise” will become more difficult to make convincing. This is discussed in the Wikipedia page for Berry’s Paradox. Your presentation essentially follows the line of reasoning given by Tarski, which Saul Kripke showed was itself a flawed explanation. You can learn there also that Gregory Chaitin showed that formalization removes the logical contradiction but that incompleteness results follow for resolution of Berry’s Paradox along these lines, and then in 1989, George Boolos used it in this way to give a simplified proof of Gödel’s first incompleteness theorem. You can read his article in the Notices of the Ansían Mathematical Society.
I think the resolution is a lot easier: the phrase simply doesn't identify a number, because it's self-contradictory. It's just a more elaborate version of "this sentence is false". Such sentences are meaningless, despite looking as though they should have a meaning. The fact that one can write well-formed but meaningless sentences in English shouldn't come as a surprise.
You can define this sentence inside a formal system, it only doesn't work because the predicate "is true" is limited in quantifier complexity, this is Tarsky's undefinability of truth.
I don't think I agree with your resolution of the paradox, but I still find it very interesting to think about. I can't quite figure out the problem with what you said.
That doesn't really resolve the paradox. Just replace "described" by "defined" to signify that descriptions are only counted when they apply to only one number, and the paradox persists. The actual way of resolving that kind of paradox is by resolving Russel's paradox, which modern set theory has already done.
I would argue that "__ is the set of all numbers mentioned in the previous step" depends on all the words in the previous step, and in the step before, and the step before ... so that, even if you meant "can be described in fewer than 20 DIFFERENT words," defining 20 would require all the numbers from 0 through 19, which is enough to exclude it. (or, at least, it would require all the numbers from 1 through 19 plus some other word that is not itself a number)
Why then could I not just as well say that every word is itself a compact form of its own definition, and thus we couldn't possibly describe any number in fewer than 20 words, since - for example, If I describe 2 as "the first prime number"; well that's not really a 4 word description because 'prime' depends on its definition as "a positive integer greater than..." and so on?
@@WrathofMath Well isn’t that how pronouns work? “Her name is Sarah” can be either true or false depending on the implied person. In logic this is usually expressed with variables, like “X’s name is Sarah” being a predicate in 1 variable. Similarly “the previous statement is false” would be something like “~P”, where P is a variable. This is different from “2 is prime” since it can be phrased as a predicate in 0 variables. I think you are conflating “context” as the logical context of a theory or predicate, i.e. values of all variable, and “context” as the set of prerequisites for understanding the meta-theory (in this case English). Context for “… in the previous step” is the value of “previous step” either in the language of formal logic or informal English that we use a replacement here. Context for “… is prime” is the definition of “prime” in the common English usage, which has nothing to do with math but rather how we use words.
I laughed when I read that but actually, in German, all natural numbers greater than 999,999 consist of more than one word. To be precise, for every power of 1000 beyond that number, we need 2 additional words, so by 10^33 - 10^6, we'd run out of our 20 word limit. Sorry for being pedantic, I'll see myself out.
8:20 "And even if we want to be generous and count the parentheses as words" If we want to be very precise, we could say that this sentence contains 19 words but only 17 WORDS (This is an extremely geeky programming joke)
The easiest resolution is to simply state that either English is not a well-defined set of mathematical axioms, or T is not a well-defined set. Pick whichever suits, but the former is probably fairest.
We could allow T to use more words. The limit of 20 was arbitrary for the sake of it being a resonable sentence length. But if we extend it to any finite sequence of words, then it is possible to create a non-context dependent sentence. "The first entry in the sequence of the recursion of the application of the union of the previous element of the sequence and the set containing that same element, starting with the empty set, which cannot be uniquely represented by a sequence of less than one hundred words in the standard Oxford English Dictionary in a way that does not depend on context."
Claim: the set of English words is actually infinite because number words are open compound words. Example: 123,456,789 is One hundred and twenty three million four hundred and fifty six thousand seven hundred and eighty nine. But we don’t say that’s 17 words, it’s actually just one big compound word describing one number. Peanut butter, for example, has two individual words which mean different things (a type of legume, a solid emulsion made from churned milk or cream) but together form an open compound word which means a singular third thing (a creamy food made from peanuts). Peanut butter is listed in the dictionary as a singular word despite containing a space. Though English does not have number words for every level of a thousand, we can use repeated words to infinitely construct them. For example, we can say a trillion or a million million.
17 was the smallest boring number. Therefore, one of my professors always used the number 17 when he had to choose a random number. Since then, 17 is no longer in the list of boring numbers, since it refers to that professor.
"The least number that cannot be described in less than 20 words" does *not* describe a specific number. This is immediately clear because assuming so leads to the paradox. Indeed it is a problem with English language because the statement is self-referential in a subtle way (it is a description in less than 20 words talking about descriptions in less than 20 words). I wish self-references would be studied more in logic, enabling us to reason despite their presence, instead of trying to avoid them via ad-hoc heuristics and hope for the best.
So many paradoxes are just a consequence of circular reference, which seems to be mostly, if not entirely, a feature of the language we use to define things. In this case, I would propose to fix the paradox by redefining the set to only allow descriptions with no relation to the set, or by extension, its compliment. So the smallest number outside the set remains outside the set.
This doesn't resolve a similar paradox where the descriptions are up to 20,000,000 words long. That's enough words to re-explain maths from scratch, build the same contradiction within it, and refer into that. I gather that's basically what Goedel constructed in his Incompleteness Theorem, but in more formal language.
@SimonClarkstone Could you build the same contradiction within it without using any circular references? I don't know that it's always possible to avoid circular references, even with formal language, but what I'm suggesting is that these paradoxes may not lie within math and logic themselves, but in the way we define things. If that's the case, it feels somewhat dishonest to call them logical paradoxes rather than acknowledge that they're just consequences of our own limitations in defining things.
Hey! Can you resolve the Greyling Nelson paradox? Dialect has a video in it. It's funny that, on confidence, Bertrand Russell is occurring all over my phone, mostly from Philosophy, but now in mathematics too 😂
The implicit assumption is that a sentence describes a number if there is a unique number that satisfy the sentence, so T is finite. But if this is your objection, then it is easy to rephrase the paradox to "The smallest number which can't be described by a sentence of less than thirty words which describe a single number".
"Let T be the set of all natural numbers with description shorter than 20 words that unique describe only a single number." The paradox is back! You can't resolve the paradox. Gödel proved that either the set is incomplete or it is inconsistent.
this works for a specific language and a specific number of words. we can define an arbitrary language with arbitrary words which have a maximally condensed meaning in few words. also, using "preceding steps" in a description is nonsensical, when you say it's less than 20 words. the whole preceding step in itself is not defined in the description, or in the language. it's a contextual definition, not part of the current description or the base definitions of the language.
When I heard 3 i just said "all natural numbers" and so when you dropped 4 i was very confused. We are using dumb words in our premesis so I can use them in the argument.
I think the only really necessarily context-dependent concept is the meaning of a phrase as a part of what it means to describe something. Everything else can be formalised
I think the paradox can still live without this "non-context-dependent" idea. Like so: "There is a least natural number which cannot be described in fewer than 100 characters".
I'm mainly confused not about the paradox and the solutions mentioned in this video, but more so in what the paradox would even accomplish. I mean, seriously, IF Berry's paradox was genuinely "true", as in there is no valid explanation and there really does exist a contradiction. Okay, cool, what then? What does that say about mathematics? Because it just seems to me like the result could only be possibly used for linguistics and maybe philosophical concepts, but if they tried to, for example, invalidate set theory in mathematics, THOSE statements would surely be non-sensical.
Natural language is internally inconsistent. Every word used in every definition in a dictionary is defined in the dictionary. All purely linguistic definitions are circular on a finite set of words. You need context outside a dictionary from experience to get anything more concrete and even then it isn't that concrete. Any attempt to do formal math USING natural language is going to butt up against this problem. That's why we create formal mathematical languages. And even then, with Godol's Theorem, we know we can't prove they are consistent. But at least we don't know they are inconsistent like we do with natural language.
the problem with Berry's paradox is far way simple: you don't even know if you can applie a way to make them discribe anything. You could say that us, humans, do understand words, but can we? we understand OUR words that are describe by each and every human differently, and even with that, words can have very different meaning, and also depend on the physical world with a lot to define (what is a "chair", what are "inside" and "outside", what is "freedom", ect.) so use the word "define" in this context have to be "define", and this is already hard to do, if not impossible. in this way, there can effectively be a way to make an infinite amount as well as a finite amount of number with it, as we can't know already if the "numbers" describe exist. I would not say that Berry's paradox is entierly solved, but with the way mathematics is done these day it seems like a nonsense because of the first-order logic.
Riddle me this Batman: 1) Take the words listed in the Oxford dictionary, set O, which are finite. 2) Set T contains every possible set of words from O with less than 20 words, which is finite. 3) Set Tn is the subset of T that contains every set of words that uniquely describes a natural number, which is finite. 4) There is a unique smallest natural number outside of Tn. 5) "Smallest natural number that cannot be uniquely described with less than 20 words from Oxford dictionary" simultaneously belongs and does not belong to Tn.
If the smallest natural number is eg 5, couldn't we say that "the smallest natural number" isn't a unique description because it can also be described as "5"?
This isn't the correct resolution of the paradox. The paradox uses "description" to mean "unique description". To clarify this, you should state the paradox inside a precise formal language first, best is second order arithmetic. Using a Godel encoding for sentences, the code of a predicate P(n) with one free variable n of natural-number type defines a unique natural number when the formal sentence S(P) defined from P as "(exists n P(n)) and (forall m (P(m) implies m=n))" is true. This uses Tarsky's definition of "true" inside a formal system, a way of recursively converting the code for S into the meaning of S. Famously, Tarsky's truth definition is limited--- you can't define it for all the sentences in a theory, you can only define it up to some limited number of quantifier alternations in front. That's key. So try to make the paradox. Consider the set C of all numbers x such that there exists a formal predicate P(n) with byte-length(S(P)) < 1,000,000,000 such that S(P) with x substituted everywhere for n inside, is true. This is a finite set, just as advertised, because there are only finitely many P. So there is a least n not in this set, and the formal description of n takes less than a billion bytes. Paradox formalized. This is ONLY not a contradiction because the truth predicate is limited in quantifier complexity. The sentence defining the paradoxical number has an additional quantifier compared to the one for which the truth predicate inside it is defined. That's the resolution, Tarsky's undefinability of truth, and there is no other resolution.
...the dictionary definition of number can be easily reworded to be less than 20 words (oxford definition 1. is 21 words). So there is no number outside the set right from the start. You can easily say "a value with real and/or imaginary components expressed in digits, symbols or words." And if something is missing, I'm certain one or two more sentences can include them.
Having watched to 5:17, my question is: Can there be a "least number not in T"? That is, since there is no way to determine or find that number therefore even though we have a description that is less than 20 words, it does not count as a description of a number. Let's see what the rest of the video says ...
The premise is that, whether we know what that particular number is or not - certainly if all premises to that point are correct - we have offered an accurate description of it. Just like I might describe a number as "the smallest prime number we have not yet found". I don't know what that number is, but I am definitely describing a particular number (we'd just have to settle the details about what 'have not found' means.
3 ways to resolve this i think: 1. T is not a set 2. No high order description 3. The map from T to the set of desciptions is not injective (this video)
Bah, I can break this. Just include all the context and definitions, and it will be a finite description and therefore we can use quine-encoding techniques to recreate the self-reference efficiently.
I use a turing complete programming language to write a program with less than n chars that will go through every program with less than n chars and store their results. Then it will give smallest number not among those results. Such program can never halt since it depends on it's own result.
"The least natural number that cannot be described in fewer than twenty-five words without referring to the description of another number" is a sentence with less than 25 words that describes a number This one does not require me to prove that every single word is non-context-dependent, because it only disallows referring to other descriptions of natural numbers, and so the paradox still persists.
This is actually fine and non-paradoxical, because your description itself *refers to other descriptions,* which means it doesn't forbid its own existence.
Even the premise that the description has to contain fewer than 20 words is arbitrary. Had your description contained say 22 words would that have really made it less viable as the solution to Barry’s paradox?
10:10 that’s the point though, yeah, you can not define a natural number without its context, and thusly T is finite. If your definition is not exclusive it’s a bad definition for establishing a paradox
You don't need to start talking about context dependence to get a revenge paradox. Just let T be the set of all natural numbers which can be uniquely described in fewer than 20 words.
I'm at 2:47. I predict that the issue will be that set theory generates paradoxes if you include self-set-referential elements in a set. Group theory FTW :)
this feels very contrived. most obviously, the definition "the least number that can be defined in less than 20 words" is 12 words, so that could easily be changed to "least number that can be defined in less than 13 words," which would make it very difficult to phrase a formal definition of a number within that. but a definition that rules out this entire solution is as simple as "the least number that can be *uniquely* defined in fewer than 20 words."
Every description in the infinite sequence of descriptions proposed, when viewed in context, does uniquely describe a single number. So the argument would come down to the validity of that strategy.
@WrathofMath OK, what about a more specific phrasing of the same then? "the least number that cannot be defined with a unique sequence of 20 words or less"
This solution of the problem is the linguistic one, but to understand it you have to go to computer science. Think of the dictionary as a set of lexical objects. A simple sentence is a directed graph of lexical objects. If you restrict yourself to simple sentences, you can guarantee that the sentence can be parsed (i.e. it has at least the possibility of meaning). If however the sentence is self-referential in any way (i.e. not a simple sentence) there is no way to guarantee that the sentence can be parsed (see the halting problem). In simple terms, if you build a set without referring to the set, you can guarantee that a fixed (possibly infinite) set exists. The moment you start referring to the set as you construct your set, your set can potentially be meaningless since it cannot resolve to a fixed result for what the set is. You can find a funny example of this in this Ryan George skit ( th-cam.com/video/hf_7xAX_fBE/w-d-xo.html ).
Obviously you want the sentences that describe numbers to be self-contained and to describe a unique number, otherwise it is discarded along with all the other nonsensical sentences. You want to be able to give it to an arbitrary idealized mathematician and have them say "Sure, I agree that that describes a unique number", without having shown them any of the other sentences. We can even go so far as to detail exactly what mathematical knowledge this mathematician knows, which we are therefore allowed to refer to in our sentences. Your "set that contains all previous iterations of this sentence" thing trivially fails this test. Not so for the original paradox sentence. I mean, it might fail that test, and that could be a legitimate refutation of the paradox, but certainly not trivially so.
okay but like... you could just increase the word limit and be much more comprehensive about everything, even construct language from scratch if you must, and the "repaired" paradox would stand; hell, it could be "repaired" in a less self-destructive way if need be, like maybe specifying "non-recursively" instead of specifically "non-context dependent words" what you're saying in the video is true, sure, but it also kinda misses the point? like, it's not that this is unsatisfying because it's "cheating" per se, it's unsatisfying because it completely ignores the actual logical foundation of the paradox and another one like it can still be constructed that would be immune to this rebuttal
The descriptions should be able to stand alone. The reason that your re-use of descriptions feels like cheating is not that it uses the term "description," but the fact that it refers to "previous steps." A disambiguated description contains all those previous steps, and quickly exceeds twenty words. Similarly, your set theory description of 3 can be written out as {{}, {{}}, {{}, {{}}}}. Now, if you really want to resolve the paradox, you would note that you can't compute the number supposedly described. But you don't. You do the equivalent of tossing the board and declaring yourself the winner.
If you insist the descriptions must be able to stand alone, the paradox is still resolved, since the description central to the paradox 'there is a least natural number which cannot be...' cannot itself stand alone, and thus doesn't qualify as a valid description. Being able to compute the number any description accords to was never a stated restriction, and if you require it as a restriction that only further emphasizes the inability of the central paradoxical description to stand alone.
@@WrathofMath "'there is a least natural number which cannot be...' cannot itself stand alone," Why not? The least natural number that cannot be // expressed as the sum of squares of three or fewer natural numbers is 7. And it is unambiguous.
@@irrelevant_noob ZFC is not concerned with ontology, the set theoretic definition of natural numbers is not an ontological claim. It is only concerned with theorems can be proven about them.
Yet another video that completely misses the point, literally none of what you said has anything to do with the underlying mathematical ideas of the Berry Paradox, rather than just with weird irrelevant linguistic nitpicks. The _actual_ Berry "Paradox" is about constructing something that's fundamentally equivalent to a Godel Numbering, and then being surprised at the Incompleteness result. The "smallest number not described in seven words" _is_ a valid statement, it's just not _decidable_ in exactly the same way that any other problem equivalent to the halting problem isn't decidable.
🎯 Key points for quick navigation: 00:00 *📚 Mathematics was experiencing a foundational crisis in the early 20th century, leading to a focus on set theory.* 00:25 *🔄 Paradoxes, like Berry's Paradox, were discovered within set theory, needing resolution.* 00:53 *🧑🔬 Berry's Paradox was described by Bertrand Russell, attributed to librarian GH Berry.* 01:07 *📖 The paradox begins with the assumption of a finite collection of English words, from which a contradiction emerges.* 02:01 *🔢 From word sequences, a set of natural numbers describable within 20 words can be identified as finite.* 03:09 *❌ A contradiction is highlighted, showing a number is both described and not described in fewer than 20 words.* 04:14 *🤔 The paradox is argued to be a linguistic problem rather than a mathematical one.* 05:19 *🧩 The paradox's mathematical premise, regarding the finiteness of descriptions, is scrutinized.* 06:13 *🗂️ Resolving the paradox involves using one description for infinitely many numbers.* 07:19 *🏗️ Set theory's approach to defining natural numbers provides a resolution through recursive definitions.* 08:41 *🌀 Descriptions for numbers must exist in context to resolve the paradox.* 09:47 *🧐 An insistence on non-context dependent descriptions leads to reevaluation but is argued to ultimately fail.* 13:42 *😲 The paradox is likened to the "boring number" paradox, illustrating a similar logical contradiction.* Made with HARPA AI
@@WrathofMath Similar to how people cite "this sentence is false" as a paradox more often than they cite "this sentence is not true", even though the latter sidesteps a trivial solution to the former. Likewise here, "it's a number" is a description that describes any number, so there's literally no paradox when expressed that way.
"Finite number of n-word sequences"
Yeah it's called a COD lobby
he said finite
Immediately had to check comments after reading it 😂
@@liteseve Same here, our minds does function similarly in many non-context (out of context) depeneded scenarios.
😅
@@baranxlr You know, the first time I read this, the "n-word" bit went over my head, but it was still funny. Imagining the posts in the lobby being so inane and meaningless that they could only be described as "a finite number of finite word sequences".
I love paradoxes which can be boiled down to "natural language isn't formal logic and that confuses me"
Berry’s Paradox was popularised by Bertrand Russell.
Then you are being fooled. This paradox can be easily translated to formal language. The correct resolution is Tarsky's undefinability of truth, and there is no other resolution.
Q: How to irritate a mathematician?
A: Create a mathematical paradox using natural language
Q: How to really irritate a mathematician?
A: Change the rules of the paradox when it is inevitably resolved
Best summary of this I've seen.
My solution to this paradox is that that sentence simply doesn't describe a natural number. Just like "set of all sets that don't contain themselves" doesn't actually describe set
Let’s try to formalise the argument a bit.
Let Words be a finite set that contains some English words
Words = {“smallest”, “the”, “number”, “natural”, “cannot”, “that”, “in”, “described”, “be”, …}
Then consider the set Sentences that consists of sequences of up to 20 words. More specifically,
Sentences = Words ∪ Words² ∪ … ∪ Words²⁰
Now, some of them will “describe a number”. We will call this subset Descriptions.
Descriptions ⊂ Sentences
This subset of sentences was chosen in such a way that every description corresponds to a single natural number. Let’s call this correspondence Means.
Means: Descriptions → ℕ
e.g., Means((“the”, “smallest”, “prime”)) = 2
The function maps short descriptions to natural numbers. Let’s call its range DescribableNumbers.
Means[Descriptions] = DescribableNumbers,
i.e. if we take all descriptions and evaluate what they represent, we will get a set of numbers that can be described in 20 words or less.
Consider the sentence Paradox.
Paradox = (“the”, “smallest”, “natural”, “number”, “that”, “cannot”, “be”, “described”, “in”, “twenty”, “words”, “or”, “less”)
We claim that Paradox ∈ Descriptions.
This is already not guaranteed by anything we said before! But let’s move on.
Then we have the number it describes, ParadoxNumber.
ParadoxNumber = Means(Paradox)
We claim that this number is in DescribableNumbers by definition. (this makes total sense)
ParadoxNumber ∈ DescribableNumbers
We also claim that this number cannot be described with 20 words or fewer and so it’s not in DescribableNumbers.
ParadoxNumber ∉ DescribableNumbers
Here we are making an assumption than Means functions similar to “common sense”, but this is not guaranteed by any of our definitions! But let’s move on.
We now have a contradiction. So let’s backtrack and see what assumptions might have caused it.
1. It is certainly true that Paradox ∈ Sentences, since it consists of the words we chose and it’s below 20 words in length.
2. It might not be the case that Paradox ∈ Descriptions. Our definition of “sentences that describe a single number” was a bit vague.
3. Assuming no other problems, it is certainly true that ParadoxNumber ∈ DescribableNumbers. This is because Paradox is a valid description and ParadoxNumber is obtained by applying the Means function.
4. It is not necessarily the case that ParadoxNumber ∉ DescribableNumbers. We didn’t really define how Means works internally. We simply assumed that it “makes sense”. It is entirely possible for it to not correspond to our intuition. For example, we didn’t exclude the possibility of
Means((“one”, “plus”, “one”)) = 42.
In conclusion, this video claims that the problem is that descriptions can correspond to multiple numbers. But even if we fix this problem, the issue does not go away. There are 2 real resolutions to the paradox.
1. The sentence we constructed does not describe a unique natural number; or
2. There is no “common sense” function that evaluates English text in the same way as we intuitively do.
you were so close... Means((“six”, “by”, “nine”)) = 42.
@@irrelevant_noob 420*
😉
Riddle me this Batman:
1) Take the words listed in the Oxford dictionary, set O, which are finite.
2) Set T contains every possible set of words from O with less than 20 words, which is finite.
3) Set Tn is the subset of T that contains every set of words that uniquely describes a natural number, which is finite.
4) There is a unique smallest natural number outside of Tn.
5) "Smallest natural number that cannot be uniquely described with less than 20 words from Oxford dictionary" simultaneously belongs and does not belong to Tn.
You didn’t prove that it does not belong in Tn.
I'm reading "uniquely describes a natural number" as "there is exactly one number that this description can refer to"
"Smallest natural number that cannot be uniquely described with less than 20 words " cannot refer to any single number, because , therefore the sentence does not belong to the set Tn, because there is no single number that it can refer to.
@kitlith Exactly! That's the conclusion I reached and I solve Russell's paradox in the same way. "The set of all sets that do not contain themselves" does not exist, only approximations exist.
@@Tata-ps4gy with Russell's paradox the problem is different. We assume (in an axiom) that for any set A, a set containing only elements of A that fulfill a certain requirement is also a valid set. So we can't just say that a set that contains every set that does not contain itself isn't a valid set. The existence of such a set is a direct consequence of the set of all sets existing. The only logical conclusion we can arrive at here is that the set of all sets doesn't exist
@@feliksporeba5851 I think it makes more sense to reject the axiom of creation of subsets from arbitrary conditions.
This doesn't seem like a good response to the paradox. It relies on an arbitrary feature of the English language, namely the fact that it requires fewer English words to describe a succession relationship than to refer to the smallest number that can't be described in less than __ words. We can easily imagine a language where this is not the case, in which the paradox would persist. There has to be a more profound and definitive way of dispelling the paradox
The way I see it, the problem is that we are mixing natural languages that are too vague with math that requires to be precise to solve problems in the first place. What does it really mean "describe"?
For instance you can write an arbitrarily large number on a board and then you could say. "The number I wrote". That's certainly less than 20 words.
Of course you could argue that with our finite ressources we can't possibly ever write a number that's arbitrarily large, but my point here is that natural language and math just don't always mix well.
It is the correct response to the paradox. As pointed out by someone else already: English is not formal logic, and mathematics cannot be formulated in English, or in any other so-called "natural language."
"the least natural number that this sentence does not describe", with extra steps.
...or if you wanna get abstract about it, it's "this sentence is false" with _several_ extra steps.
T just isn't a set in the first place. This is just a rephrasal of Russel's paradox where they found a contradiction in unrestricted comprehension.
You are completely wrong. Every subset of natural numbers is a set. This follows from the axiom of specification.
@@fullfungoIt’s actually any subset that can be defined using a property expressible in first-order logic.
Until you define terms like “description” and “words” in first-order logic, you cannot claim that sets defined using those terms exist.
Now such sets could exist, but you need to define the words “described” and “words” in FOL before you can prove it.
The way this is described is dependent upon what one means by a word and, as Sean pointed out early on, by a description. There’s a reformulation of Berry’s Paradox in formal languages that makes it clear that your objection doesn’t hold water.
@@ethannguyen2754
I think that you mean “first order property”, because “General first order property” comes with its own set of issues department from Berry’ paradox.
@@fullfungo the comment from @ethanguyen2754 is telling you that you were "completely wrong" (whatever that means). But @ethos8863 missed a beat too, which was nicely summarized by @ethanguyen2754 - Berry's Paradox _would be_ a case use of unrestricted comprehension _if_ "described by" was a condition in the formal language, but it is not, so we never even make it to unrestricted comprehension. So T is not a valid set.
A logical notion one could use to define a set would be "computable with this Turing machine" (give the Turing machine) - but with this notion there is no way to get a paradox.
Your argument about the fact that a sentence can describe several number doesn't really "solve the paradox" : The implicit assumption is that a sentence describes a number if there is a unique number that satisfies the sentence. If you prefer, it is easy to rephrase the paradox to "The smallest number which can't be described by a sentence of fewer than thirty words which describe a single number".
To solution to the paradox is that (and the paradox is in fact the proof of this fact) no language can do the following three things at the same time: (1) talk about itself, (2) is rich enough to formulate a sentence like the one in Berry's paradox, and (3) is complete, that is for every sentence you can always objectively decide if the sentence is true,false or meaningless. You need this to make sense of what it means that any sentence either "describe a number or don't"
"English" (whatever that means) does (1) and (2), but not (3): there are sentences for which it can't be objectively decided if they describe a number or not, or which number they describe, and so you can build the set "T".
But the interesting point is that this observation is Godel's incompleteness theorem. Godel's theorem is more formal of course, but that is essentially what it says: Any language which his "rich enough" (allow to basic formulate arithmetic) and "can talk about itself" (for e.g. formulate arithmetic and is recursively enumerable) must be incomplete.
2:20
Ive heard this paradox before, but this has never occurred to me until now: why would we assume that T is a finite set?
Here's an english language description of a natural number: "A number less than or equal to infinity". Twelve words. Therefore, all natural numbers are describable in less than 20 words, and T is NOT a finite set
we have a dictionary with all the worlds which are finite (assumption), lets assume the number of available words are x, we can only arrange words in a string of 20 therefore you have x^20 posible arrangements of words incluiding the nonsensical, the thing is that the description should describe a number and only that number, theres a set of numbers that fit your description
You need to describe each natural number specifically. Your definition does not describe the number 341245.
You can leave out the ‘or equal to’ part for a natural number.
@@gabrielgauchez9435 I mean, the video went on to basically make the same argument I was getting at but, why do we assume that a finite string of 20 words would describe one-and-only-one number? I mean, the paradox itself kind of illustrates this: "the smallest number not describable in under 20 words" does not describe a fixed specific number
@@rzeqdw some would describe a number, some would describe a certain a mount of number some would be griberish the string of 20 words is just puting an upper limit at the amount of number that can be uniquely described, you wont be able to describe more numbers uniquely than the amount of descriptions you can make, "the samllest number you can describe with 20 words" is something that you would think it uniquely describes a number because if it describes a bunch of numbers the smallest part would make you pick the one thats smallest, thats the paradox the point they make in the video is a way to solve it thats another topic
The problem is that we're letting the set be self referential. It's easy to construct "The first natural number not in this set" or "2 if all numbers in the set are multiplied together to make an odd number, otherwise 1".
If the statement in question defines a number then it leads to a contradiction - the result would be both in T and not in T. Therefore the statement does not define a number, any more than does the statement "One more than the number defined by this statement".
Dictionary mapping:
0 = {}
1 = {0}
2 = {0, 1}
...
T is set of natural numbers whose description (number of elements in its representation) is of length n, where 0
How is this a paradox?
@fullfungo I'm thinking it is not, and yet it is similar to what is presented as possibly paradoxical. What do you think?
@@waylonbarrett3456I also think it’s not a paradox
A description that refers tk a previous steo isnt a standalone description; we wpuld need tp include the wording of the previous step to resolve it, and so that should count as part of our alloted word count.
But the central paradoxical description 'there is a least number which can be described...' cannot stand alone either. By stating the arbitrary rule that the description must standalone, that only further emphasizes the fact that 'there is a least number which can be described...' cannot be fully understood alone.
Pretty good. But I've always preferred the resolution that points out to "describe" or "name" a number is not a well-defined procedure. A sharper Berry Paradox is to consider the least algorithm expressed in a given formal language that can compute something. It is then impossible to generate a paradox. So the original Berry Paradox is not about mathematics, it is about informal language. Informal languages admit all sorts of paradoxes, because the meanings are not sharp. There is not sharp meaning of "described". There is a sharp meaning of "computed" but no paradox possible in that case.
Some things come to mind.
We cannot ban the explicit reference to whether and which other entries are the set, because the paradoxical entry is formulated as such.
We can, however, clean up the relation of word sequences to numbers.
Note that we are allowing anywhere from 0 to 19 words in a sequence in the set of word sequences. We want each element to be associated with at most one natural number, and we want to be clear and rigorous to find the spot where the paradox happens, or an assumption is contradicted.
We are given a dictionary D, a set of words. This is a finite set of size n.
We generate the set W, being a set of all possible ordered lists of words that contain fewer than 20 words. This set is finite with size (n^20 - 1)/(n - 1).
Logically speaking, for each element of W, it either describes one or numbers, or it does not.
We create the relation t(x) from W -> Z+, relating each word-sequence to a natural number with the following rule:
If x cannot describe a number, output 0.
If x can describe one or more numbers, output the least natural number that it can describe.
Let T be the set of all outputs of t(x) over all x in W; that is, T is now the set of all natural numbers that our dictionary can describe in less than 20 words, and the element 0.
Since no element of W can relate to more than one natural number, T is finite.
Consider the word sequence k = "The least natural number that cannot be described in less than twenty words." This 13 word sequence is in W.
This generates the paradox as described in the video without ambiguity of a word-list having multiple number outputs.
There are two red flags, and they both relate to the process of 'describing'.
There are several lists of words that seem to describe a number, but are problematic. For example, g = "The greatest number that can be described in nineteen words or less plus one." This is a 14 word string, so must be in T, but it describes the largest element of T plus 1. That is, t(g) > t(x) for all x in W. Meaning t(g) > t(g), which is impossible.
The problem here is that the 'paradox' tries to claim that when a phrase that seems to describe describe a number generates an impossible result, we cannot determine its result, but we can.
When we generate W, the phrase "The least natural number that cannot be described in less than twenty words." is in there. It seems like it describes a number, but when we examine it, it becomes clear that it does not actually do so; If we assume k describes any number y, we conclude it doesn't describe the number y. Contradiction.
This contradiction invalidates the case - that is, "The least natural number that cannot be described in less than twenty words." *does not describe one or more natural numbers*. It looks like it should, but it actually doesn't, since no natural number actually is described by it. Therefore it outputs 0. The same would be true for the 'g' statement above; since it any natural number we want to claim it describes ends up being a number it doesn't actually describe, it cannot describe any number, and so outputs to 0.
(We don't actually need to output to 0, but it's a little cleaner if every possible input has an output).
As such, the system described remains consistent, without these paradoxes. When a statement that seems to describe a number actually fails to describe any number at all, then that statement doesn't actually describe a number, so outputs 0. If the statement actually did describe one or more numbers, then it would output the smallest of those numbers to include in T, which remains a finite set.
TL;DR - The paradox does not work, because the phrase "The least natural number that cannot be described in less than twenty words." is a phrase that does not actually describe any number (even though it *looks* like it should), and so doesn't add anything to the set. It's as meaningless to the set contents as "purple banana Eifel Tower monkey spleen".
You have created a nice description not of a given number but of the successor operation. And now hat the successor operation is properly defined, it can be used as 1 word. Then we can define all natural numbers in less than 20 words as:
The Number is the successor of the number . (10 words) with a special definition for 0 to avoid iterating out of the domain of the successor operation.
I think that if you formulate the paradox in a formal language, your approach to explaining that it “has a flawed premise” will become more difficult to make convincing. This is discussed in the Wikipedia page for Berry’s Paradox. Your presentation essentially follows the line of reasoning given by Tarski, which Saul Kripke showed was itself a flawed explanation. You can learn there also that Gregory Chaitin showed that formalization removes the logical contradiction but that incompleteness results follow for resolution of Berry’s Paradox along these lines, and then in 1989, George Boolos used it in this way to give a simplified proof of Gödel’s first incompleteness theorem. You can read his article in the Notices of the Ansían Mathematical Society.
I think the resolution is a lot easier: the phrase simply doesn't identify a number, because it's self-contradictory. It's just a more elaborate version of "this sentence is false". Such sentences are meaningless, despite looking as though they should have a meaning. The fact that one can write well-formed but meaningless sentences in English shouldn't come as a surprise.
You can define this sentence inside a formal system, it only doesn't work because the predicate "is true" is limited in quantifier complexity, this is Tarsky's undefinability of truth.
I don't think I agree with your resolution of the paradox, but I still find it very interesting to think about. I can't quite figure out the problem with what you said.
That doesn't really resolve the paradox. Just replace "described" by "defined" to signify that descriptions are only counted when they apply to only one number, and the paradox persists. The actual way of resolving that kind of paradox is by resolving Russel's paradox, which modern set theory has already done.
I would argue that "__ is the set of all numbers mentioned in the previous step" depends on all the words in the previous step, and in the step before, and the step before ... so that, even if you meant "can be described in fewer than 20 DIFFERENT words," defining 20 would require all the numbers from 0 through 19, which is enough to exclude it. (or, at least, it would require all the numbers from 1 through 19 plus some other word that is not itself a number)
Why then could I not just as well say that every word is itself a compact form of its own definition, and thus we couldn't possibly describe any number in fewer than 20 words, since - for example, If I describe 2 as "the first prime number"; well that's not really a 4 word description because 'prime' depends on its definition as "a positive integer greater than..." and so on?
@@WrathofMath Well isn’t that how pronouns work?
“Her name is Sarah” can be either true or false depending on the implied person. In logic this is usually expressed with variables, like “X’s name is Sarah” being a predicate in 1 variable.
Similarly “the previous statement is false” would be something like “~P”, where P is a variable.
This is different from “2 is prime” since it can be phrased as a predicate in 0 variables.
I think you are conflating “context” as the logical context of a theory or predicate, i.e. values of all variable, and “context” as the set of prerequisites for understanding the meta-theory (in this case English).
Context for “… in the previous step” is the value of “previous step” either in the language of formal logic or informal English that we use a replacement here.
Context for “… is prime” is the definition of “prime” in the common English usage, which has nothing to do with math but rather how we use words.
So basically, you're performing induction within the set. Brilliant.
T is actually the set of all natural numbers. Just use German.
I laughed when I read that but actually, in German, all natural numbers greater than 999,999 consist of more than one word.
To be precise, for every power of 1000 beyond that number, we need 2 additional words, so by 10^33 - 10^6, we'd run out of our 20 word limit.
Sorry for being pedantic, I'll see myself out.
@j0code Thanks! I figured something like that.
If the description doesn't have to be unique, then all numbers can be described as "an even number" and "an odd number"
I thought you were going for the axiom of foundation, or the axiom of schema of specification, but this works too.
8:20 "And even if we want to be generous and count the parentheses as words"
If we want to be very precise, we could say that this sentence contains 19 words but only 17 WORDS
(This is an extremely geeky programming joke)
The easiest resolution is to simply state that either English is not a well-defined set of mathematical axioms, or T is not a well-defined set.
Pick whichever suits, but the former is probably fairest.
We could allow T to use more words. The limit of 20 was arbitrary for the sake of it being a resonable sentence length. But if we extend it to any finite sequence of words, then it is possible to create a non-context dependent sentence.
"The first entry in the sequence of the recursion of the application of the union of the previous element of the sequence and the set containing that same element, starting with the empty set, which cannot be uniquely represented by a sequence of less than one hundred words in the standard Oxford English Dictionary in a way that does not depend on context."
"A number with a value less than infinity."
There. I just described all numbers with less then twenty words.
Claim: the set of English words is actually infinite because number words are open compound words.
Example: 123,456,789 is One hundred and twenty three million four hundred and fifty six thousand seven hundred and eighty nine. But we don’t say that’s 17 words, it’s actually just one big compound word describing one number.
Peanut butter, for example, has two individual words which mean different things (a type of legume, a solid emulsion made from churned milk or cream) but together form an open compound word which means a singular third thing (a creamy food made from peanuts). Peanut butter is listed in the dictionary as a singular word despite containing a space.
Though English does not have number words for every level of a thousand, we can use repeated words to infinitely construct them. For example, we can say a trillion or a million million.
17 was the smallest boring number. Therefore, one of my professors always used the number 17 when he had to choose a random number. Since then, 17 is no longer in the list of boring numbers, since it refers to that professor.
What's the smallest boring number now?
@@WrathofMath I was thinking of 29.
"The least number that cannot be described in less than 20 words" does *not* describe a specific number. This is immediately clear because assuming so leads to the paradox. Indeed it is a problem with English language because the statement is self-referential in a subtle way (it is a description in less than 20 words talking about descriptions in less than 20 words).
I wish self-references would be studied more in logic, enabling us to reason despite their presence, instead of trying to avoid them via ad-hoc heuristics and hope for the best.
"Any number" is less than 20 words, and is a description matching infinitely many numbers. Hence, set T is infinite.
So many paradoxes are just a consequence of circular reference, which seems to be mostly, if not entirely, a feature of the language we use to define things.
In this case, I would propose to fix the paradox by redefining the set to only allow descriptions with no relation to the set, or by extension, its compliment. So the smallest number outside the set remains outside the set.
This doesn't resolve a similar paradox where the descriptions are up to 20,000,000 words long. That's enough words to re-explain maths from scratch, build the same contradiction within it, and refer into that. I gather that's basically what Goedel constructed in his Incompleteness Theorem, but in more formal language.
@SimonClarkstone Could you build the same contradiction within it without using any circular references?
I don't know that it's always possible to avoid circular references, even with formal language, but what I'm suggesting is that these paradoxes may not lie within math and logic themselves, but in the way we define things. If that's the case, it feels somewhat dishonest to call them logical paradoxes rather than acknowledge that they're just consequences of our own limitations in defining things.
Hey! Can you resolve the Greyling Nelson paradox? Dialect has a video in it. It's funny that, on confidence, Bertrand Russell is occurring all over my phone, mostly from Philosophy, but now in mathematics too 😂
*on *Bertrand
@@irrelevant_noob 👍
The problem is that ultimately, the description is self-referential. It's a verbal equivalent of the Grandfather Paradox.
The implicit assumption is that a sentence describes a number if there is a unique number that satisfy the sentence, so T is finite. But if this is your objection, then it is easy to rephrase the paradox to "The smallest number which can't be described by a sentence of less than thirty words which describe a single number".
Simple definition: “___ is a member of the natural numbers”
thus T contains all natural numbers (assuming non-uniqueness in definitions)
"Let T be the set of all natural numbers with description shorter than 20 words that unique describe only a single number."
The paradox is back!
You can't resolve the paradox. Gödel proved that either the set is incomplete or it is inconsistent.
this works for a specific language and a specific number of words.
we can define an arbitrary language with arbitrary words which have a maximally condensed meaning in few words.
also, using "preceding steps" in a description is nonsensical, when you say it's less than 20 words.
the whole preceding step in itself is not defined in the description, or in the language. it's a contextual definition, not part of the current description or the base definitions of the language.
"The largest non-negative number less than 0"
When I heard 3 i just said "all natural numbers" and so when you dropped 4 i was very confused.
We are using dumb words in our premesis so I can use them in the argument.
I think the only really necessarily context-dependent concept is the meaning of a phrase as a part of what it means to describe something. Everything else can be formalised
I think the paradox can still live without this "non-context-dependent" idea.
Like so: "There is a least natural number which cannot be described in fewer than 100 characters".
There is a least natural number which cannot be described in fewer than twenty non recursive words.
I'm mainly confused not about the paradox and the solutions mentioned in this video, but more so in what the paradox would even accomplish.
I mean, seriously, IF Berry's paradox was genuinely "true", as in there is no valid explanation and there really does exist a contradiction. Okay, cool, what then? What does that say about mathematics?
Because it just seems to me like the result could only be possibly used for linguistics and maybe philosophical concepts, but if they tried to, for example, invalidate set theory in mathematics, THOSE statements would surely be non-sensical.
I'm not sure if the paradox is meant to "accomplish" anything other than just being a curiosity.
Natural language is internally inconsistent. Every word used in every definition in a dictionary is defined in the dictionary. All purely linguistic definitions are circular on a finite set of words. You need context outside a dictionary from experience to get anything more concrete and even then it isn't that concrete. Any attempt to do formal math USING natural language is going to butt up against this problem. That's why we create formal mathematical languages. And even then, with Godol's Theorem, we know we can't prove they are consistent. But at least we don't know they are inconsistent like we do with natural language.
There’s an infinite number of natural numbers that can be described in two words-
It’s even.
That describes an infinite number of even numbers.
"it's finite"
That also describes an infinite number of... numbers
Yes but it doesn't describe any particular number. Every description in the recursive argument, in context, describes one single number.
the problem with Berry's paradox is far way simple: you don't even know if you can applie a way to make them discribe anything. You could say that us, humans, do understand words, but can we? we understand OUR words that are describe by each and every human differently, and even with that, words can have very different meaning, and also depend on the physical world with a lot to define (what is a "chair", what are "inside" and "outside", what is "freedom", ect.) so use the word "define" in this context have to be "define", and this is already hard to do, if not impossible. in this way, there can effectively be a way to make an infinite amount as well as a finite amount of number with it, as we can't know already if the "numbers" describe exist. I would not say that Berry's paradox is entierly solved, but with the way mathematics is done these day it seems like a nonsense because of the first-order logic.
"Preceding steps" is cheating because the description is no longer self-contained.
What about the phrase “The least number that cannot be described in fewer than twenty words without referrring to other descriptions”?
Riddle me this Batman:
1) Take the words listed in the Oxford dictionary, set O, which are finite.
2) Set T contains every possible set of words from O with less than 20 words, which is finite.
3) Set Tn is the subset of T that contains every set of words that uniquely describes a natural number, which is finite.
4) There is a unique smallest natural number outside of Tn.
5) "Smallest natural number that cannot be uniquely described with less than 20 words from Oxford dictionary" simultaneously belongs and does not belong to Tn.
@ ah, “uniquely” is good
@@Tata-ps4gy no, uniquely is NOT good. The way you phrased it, 3 and 5 mean that no number can have multiple descriptions... Isn't English fun? 🤪
@irrelevant_noob You made a little fallacy. 3 has multiple descriptions. But the description "three" uniquely describes the number three.
If the smallest natural number is eg 5, couldn't we say that "the smallest natural number" isn't a unique description because it can also be described as "5"?
This isn't the correct resolution of the paradox. The paradox uses "description" to mean "unique description". To clarify this, you should state the paradox inside a precise formal language first, best is second order arithmetic. Using a Godel encoding for sentences, the code of a predicate P(n) with one free variable n of natural-number type defines a unique natural number when the formal sentence S(P) defined from P as "(exists n P(n)) and (forall m (P(m) implies m=n))" is true. This uses Tarsky's definition of "true" inside a formal system, a way of recursively converting the code for S into the meaning of S. Famously, Tarsky's truth definition is limited--- you can't define it for all the sentences in a theory, you can only define it up to some limited number of quantifier alternations in front. That's key.
So try to make the paradox. Consider the set C of all numbers x such that there exists a formal predicate P(n) with byte-length(S(P)) < 1,000,000,000 such that S(P) with x substituted everywhere for n inside, is true. This is a finite set, just as advertised, because there are only finitely many P. So there is a least n not in this set, and the formal description of n takes less than a billion bytes. Paradox formalized. This is ONLY not a contradiction because the truth predicate is limited in quantifier complexity. The sentence defining the paradoxical number has an additional quantifier compared to the one for which the truth predicate inside it is defined. That's the resolution, Tarsky's undefinability of truth, and there is no other resolution.
...the dictionary definition of number can be easily reworded to be less than 20 words (oxford definition 1. is 21 words). So there is no number outside the set right from the start. You can easily say "a value with real and/or imaginary components expressed in digits, symbols or words." And if something is missing, I'm certain one or two more sentences can include them.
Having watched to 5:17, my question is: Can there be a "least number not in T"? That is, since there is no way to determine or find that number therefore even though we have a description that is less than 20 words, it does not count as a description of a number.
Let's see what the rest of the video says ...
The premise is that, whether we know what that particular number is or not - certainly if all premises to that point are correct - we have offered an accurate description of it. Just like I might describe a number as "the smallest prime number we have not yet found". I don't know what that number is, but I am definitely describing a particular number (we'd just have to settle the details about what 'have not found' means.
3 ways to resolve this i think:
1. T is not a set
2. No high order description
3. The map from T to the set of desciptions is not injective (this video)
my first thought was make the description the equation that makes it, there are probably be a description for every number under 20 words,
Bah, I can break this. Just include all the context and definitions, and it will be a finite description and therefore we can use quine-encoding techniques to recreate the self-reference efficiently.
I use a turing complete programming language to write a program with less than n chars that will go through every program with less than n chars and store their results. Then it will give smallest number not among those results. Such program can never halt since it depends on it's own result.
"The least natural number that cannot be described in fewer than twenty-five words without referring to the description of another number" is a sentence with less than 25 words that describes a number
This one does not require me to prove that every single word is non-context-dependent, because it only disallows referring to other descriptions of natural numbers, and so the paradox still persists.
How are you defining "without reference to other numbers," though? That seems ill-defined (and I mean rigorously. This is foundations after all :P).
This is actually fine and non-paradoxical, because your description itself *refers to other descriptions,* which means it doesn't forbid its own existence.
Even the premise that the description has to contain fewer than 20 words is arbitrary. Had your description contained say 22 words would that have really made it less viable as the solution to Barry’s paradox?
Not in my opinion. Whatever the stated word restriction is, be it 20, 30, or any other number, nothing really changes.
@ exactly.
Natural Languages are not formal languages ... Solved ...
10:10 that’s the point though, yeah, you can not define a natural number without its context, and thusly T is finite.
If your definition is not exclusive it’s a bad definition for establishing a paradox
Go with "less than 200 characters."
replace "describe" by "uniquely describe" and the workaround fails I think
You don't need to start talking about context dependence to get a revenge paradox. Just let T be the set of all natural numbers which can be uniquely described in fewer than 20 words.
I'm at 2:47. I predict that the issue will be that set theory generates paradoxes if you include self-set-referential elements in a set. Group theory FTW :)
“finite number of n word sequences”
i would recommend muting your mic the next time you game
this feels very contrived.
most obviously, the definition "the least number that can be defined in less than 20 words" is 12 words, so that could easily be changed to "least number that can be defined in less than 13 words," which would make it very difficult to phrase a formal definition of a number within that.
but a definition that rules out this entire solution is as simple as "the least number that can be *uniquely* defined in fewer than 20 words."
Every description in the infinite sequence of descriptions proposed, when viewed in context, does uniquely describe a single number. So the argument would come down to the validity of that strategy.
@WrathofMath OK, what about a more specific phrasing of the same then? "the least number that cannot be defined with a unique sequence of 20 words or less"
Doesn’t this sort of hint that the descriptions have to be bijective, surjective and err..? Injective? To be a proper set?
This solution of the problem is the linguistic one, but to understand it you have to go to computer science. Think of the dictionary as a set of lexical objects. A simple sentence is a directed graph of lexical objects. If you restrict yourself to simple sentences, you can guarantee that the sentence can be parsed (i.e. it has at least the possibility of meaning). If however the sentence is self-referential in any way (i.e. not a simple sentence) there is no way to guarantee that the sentence can be parsed (see the halting problem). In simple terms, if you build a set without referring to the set, you can guarantee that a fixed (possibly infinite) set exists. The moment you start referring to the set as you construct your set, your set can potentially be meaningless since it cannot resolve to a fixed result for what the set is. You can find a funny example of this in this Ryan George skit ( th-cam.com/video/hf_7xAX_fBE/w-d-xo.html ).
What about the set of words `all natural numbers` definately less than 20, definitely describes an infinite number if items.
Is there a full version of the ending song? I love it....
th-cam.com/video/x2wm5RxpHKA/w-d-xo.html
Just type in "Super Mario 64 credits song"
1:40 "n word" 💀
🌚
there is a finite number of n-word passes in this planet
Obviously you want the sentences that describe numbers to be self-contained and to describe a unique number, otherwise it is discarded along with all the other nonsensical sentences. You want to be able to give it to an arbitrary idealized mathematician and have them say "Sure, I agree that that describes a unique number", without having shown them any of the other sentences. We can even go so far as to detail exactly what mathematical knowledge this mathematician knows, which we are therefore allowed to refer to in our sentences.
Your "set that contains all previous iterations of this sentence" thing trivially fails this test. Not so for the original paradox sentence. I mean, it might fail that test, and that could be a legitimate refutation of the paradox, but certainly not trivially so.
I love the SM64 music near the end!!!!! ~13:45
Timeless!
All paradoxes arise solely in language. There are no paradoxes in reality.😊
okay but like... you could just increase the word limit and be much more comprehensive about everything, even construct language from scratch if you must, and the "repaired" paradox would stand; hell, it could be "repaired" in a less self-destructive way if need be, like maybe specifying "non-recursively" instead of specifically "non-context dependent words"
what you're saying in the video is true, sure, but it also kinda misses the point? like, it's not that this is unsatisfying because it's "cheating" per se, it's unsatisfying because it completely ignores the actual logical foundation of the paradox and another one like it can still be constructed that would be immune to this rebuttal
The descriptions should be able to stand alone. The reason that your re-use of descriptions feels like cheating is not that it uses the term "description," but the fact that it refers to "previous steps." A disambiguated description contains all those previous steps, and quickly exceeds twenty words. Similarly, your set theory description of 3 can be written out as {{}, {{}}, {{}, {{}}}}.
Now, if you really want to resolve the paradox, you would note that you can't compute the number supposedly described. But you don't. You do the equivalent of tossing the board and declaring yourself the winner.
If you insist the descriptions must be able to stand alone, the paradox is still resolved, since the description central to the paradox 'there is a least natural number which cannot be...' cannot itself stand alone, and thus doesn't qualify as a valid description. Being able to compute the number any description accords to was never a stated restriction, and if you require it as a restriction that only further emphasizes the inability of the central paradoxical description to stand alone.
@@WrathofMath
"'there is a least natural number which cannot be...' cannot itself stand alone,"
Why not? The least natural number that cannot be // expressed as the sum of squares of three or fewer natural numbers is 7. And it is unambiguous.
All natural numbers.
Wow I cannot believe I didn't see an N word joke in the top four comments
I really like your video production and your presentation skills. It would be fun to collaborate sometime.
Dammit Berry!
Ahsh, symbols with no meaning
Dictionary: all binary strings of length n, where 1 < n
How exactly is this a paradox?
@fullfungo It may not be
1:45 Whit people shouldn't use the N-Word... or sequences derived from the N-Word.
This video is not boring! 🎉😊
So it's like every other video? That's pretty boring.
Zero is not the empty set, zero is the span of the empty set.
By definition, zero *_is_* the empty set. The fact that it _also_ equals its span is coincidental.
@@irrelevant_noob ZFC is not concerned with ontology, the set theoretic definition of natural numbers is not an ontological claim. It is only concerned with theorems can be proven about them.
1:40 Dude. You said the "n-word"! 🤦🏼♂️
N is (n-1) + 1 seems fine to me
Yet another video that completely misses the point, literally none of what you said has anything to do with the underlying mathematical ideas of the Berry Paradox, rather than just with weird irrelevant linguistic nitpicks. The _actual_ Berry "Paradox" is about constructing something that's fundamentally equivalent to a Godel Numbering, and then being surprised at the Incompleteness result. The "smallest number not described in seven words" _is_ a valid statement, it's just not _decidable_ in exactly the same way that any other problem equivalent to the halting problem isn't decidable.
It sounds like a turing halting problem. Is the English language turing-complete?
Not relevant.
I don't think your argument is very good
this is stupid, and not a solution
🎯 Key points for quick navigation:
00:00 *📚 Mathematics was experiencing a foundational crisis in the early 20th century, leading to a focus on set theory.*
00:25 *🔄 Paradoxes, like Berry's Paradox, were discovered within set theory, needing resolution.*
00:53 *🧑🔬 Berry's Paradox was described by Bertrand Russell, attributed to librarian GH Berry.*
01:07 *📖 The paradox begins with the assumption of a finite collection of English words, from which a contradiction emerges.*
02:01 *🔢 From word sequences, a set of natural numbers describable within 20 words can be identified as finite.*
03:09 *❌ A contradiction is highlighted, showing a number is both described and not described in fewer than 20 words.*
04:14 *🤔 The paradox is argued to be a linguistic problem rather than a mathematical one.*
05:19 *🧩 The paradox's mathematical premise, regarding the finiteness of descriptions, is scrutinized.*
06:13 *🗂️ Resolving the paradox involves using one description for infinitely many numbers.*
07:19 *🏗️ Set theory's approach to defining natural numbers provides a resolution through recursive definitions.*
08:41 *🌀 Descriptions for numbers must exist in context to resolve the paradox.*
09:47 *🧐 An insistence on non-context dependent descriptions leads to reevaluation but is argued to ultimately fail.*
13:42 *😲 The paradox is likened to the "boring number" paradox, illustrating a similar logical contradiction.*
Made with HARPA AI
The video already has chapters. This is useless.
I wince every time you say "described" when you mean "specified".
I would have used a more particular word, like specified, but for the fact that statements of this paradox historically use the word described.
@@WrathofMath Similar to how people cite "this sentence is false" as a paradox more often than they cite "this sentence is not true", even though the latter sidesteps a trivial solution to the former. Likewise here, "it's a number" is a description that describes any number, so there's literally no paradox when expressed that way.
Has anyone here ever touched a girl?
Me🙋♀️ I’m a girl, btw
@@fullfungo You do realise what you've done to all the others on the site, don't you?
@@caloricphlogistonandthelum4008?