Really grateful for your choice to share your knowledge for free here. Truly, so nice to feel like I'm in math class again, without bleeding out my ear to pay for it.
I came to this video from a literary analysis and have no real background in maths to speak of, so I was not expecting to understand much After watching the entire thing I much say that I'm impressed at how much I understood. The explanations were thorough and clear enough that I managed to get my head around most of them without much issue. Than you for an interesting and well made video!
Two sets are equal ↔ they have the same elements. 0:30 There exist Intentional and Extentional definitions of sets. { 0>x>10 | x%2=0 } {2, 4, 6, 8} The first set is defined Intentionally, by some function, the second set, Extensionally, by listing out all of its elements. Extensionality says these two are one and the same. An unintuitive result, is that to Extensionality, Bubble Sort and Quick Sort are exactly the same. 3:45 {a, a, b} = {a, b} = {b, a} 4:15 Without Extensionality, you can get ordered sets or multisets, which can be useful, e.g the prime factors of 12 are {2, 2, 3}, but everything achievable without Extensionality is also achievable with it. An ordered pair (a,b) is described as {a, {a,b}}. "Multisets are functions from a set to cardinal numbers." (I assume this means a function assigning 2 to the first element of {2,3} and 1 to the second, in order to represent 12's factors.) 8:45 We can drop Extensionality by just defining our symbol "=" to mean "having the same elements. Then: 1."=" must be transitive, symmetric, and reflexive. a=b ∧ b=c → a=c a=b → b=a a=a So far so good. 2."=" must allow for substitution. If a=b you can substitute a for b. a∈c → b∈c c∈a → c∈b and without Extensionality, this line does not follow from our definition of "=", therefore it must be a new axiom, so this dropping of Extensionality does not really give us anything. 11:30 Different ways to handle equality: 1."=" as part of the underlying logic (as a member of the language of FOL). 2.a=b if a is in all and only the sets in which b is. In other words, a=b if a and b have the exact same properties. 3.Extensionality. These definitions are mostly interchangeable.
A fascinating lecture! It raises many deep questions, for me, at least. First, I don't see why it is claimed that the second part of (2) here (see 10:52) does not follow from the definition of equality being worked with, and that it therefore needs to be further stipulated. Why is "substitutablility" natural to the former and not the latter? Second - and maybe this is where I'm missing something significant - what is the difference between the Axiom of Extension and a definition of equality you propose? Why is the definition treated as distinct from the axiom? The Axiom and the Definition would seem to be intensionally identical! There's something ontologically principled going on here which hasn't been made explicit (or that I just don't see).
10:35 I didn't really understand why this doesn't follow from our definition. If a and b have the same elements, and a \in c, then c={a, other stuff} but then also c={b, other stuff} right since a and b are the same?
Suppose we were to instead use an intensional definition of sets, so that A := "primes of the form 4n+1 or 4n+2" and B:= "primes that are the sum of two squares" were not actually «the same set». We could then define the symbol "=" such that A = B iff A and B contain the same elements. However, because we are using an intensional definition of sets, {A} and {B} are not actually the same set (despite satisfying the equivalence relation "="). If A = B, then it *is* still the case that a ∈ A ⟹ (∃b such that b ∈ B and a = b), but it is *not* necessarily the case that a ∈ A ⟹ a ∈ B. Another way of interpreting the axiom of extensionality that might be more intuitive is that it's basically saying «a and b have the same elements implies a and b are in the same sets», or «∀A∀B(∀X(X∈A⇔X∈B)⟹∀Y(A∈Y⇔B∈Y))». You can then use A=B as an abbreviation for ∀A∀B(∀X(X∈A⇔X∈B)).
Suppose we drop the extensionality axiom, then {1,2} and {1,1,2} are two different sets (at this point). We then define a relation R, R(A, B) if A, B have the same elements. So R({1,2}, {1,1,2}). We want to check whether R satisfies all properties of =: (1) R is an equivalence relation (yes), (2) For any R(A, B), x in A => x in B (yes), (3) For any R(A, B), A in C => B in C, but we stuck, {1,2} in C does not mean {1,1,2} in C. Then we make this axiom so we can make this deduction go through.
@@ApriiSR Thanks you, this makes it more clear. Indeed without the axiom of extensionality there is a difference between satisfying the '=' relation and actually 'being the same set'.
I guess the question has been answered, but just wanted to point out: it may have seemed to follow from the definition because you were thinking of the elements of a being in c, instead of a being in c.
Thank you for these wonderful lectures. A question: I don't understand how dropping Axiom of Extensionality permits the existence of multisets. Either an element a is a member of a set S or it isn't. If it is, then "writing it again" in the extension adds no new information. For the second copy to mean anything would entail writing "something different" - but without the Axiom of Extensionality, we don't know what constitutes that difference, since we don't know what constitutes equality. You would need some sort of index or mark to distinguish multiple copies, but indices are not a part of the definition of sets. (A similar issue arises with ordered "sets".) This issue, interestingly, is directly connected with an issue of great importance in linguistics, specifically in the question of movement in minimalist theories of syntax, which use the mathematics of sets to model constituency.
The rooted trees are very interesting. It reminds me of how computer scientists encode algebraic structures using bitstrings. (/ trees of pointers, relational database tables, etc.) It's just implemented with these rooted unlabeled trees, instead. With the interesting "horizontal idempotence" property coming from extensionality. Similar to something you'd see in a CRDT. A CRDT! It IS a CRDT! Implemented using journal articles! Ha, now THAT'S a fresh interpretation.
Can someone pls explain how including the Axiom of Extensionality (saying two sets are equal iff they have the same elements) is bypassing the problem that arises when we attempt to define equality of sets by saying a = b if they have the same elements?
Despite the definition of a=b, is not consistent with the properties of equality ( i.e. as mentioned in the video a ∈c , but b ∈c may not be true take a={1,1,2} , b={1,2}, c={{1,1,2}, {8}}), we take a=b as an axiom if they have same elements.
Hi, Prof. Borcherds. First of all, thank you for these lectures! Almost nobody has done in-depth coverage of ZF(C) except perhaps from a purely 'standard curriculum' perspective. Wonderful to hear a more complete and nuanced perspective like yours! May I suggest, if possible, to record/upload in at least 720p video resolution? So far, I haven't actually had any problems with your 480p videos, so this is not really an urgent request; it's more like a 'future-proofing' request, so that your videos will 'stand the test of time' longer (at least in a visual sense). The potential problem with 480p is that writing or drawings, especially if they become small or detailed, can be harder to make out to the point of being almost illegible. And if the resolution isn't there to support it, you can't even 'zoom in' or 'go full screen' to try to improve the legibility. In contrast, most writing in 720p is at least legible, and if it's illegible when displayed on a small screen, can at least be zoomed or scaled up to make it more legible. Again, I have not actually experienced this problem on any of your videos I've seen so far. However, it is a common problem on 480p videos out there which feature writing or drawing, so I'm really just trying to 'forewarn' you of a potential problem. Your writing tends to be large enough to be quite legible, so maybe it will never be a problem. However, I'm reminded of the many lectures I've watched which were recorded when TH-cam first started out, and some of them were recorded with 240p, even up to 480p, and even though the writing was large and on a chalk board, it is sometimes very poorly legible. But since no higher resolution is available, those lectures cannot be improved and will remain stuck at that low visual quality for the rest of time. It's hard to predict how video technology will progress, and especially how videos will be used in the future (e.g. imagine your lectures being projected onto a big screen for a large lecture-room audience with some people sitting far in the back; who knows?). These days, 720p seems to be maybe-not-quite the 'minimum' standard, but on the other hand 480p seems to be maybe-not-quite 'on the outs'. Seems to me to be a bit of a transitional period. I mean, 720p isn't even considered 'high definition' (HD) on TH-cam anymore! (And virtually nobody goes as low as 240p, as used to be quite common actually.) I think if you upgraded to 720p, your videos (their visual quality at least) will stand the test of time much longer than the current 480p, and at the same time I don't think 720p will be that much more difficult to work with on your end, with recording, editing, uploading, and archiving/backing-up. Might require an upgrade of a camera, although I believe even the cheapest of webcams will be capable of 720p these days. Again, this is just an idea/suggestion. Sorry for such a long comment on such a basic topic. Sometimes I find it difficult to express myself succinctly. Cheers!
I understood you to say the three sets at the beginning were the same. If you did, I'm missing something important. How can '2' (which is indeed prime) be a member of a set whose elements are the sum of two squares? - unless you count '1' as prime, which is not traditional.
2 is a member of a set whose elements are prime (2 is prime) and AT THE SAME TIME a sum of two squares (2=1^2 + 1^2). The two squares don't have to be prime tho
Really grateful for your choice to share your knowledge for free here. Truly, so nice to feel like I'm in math class again, without bleeding out my ear to pay for it.
I came to this video from a literary analysis and have no real background in maths to speak of, so I was not expecting to understand much After watching the entire thing I much say that I'm impressed at how much I understood. The explanations were thorough and clear enough that I managed to get my head around most of them without much issue. Than you for an interesting and well made video!
Two sets are equal ↔ they have the same elements.
0:30 There exist Intentional and Extentional definitions of sets.
{ 0>x>10 | x%2=0 }
{2, 4, 6, 8}
The first set is defined Intentionally, by some function, the second set, Extensionally, by listing out all of its elements. Extensionality says these two are one and the same.
An unintuitive result, is that to Extensionality, Bubble Sort and Quick Sort are exactly the same.
3:45 {a, a, b} = {a, b} = {b, a}
4:15 Without Extensionality, you can get ordered sets or multisets, which can be useful, e.g the prime factors of 12 are {2, 2, 3}, but everything achievable without Extensionality is also achievable with it.
An ordered pair (a,b) is described as {a, {a,b}}.
"Multisets are functions from a set to cardinal numbers." (I assume this means a function assigning 2 to the first element of {2,3} and 1 to the second, in order to represent 12's factors.)
8:45 We can drop Extensionality by just defining our symbol "=" to mean "having the same elements. Then:
1."=" must be transitive, symmetric, and reflexive.
a=b ∧ b=c → a=c
a=b → b=a
a=a
So far so good.
2."=" must allow for substitution. If a=b you can substitute a for b.
a∈c → b∈c
c∈a → c∈b and without Extensionality, this line does not follow from our definition of "=", therefore it must be a new axiom, so this dropping of Extensionality does not really give us anything.
11:30 Different ways to handle equality:
1."=" as part of the underlying logic (as a member of the language of FOL).
2.a=b if a is in all and only the sets in which b is. In other words, a=b if a and b have the exact same properties.
3.Extensionality.
These definitions are mostly interchangeable.
Thank you so much professor. Your knowledge is blessing to the world. May Allah keep you in good health and mind always.
A fascinating lecture! It raises many deep questions, for me, at least. First, I don't see why it is claimed that the second part of (2) here (see 10:52) does not follow from the definition of equality being worked with, and that it therefore needs to be further stipulated. Why is "substitutablility" natural to the former and not the latter? Second - and maybe this is where I'm missing something significant - what is the difference between the Axiom of Extension and a definition of equality you propose? Why is the definition treated as distinct from the axiom? The Axiom and the Definition would seem to be intensionally identical! There's something ontologically principled going on here which hasn't been made explicit (or that I just don't see).
10:35 I didn't really understand why this doesn't follow from our definition. If a and b have the same elements, and a \in c, then c={a, other stuff} but then also c={b, other stuff} right since a and b are the same?
Yes that part struck me as unclear also.
Suppose we were to instead use an intensional definition of sets, so that A := "primes of the form 4n+1 or 4n+2" and B:= "primes that are the sum of two squares" were not actually «the same set». We could then define the symbol "=" such that A = B iff A and B contain the same elements. However, because we are using an intensional definition of sets, {A} and {B} are not actually the same set (despite satisfying the equivalence relation "=").
If A = B, then it *is* still the case that a ∈ A ⟹ (∃b such that b ∈ B and a = b), but it is *not* necessarily the case that a ∈ A ⟹ a ∈ B.
Another way of interpreting the axiom of extensionality that might be more intuitive is that it's basically saying «a and b have the same elements implies a and b are in the same sets», or «∀A∀B(∀X(X∈A⇔X∈B)⟹∀Y(A∈Y⇔B∈Y))». You can then use A=B as an abbreviation for ∀A∀B(∀X(X∈A⇔X∈B)).
Suppose we drop the extensionality axiom, then {1,2} and {1,1,2} are two different sets (at this point). We then define a relation R, R(A, B) if A, B have the same elements. So R({1,2}, {1,1,2}). We want to check whether R satisfies all properties of =: (1) R is an equivalence relation (yes), (2) For any R(A, B), x in A => x in B (yes), (3) For any R(A, B), A in C => B in C, but we stuck, {1,2} in C does not mean {1,1,2} in C. Then we make this axiom so we can make this deduction go through.
@@ApriiSR Thanks you, this makes it more clear. Indeed without the axiom of extensionality there is a difference between satisfying the '=' relation and actually 'being the same set'.
I guess the question has been answered, but just wanted to point out: it may have seemed to follow from the definition because you were thinking of the elements of a being in c, instead of a being in c.
Thank you for these wonderful lectures. A question: I don't understand how dropping Axiom of Extensionality permits the existence of multisets. Either an element a is a member of a set S or it isn't. If it is, then "writing it again" in the extension adds no new information. For the second copy to mean anything would entail writing "something different" - but without the Axiom of Extensionality, we don't know what constitutes that difference, since we don't know what constitutes equality. You would need some sort of index or mark to distinguish multiple copies, but indices are not a part of the definition of sets. (A similar issue arises with ordered "sets".) This issue, interestingly, is directly connected with an issue of great importance in linguistics, specifically in the question of movement in minimalist theories of syntax, which use the mathematics of sets to model constituency.
The rooted trees are very interesting. It reminds me of how computer scientists encode algebraic structures using bitstrings. (/ trees of pointers, relational database tables, etc.)
It's just implemented with these rooted unlabeled trees, instead. With the interesting "horizontal idempotence" property coming from extensionality. Similar to something you'd see in a CRDT.
A CRDT! It IS a CRDT! Implemented using journal articles! Ha, now THAT'S a fresh interpretation.
Can someone pls explain how including the Axiom of Extensionality (saying two sets are equal iff they have the same elements) is bypassing the problem that arises when we attempt to define equality of sets by saying a = b if they have the same elements?
The difference is between if and iff. Losing the other direction means you can't prove statements like the one 11:20.
Despite the definition of a=b, is not consistent with the properties of equality ( i.e. as mentioned in the video a ∈c , but b ∈c may not be true take a={1,1,2} , b={1,2}, c={{1,1,2}, {8}}), we take a=b as an axiom if they have same elements.
Hi, Prof. Borcherds. First of all, thank you for these lectures! Almost nobody has done in-depth coverage of ZF(C) except perhaps from a purely 'standard curriculum' perspective. Wonderful to hear a more complete and nuanced perspective like yours!
May I suggest, if possible, to record/upload in at least 720p video resolution? So far, I haven't actually had any problems with your 480p videos, so this is not really an urgent request; it's more like a 'future-proofing' request, so that your videos will 'stand the test of time' longer (at least in a visual sense).
The potential problem with 480p is that writing or drawings, especially if they become small or detailed, can be harder to make out to the point of being almost illegible. And if the resolution isn't there to support it, you can't even 'zoom in' or 'go full screen' to try to improve the legibility. In contrast, most writing in 720p is at least legible, and if it's illegible when displayed on a small screen, can at least be zoomed or scaled up to make it more legible.
Again, I have not actually experienced this problem on any of your videos I've seen so far. However, it is a common problem on 480p videos out there which feature writing or drawing, so I'm really just trying to 'forewarn' you of a potential problem.
Your writing tends to be large enough to be quite legible, so maybe it will never be a problem. However, I'm reminded of the many lectures I've watched which were recorded when TH-cam first started out, and some of them were recorded with 240p, even up to 480p, and even though the writing was large and on a chalk board, it is sometimes very poorly legible. But since no higher resolution is available, those lectures cannot be improved and will remain stuck at that low visual quality for the rest of time.
It's hard to predict how video technology will progress, and especially how videos will be used in the future (e.g. imagine your lectures being projected onto a big screen for a large lecture-room audience with some people sitting far in the back; who knows?). These days, 720p seems to be maybe-not-quite the 'minimum' standard, but on the other hand 480p seems to be maybe-not-quite 'on the outs'. Seems to me to be a bit of a transitional period. I mean, 720p isn't even considered 'high definition' (HD) on TH-cam anymore! (And virtually nobody goes as low as 240p, as used to be quite common actually.)
I think if you upgraded to 720p, your videos (their visual quality at least) will stand the test of time much longer than the current 480p, and at the same time I don't think 720p will be that much more difficult to work with on your end, with recording, editing, uploading, and archiving/backing-up. Might require an upgrade of a camera, although I believe even the cheapest of webcams will be capable of 720p these days.
Again, this is just an idea/suggestion. Sorry for such a long comment on such a basic topic. Sometimes I find it difficult to express myself succinctly. Cheers!
I understood you to say the three sets at the beginning were the same. If you did, I'm missing something important. How can '2' (which is indeed prime) be a member of a set whose elements are the sum of two squares? - unless you count '1' as prime, which is not traditional.
2 is a member of a set whose elements are prime (2 is prime) and AT THE SAME TIME a sum of two squares (2=1^2 + 1^2). The two squares don't have to be prime tho
@@andreaemanuele7293 To take it just a bit further: 17 = 16 +1, 5 = 4+1 and one is still not a prime and in the case of 17 neither is 4.
@@andreaemanuele7293 Thanks, Andrea! //David
@@terryendicott2939 ... and thanks to you, Terry! //David.
@@terryendicott2939 Indeed! :)
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeee