One of the best lectures I've ever listened too. Not for the total beginner, but if you have an armchair degree from popular youtuber/channels in some set theory, mathematical logic, and real analysis, everything should make sense, even if we don't understand it to any degree beyond the superficial.
I'm a Lawyer and have no formal education on Mathematics. And for some reason I'm hooked into these professor's series of Lectures. Very good! Came here for the philosophy, ended up staying for the Math. I'm definetly going deeper in this amazing field. Thank you, Sir.
@@nikolaykolev5125100%. My calculus professor didn’t do a lot of proofs (as is kind of the norm now) but when she did go over them, she would comment that if you enjoyed the process she would recommend you maybe go into law. It’s a very similar thinking pattern by making tight deductions off of legal axioms. The LSAT is mostly comprised of rigorous analytical thinking. Bertrand Russell thought of Law as a deductive science, including only it, religion, and mathematics to that set.
For me this was the first series of lectures on *any* academic subject where I did not miss a single video -- either live or later on YT -- since I first enrolled at University in 2015. I finally got a better picture of how various things fit together in the Philosophy of Mathematics. Great content -- I'll definitely read the book in February!
Superb series of lecture - thank you. I wish you said something about Large Cardinals beyond Choice and Woodin's program. But I guess time limits the number of topics.
@1;23:00 this was lovely, had not heard my affinity with platonism expressed so well before. I am definitely a platonist, but I like this plurality idea of different conceptions of "set", and it follows there could be different conceptions of ℝ. Just like your analogy to non-Euclidean geometry. But it is a fascinating analogy. Non-Euclidean geometries have such clear concrete manifestations (any curved surface or hypersurface). What I've never heard from a set theorist is a similar obvious concrete alternative concept of "set", only different but _very abstract _*_in_*_ their difference_ axiom schemas.
Thanks for this great lecture! I will go through the book and the lectures simultaneously. It's great that anyone in the world can have access to this high-quality material.
@1:14:00 no? Not necessarily? Woodin's V=Ultimate-L conjecture would render this conception immune to Cohen forcing. It seemed pretty simple. It is _only_ Cohen forcing that gives rise to models where CH is false. I think this is a terrific possible resolution for CH. It does not imply there is only one unique set theoretic platonic universe, just that there is this nice rigid one!
Great series overall! You mentioned in the Q&A section about picking a path in an infinite tree and its relationship to a weaker variant of the axiom of choice. I bring it up because I remember going to a conference in my masters year (2014) and seeing a talk by an Oxford PhD student saying that he was working on showing that this graph theoretic principle (or one very similar; it's been a while and my memory's not great!) is equivalent to the axiom of choice. My question is, is what you mentioned (its relation to a weaker version of the axiom) the product of his research? It would be very interesting to hear how the story from so many years ago ended up! Thanks again for this incredible lecture series. Can't wait for the book to come out next year!
I'm so glad you enjoyed the series. The principle I was referring to about picking a path through a tree is known as the principle of Dependent Choices, which has been known to be strictly stronger than countable choice and strictly weaker than the full axiom of choice. I think that must have been known by the 1970s, and so I think it must be different from the graph-theoretic principle you mention.
Man I am drunk and I have no idea what this is about, but I feel so smart listening to this and kind of picking up on stuff. I don't know why TH-cam suggested me this but I am listening for like an hour now, "general comprehension principle" man, you rock!
Professor Hamkins, hello! I found your channel via your wonderful discussion with Daniel Rubin! I am curious, could you shed some light on why it is that Cantor's diagonal argument is used to suggest the "uncountability" of certain infinite sets when the same reasoning applies to very small finite sets which can clearly be counted? I am trying to wrap my mind around why having 2^n combinations of numbers with n digits is in any way more or less "countable" than something else, and what is different in this regard for small finite sets and infinite sets? (For example, the set of all 3-digit binary numbers 000, 001, etc.. contains 2^3= 8 elements, there is nothing at all surprising that there are more than three numbers on this list.. and clearly the inverted diagonal number "111" (for example, or whatever you'd like) is on the list. Three and eight cannot be put into one to one correspondence, yet we can all count to 8. If we allow the use of infinity, the same is true for an infinite list.. sure, for numbers with omega digits, the list of all of them will be more than omega, but the inverted diagonal number, omega-digits long will still be on the list, and we can clearly define it's position if the rest of the list is organized in a way that allows communication. Why is one of the numbers considered to be "uncountable"? Is eight an "uncountable" finite number if it is written as 2^3? Etc). Hope that makes sense.
This is an interesting question! The answer is that to be countable means to be placeable in a one-to-one correspondence with a set of natural numbers, and so Cantor's proof shows that the real numbers are uncountable. His proof also shows just the same, as you observe, that the number of subsets of an n-element set cannot be placed into one-to-one correspondence with n. That is, the argument doesn't show that they are not countable, but rather that they are not countable in n steps. Similarly, the reals ℝ are not countable in ℕ steps, where ℕ is the set of natural numbers. And that is what it means to be "uncountable".
@@joeldavidhamkins5484 Thank you for your response! I understand that Cantor redefined "uncountable" for ℝ to mean "not countable in ℕ steps", but what I am not understanding is why he thinks not being countable in ℕ steps is now some different special new category of uncountability or a higher type of level of cardinality while not applying the same logic to large finite sets or even very small finite sets. It is very obvious that 8 is not countable in 3 steps, but we don't take this to mean that 8 is in some totally new level of countability. Everyone I have talked to about this, and everything I have read seems to take for granted that Cantor's diagonal argument self-evidently proves that ℝ is in some distinct category of countability or cardinality, but I cannot see how he has proved anything other than just that a list of numbers n digits long must be 2^n terms long, which is true for finite and infinite numbers, natural numbers and reals. If the number of digits is defined up to n digits, the inverted diagonal number will be n-digits long, and will not have a defined 2^nth digit. If we define the inverted diagonal number to be 2^n digits long, then, if the other sequences are not also 2^n digits long, we will not be comparing similar objects. An inverted diagonal number 2^n digits long will absolutely be on the list enumerating all combinations of digits, and will not lead to a contradiction, but all lists of sequences will be longer than the number of digits for each sequence. The example given on the Wikipedia page for Cantor's Diagonal Argument claims that the sequence "S = 10111010011.." "cannot occur anywhere in the enumeration of sequences above" .. but if we continue the list of sequence in binary order, sequence S is just s1491. Cantor has not defined a 1491st digit for S, and the way he has set up the list, if he does give a 1491st digit of S and changes it to be different from what I have claimed, I can still find that new sequence which is now 1491 digits long, but the list must be expanded to 2^1491 digits. The claim that S cannot occur in the enumeration is false, and the trick is just that for any sequence of digits, the list has to be 2^n long, so if I find a sequence with those digits, it can be claimed "well what about the same sequence with 2^n digits rather than just n". Any of those sequences given so far can also be interpreted as a representation of some sequence of natural numbers just as much as they could represent decimal digits of real numbers, so why has Cantor convinced himself he has proven anything about ℝ vs ℕ in particular, and why does he think he has proved anything about infinite numbers that is not also already true about finite numbers?
@@MrLove8736 in your formulation, Cantor showed that there is no 1-1 mapping from omega to 2^omega. In the case of 3 to 2^3 that's obvious, but in the infinite case it is not intuitively obvious.
Professor, i read somewhere that you're a pluralist on the philosophy of set theory, i am just not so sure what this view means exactly: Do you think the cumulative hierarchy ( as defined by the successive application of power sets starting at the empty set) defines multiple universes ( therefore not having an intended model ), each of which is true in some sense or do you believe the cumulative hierarchy defines a single universe, while holding it is not "special" and others - such as the constructible universe - are true in some sense as well?
My view is that we have multiple incompatible and incomparable notions of set, which each give rise to their own cumulative hierarchy, which may not fit together in a coherent way. You can read more about my views in my article, The set-theoretic multiverse: jdh.hamkins.org/themultiverse/.
Regarding the formulation of the cumulative hierarchy, atemporal or not, is it possible to symmetrically reverse it and start(or ongoing without a single starting point) from the unbounded transfinite levels in order to reach ω and ultimately the empty set? In this context, are finite sets inaccessible from above? The true question I believe is what is a step and what really happens at a limit stage.
Just a quick note, which I got from Hugh Woodin. Given two doors you can choose one to go through, forbidding forever entry to the other. One leads to an Oracle for Set Theory, the other to an Oracle for Number Theory (and only NT). No one in their right mind is ever going to choose the NT door, since the ST Oracle can already tell you if Number Theory is consistent or not. The number theory Oracle cannot tell you the converse. If the ST Oracle tells you ST is inconsistent, then you can continue on wherever you go developing NT with no worries. But your stupid cousin who took the NT door will never know whether the greater richness of ST could have been a thing.
Actually, if set theory is consistent (which is artihmetical fact, therefore oracle will know that), then you can define an interpretation of set theory within number theory (this is arithmetized completeness theorem). Moreover, if you want to use such emulated set theory to prove results in number theory, you just ask an oracle about every single instance of arithmetical reflection that you are interested in. An instance of arithmetical reflection is a sentence "if phi holds in interpretation of set theory, then phi". Phi is an arithmetical sentence here. Also, i assume that we ask an oracle only questions expressible in finitary first order logic, as knowing which arbitrarily infinite questions can be asked would already require some set-theoretic oracle (that is the problem of large cardinals). Sorry i am responding to your comments so often. They are good.
When you comment on something, you should have an idea of what you are talking about. Clearly, you don't. The infinities of set theory are not a mere "unboundedness". Precisely, Aristotle considered that an "actual" infinity (as opposed to a mere potential infinity, to which "unboundedness" refers) was contradictory. With Bolzano and Cantor, the actual infinities got rehabilitated --but of course they introduced a new ingredient, which was not present in the Greek tradition
@@AlainGenestier-tj8fn Clearly you cannot read English. Aristotle did not know set theory. My comment was about the lack of necessity of set theory - not about how the Greeks understood infinity.
Aristotle didn't know set theory": of course, and that's obviously not what I claimed (if it is what you choose to understand, that's you problem!). "Lack of necessity": for what ? Modern math needs set theory (or more or less equivalent alternatives: type theory, category theory, which also deal with actual infinities). If you didn't notice, math has made progress since the Greeks. And so has our understanding of the concept of infinity. You are not supposed to accept those theories, but at least if you want to criticize them, don't make a superficial and uninformed criticism.@@peterwaksman9179
@@peterwaksman9179 Where did I claim that Aristotle knew about set theory? It's that's what you understood, your reading skills are questionable. On the contrary, Aristotle thought that it was inconsistant to consider infinite totalities. A set theory with only finite sets would be a quite rudimentary theory, and set theory is expressely built against this Aristotelian taboo of infinite totalities. "The lack of necessity of set theory": well, set theory is necessary for modern math. Or, you at least need o theory (like type theory, ...) admitting actual infinities, against Aristotle's prescription. In case you didn't notice, math has made progress since the Greeks. You are not committed to "accept" set theories (or even actual infinities). But if you want to criticize them, make at least an informed criticism!
Many highly educated people can't see the forest for the trees. Too "strong/rigorous" demonstrations destroy any useful applications. They are a bit like ants that spin around. The hard part is not to be “rigorous”-that can be done by robots-but to set from the beginning the context/axioms and to see where you can go with the results and how to use them. The real is so vast and dense that we can start from anywhere, construct logically some model/structure/theory, that it is self-consistent 99% or more. Nevertheless, it could be completely not fitted for practical use, because it is not compatible with our reality. You can't trust maths as a dogma; this is intellectual fanaticism. Or, as Feynman put it ..., i let u find the citation yourself. Mathematics is tools we have being invented and continues to be invented, not a gift from GOD :)
One of the best lectures I've ever listened too. Not for the total beginner, but if you have an armchair degree from popular youtuber/channels in some set theory, mathematical logic, and real analysis, everything should make sense, even if we don't understand it to any degree beyond the superficial.
I'm a Lawyer and have no formal education on Mathematics. And for some reason I'm hooked into these professor's series of Lectures. Very good! Came here for the philosophy, ended up staying for the Math. I'm definetly going deeper in this amazing field. Thank you, Sir.
May be because legal arguments have common with logical deduction and balance of probabilities which in turn relate to sets operations.
@@nikolaykolev5125100%. My calculus professor didn’t do a lot of proofs (as is kind of the norm now) but when she did go over them, she would comment that if you enjoyed the process she would recommend you maybe go into law. It’s a very similar thinking pattern by making tight deductions off of legal axioms. The LSAT is mostly comprised of rigorous analytical thinking. Bertrand Russell thought of Law as a deductive science, including only it, religion, and mathematics to that set.
For me this was the first series of lectures on *any* academic subject where I did not miss a single video -- either live or later on YT -- since I first enrolled at University in 2015. I finally got a better picture of how various things fit together in the Philosophy of Mathematics. Great content -- I'll definitely read the book in February!
I'm so glad that you found them worthwhile. Let me add that your own interview content is outstanding!
Have you finished reading the book?
Start at 4:47 for the full-screen speaker view. Apologies for not getting this right from the start.
I really love that drawing about removing isolated points! Such a great series!
Superb series of lecture - thank you. I wish you said something about Large Cardinals beyond Choice and Woodin's program. But I guess time limits the number of topics.
Nice, this lecture only took me 2 weeks to get through. The Gödel one was much harder, over a month! Thank you very much for putting them online!
What a great series of lectures. Thanks for making them public.
@1;23:00 this was lovely, had not heard my affinity with platonism expressed so well before. I am definitely a platonist, but I like this plurality idea of different conceptions of "set", and it follows there could be different conceptions of ℝ. Just like your analogy to non-Euclidean geometry. But it is a fascinating analogy. Non-Euclidean geometries have such clear concrete manifestations (any curved surface or hypersurface). What I've never heard from a set theorist is a similar obvious concrete alternative concept of "set", only different but _very abstract _*_in_*_ their difference_ axiom schemas.
Thank you for publishing these lectures. I found them to be very illuminating.
Thanks for this great lecture! I will go through the book and the lectures simultaneously.
It's great that anyone in the world can have access to this high-quality material.
Thank you for all the enlightening lectures in this series!
Great lecture series! Thanks for posting them.
I am a lawyer too. Its amazing how this lecture reiterates Socrates' parmenidis. A real and genuine platonist view.
@1:33:00 isn't a shorter answer that Peano Arithmetic does not admit power sets. So you do not have ℝ .
@1:14:00 no? Not necessarily? Woodin's V=Ultimate-L conjecture would render this conception immune to Cohen forcing. It seemed pretty simple. It is _only_ Cohen forcing that gives rise to models where CH is false. I think this is a terrific possible resolution for CH. It does not imply there is only one unique set theoretic platonic universe, just that there is this nice rigid one!
Many iterated forcing extensions different from cohen's render CH false.
Great series overall! You mentioned in the Q&A section about picking a path in an infinite tree and its relationship to a weaker variant of the axiom of choice. I bring it up because I remember going to a conference in my masters year (2014) and seeing a talk by an Oxford PhD student saying that he was working on showing that this graph theoretic principle (or one very similar; it's been a while and my memory's not great!) is equivalent to the axiom of choice. My question is, is what you mentioned (its relation to a weaker version of the axiom) the product of his research? It would be very interesting to hear how the story from so many years ago ended up!
Thanks again for this incredible lecture series. Can't wait for the book to come out next year!
I'm so glad you enjoyed the series. The principle I was referring to about picking a path through a tree is known as the principle of Dependent Choices, which has been known to be strictly stronger than countable choice and strictly weaker than the full axiom of choice. I think that must have been known by the 1970s, and so I think it must be different from the graph-theoretic principle you mention.
@@joeldavidhamkins5484 I must be half-remembering then, no worries!
Man I am drunk and I have no idea what this is about, but I feel so smart listening to this and kind of picking up on stuff. I don't know why TH-cam suggested me this but I am listening for like an hour now, "general comprehension principle" man, you rock!
Professor Hamkins, hello! I found your channel via your wonderful discussion with Daniel Rubin!
I am curious, could you shed some light on why it is that Cantor's diagonal argument is used to suggest the "uncountability" of certain infinite sets when the same reasoning applies to very small finite sets which can clearly be counted? I am trying to wrap my mind around why having 2^n combinations of numbers with n digits is in any way more or less "countable" than something else, and what is different in this regard for small finite sets and infinite sets? (For example, the set of all 3-digit binary numbers 000, 001, etc.. contains 2^3= 8 elements, there is nothing at all surprising that there are more than three numbers on this list.. and clearly the inverted diagonal number "111" (for example, or whatever you'd like) is on the list. Three and eight cannot be put into one to one correspondence, yet we can all count to 8. If we allow the use of infinity, the same is true for an infinite list.. sure, for numbers with omega digits, the list of all of them will be more than omega, but the inverted diagonal number, omega-digits long will still be on the list, and we can clearly define it's position if the rest of the list is organized in a way that allows communication. Why is one of the numbers considered to be "uncountable"? Is eight an "uncountable" finite number if it is written as 2^3? Etc). Hope that makes sense.
This is an interesting question! The answer is that to be countable means to be placeable in a one-to-one correspondence with a set of natural numbers, and so Cantor's proof shows that the real numbers are uncountable. His proof also shows just the same, as you observe, that the number of subsets of an n-element set cannot be placed into one-to-one correspondence with n. That is, the argument doesn't show that they are not countable, but rather that they are not countable in n steps. Similarly, the reals ℝ are not countable in ℕ steps, where ℕ is the set of natural numbers. And that is what it means to be "uncountable".
@@joeldavidhamkins5484 Thank you for your response! I understand that Cantor redefined "uncountable" for ℝ to mean "not countable in ℕ steps", but what I am not understanding is why he thinks not being countable in ℕ steps is now some different special new category of uncountability or a higher type of level of cardinality while not applying the same logic to large finite sets or even very small finite sets. It is very obvious that 8 is not countable in 3 steps, but we don't take this to mean that 8 is in some totally new level of countability. Everyone I have talked to about this, and everything I have read seems to take for granted that Cantor's diagonal argument self-evidently proves that ℝ is in some distinct category of countability or cardinality, but I cannot see how he has proved anything other than just that a list of numbers n digits long must be 2^n terms long, which is true for finite and infinite numbers, natural numbers and reals. If the number of digits is defined up to n digits, the inverted diagonal number will be n-digits long, and will not have a defined 2^nth digit. If we define the inverted diagonal number to be 2^n digits long, then, if the other sequences are not also 2^n digits long, we will not be comparing similar objects. An inverted diagonal number 2^n digits long will absolutely be on the list enumerating all combinations of digits, and will not lead to a contradiction, but all lists of sequences will be longer than the number of digits for each sequence.
The example given on the Wikipedia page for Cantor's Diagonal Argument claims that the sequence "S = 10111010011.." "cannot occur anywhere in the enumeration of sequences above"
.. but if we continue the list of sequence in binary order, sequence S is just s1491.
Cantor has not defined a 1491st digit for S, and the way he has set up the list, if he does give a 1491st digit of S and changes it to be different from what I have claimed, I can still find that new sequence which is now 1491 digits long, but the list must be expanded to 2^1491 digits.
The claim that S cannot occur in the enumeration is false, and the trick is just that for any sequence of digits, the list has to be 2^n long, so if I find a sequence with those digits, it can be claimed "well what about the same sequence with 2^n digits rather than just n".
Any of those sequences given so far can also be interpreted as a representation of some sequence of natural numbers just as much as they could represent decimal digits of real numbers, so why has Cantor convinced himself he has proven anything about ℝ vs ℕ in particular, and why does he think he has proved anything about infinite numbers that is not also already true about finite numbers?
@@MrLove8736 in your formulation, Cantor showed that there is no 1-1 mapping from omega to 2^omega. In the case of 3 to 2^3 that's obvious, but in the infinite case it is not intuitively obvious.
Professor, i read somewhere that you're a pluralist on the philosophy of set theory, i am just not so sure what this view means exactly:
Do you think the cumulative hierarchy ( as defined by the successive application of power sets starting at the empty set) defines multiple universes ( therefore not having an intended model ), each of which is true in some sense or do you believe the cumulative hierarchy defines a single universe, while holding it is not "special" and others - such as the constructible universe - are true in some sense as well?
My view is that we have multiple incompatible and incomparable notions of set, which each give rise to their own cumulative hierarchy, which may not fit together in a coherent way. You can read more about my views in my article, The set-theoretic multiverse: jdh.hamkins.org/themultiverse/.
Regarding the formulation of the cumulative hierarchy, atemporal or not, is it possible to symmetrically reverse it and start(or ongoing without a single starting point) from the unbounded transfinite levels in order to reach ω and ultimately the empty set? In this context, are finite sets inaccessible from above? The true question I believe is what is a step and what really happens at a limit stage.
The class of all ordinals is well-founded so any descending sequence from Ord to the empty set must be finitely long
Thanks for a good review of set theory. I think about set theory a lot as well as the philosophy of mathematics.
Incredible
Just a quick note, which I got from Hugh Woodin. Given two doors you can choose one to go through, forbidding forever entry to the other. One leads to an Oracle for Set Theory, the other to an Oracle for Number Theory (and only NT). No one in their right mind is ever going to choose the NT door, since the ST Oracle can already tell you if Number Theory is consistent or not. The number theory Oracle cannot tell you the converse. If the ST Oracle tells you ST is inconsistent, then you can continue on wherever you go developing NT with no worries. But your stupid cousin who took the NT door will never know whether the greater richness of ST could have been a thing.
Actually, if set theory is consistent (which is artihmetical fact, therefore oracle will know that), then you can define an interpretation of set theory within number theory (this is arithmetized completeness theorem).
Moreover, if you want to use such emulated set theory to prove results in number theory, you just ask an oracle about every single instance of arithmetical reflection that you are interested in.
An instance of arithmetical reflection is a sentence "if phi holds in interpretation of set theory, then phi". Phi is an arithmetical sentence here.
Also, i assume that we ask an oracle only questions expressible in finitary first order logic, as knowing which arbitrarily infinite questions can be asked would already require some set-theoretic oracle (that is the problem of large cardinals).
Sorry i am responding to your comments so often. They are good.
Erin is there!
Hello Erin!!
Thanks
Infinity in no way depends on a set theory analysis. Un-boundedness was probably understood by the Greeks.
When you comment on something, you should have an idea of what you are talking about. Clearly, you don't.
The infinities of set theory are not a mere "unboundedness". Precisely, Aristotle considered that an "actual" infinity (as opposed to a mere potential infinity, to which "unboundedness" refers) was contradictory. With Bolzano and Cantor, the actual infinities got rehabilitated --but of course they introduced a new ingredient, which was not present in the Greek tradition
@@AlainGenestier-tj8fn Clearly you cannot read English. Aristotle did not know set theory. My comment was about the lack of necessity of set theory - not about how the Greeks understood infinity.
Aristotle didn't know set theory": of course, and that's obviously not what I claimed (if it is what you choose to understand, that's you problem!).
"Lack of necessity": for what ? Modern math needs set theory (or more or less equivalent alternatives: type theory, category theory, which also deal with actual infinities).
If you didn't notice, math has made progress since the Greeks. And so has our understanding of the concept of infinity.
You are not supposed to accept those theories, but at least if you want to criticize them, don't make a superficial and uninformed criticism.@@peterwaksman9179
@@peterwaksman9179 Where did I claim that Aristotle knew about set theory? It's that's what you understood, your reading skills are questionable.
On the contrary, Aristotle thought that it was inconsistant to consider infinite totalities. A set theory with only finite sets would be a quite rudimentary theory, and set theory is expressely built against this Aristotelian taboo of infinite totalities.
"The lack of necessity of set theory": well, set theory is necessary for modern math. Or, you at least need o theory (like type theory, ...) admitting actual infinities, against Aristotle's prescription. In case you didn't notice, math has made progress since the Greeks.
You are not committed to "accept" set theories (or even actual infinities). But if you want to criticize them, make at least an informed criticism!
Set theory
This stuff is so beyond me. I feel so stupid.
You can always read more about the topic!
@Adam-rt2ir yay.
You are not alone.
i want to know what Satan think of mathematics
Many highly educated people can't see the forest for the trees. Too "strong/rigorous" demonstrations destroy any useful applications. They are a bit like ants that spin around. The hard part is not to be “rigorous”-that can be done by robots-but to set from the beginning the context/axioms and to see where you can go with the results and how to use them. The real is so vast and dense that we can start from anywhere, construct logically some model/structure/theory, that it is self-consistent 99% or more. Nevertheless, it could be completely not fitted for practical use, because it is not compatible with our reality. You can't trust maths as a dogma; this is intellectual fanaticism. Or, as Feynman put it ..., i let u find the citation yourself. Mathematics is tools we have being invented and continues to be invented, not a gift from GOD :)
Math is definitely not invented. Language, axioms, rules are invented, but they are set up to capture truth.
Perhaps the most poorly produced math video i've ever een