Before this lecture, I genuinely thought there was absolutely nothing interesting to say about foundation axiom. Now I know better. Thank you.

I never knew that simply by asking what happens when you don't assume one axiom could result in such interesting mathematics! Amazing video!

what a treasure!!

Thank you so so so so so so so so much for putting this out for free like this

This is so exciting. Thank you for making this available to the world!

I’m very thankful to Professor Borcherds for putting this series like this so it’s much easier to get started. Working from the naive/informal definition and discussion and giving more details while going through various of bizarre examples really give a sense of why are these necessary to be described in 1st order logic with such strictness. I’m really excited to see the next after another one.

@11:45 This is very confusing. So under ZFC+(ZFC is inconsistent), we are given all the axioms of ZFC and we are also given the existential fact that somewhere out there there is a derivation of False from the axioms, but obviously we are not told what it is, and the claim that this is a consistent theory means that we will go on forever never actually deriving a contradiction though always cynically believing there is some way to do so. In a model M of ZFC + (ZFC is inconsistent), there is some nonstandard integer encoding a derivation of a contradiction, which is something like somebody having an infinitely long proof of a contradiction that might be right but nobody knows because they can never read it all.

10:44 ok, but what if we add an extra constant C and for each n we add an axiom c_n \in C? Wouldn't the new theory be still finitely satisfiable, but inconsistent? EDIT: found the loophole. The standard form of foundation I know, that I thought was troubling, says that for every set x there's an element y \in x disjoint from x. I thought this would give a contradiction for x=C, but it doesn't necessarily as C may contain (and indeed must contain) elements other than the c_i's. I feel like there are more things to explore, such as trying to add for each term t not containing the new constants an axiom stating that t is not in C. I feel like this could go somewhere interesting.

Prof. Borcherds could you share any bibliography to learn more about all these aspects? This is such an amazing new source of learnings for me. Thank you for these amazing videos!

@An-ht8so

2 жыл бұрын

The french reference is the book by Krivine, and it's fantastic. I would assume that its translation : "Introduction to axiomatic set theory" is a must-have as well.

@quantumcarlos

2 жыл бұрын

@@An-ht8so Thanks so much for the reference!

10:31 There are some points that trouble me here. 1. Why can we not use Replacement to construct c within M? 2. I had imagined the argument that 'inclusion' in the model M cannot be interpreted as 'inclusion' in the universe U which contains the model M, otherwise we would still have a contradiction in U (assuming that the universe we are working in satisfies well-foundedness). Is this how the argument in the video works or am I messing something up here?

@oh3991

2 жыл бұрын

Sorry I don't get your 2. but this may answer the both: Axioms we have declared(ZF+something) only works in "elements" in "M" thus you can't use the M as the starting point of the set of the comprehension axiom etc...

@midnightposting

2 жыл бұрын

I also would think that C={c_0,c_1,c_2,...} is definible in M as, by the axiom of pairing, each set {c_0}, {c_1}, {c_2},... is in M and then their union C is in M. But probably the problem is that we cannot take the union as it goes over the set {0,1,2,...} and maybe this set is actually not in M as it may be (this is the part I am less sure) a non-standard copy of the natural numbers which does not live in M.

Hi Prof. Borcherds! I wonder if you have any interest in doing any sort of lectures (or just informal videos, perhaps) covering some of the _history_ of the mathematics (e.g. the people involved, how the maths were developed and what relation they have to other mathematics that were being developed at the time)? For instance, you mentioned in this video that Zermelo didn't include the Axiom of Foundation in his 1908 version, but did later in 1930. That raised a few questions in my mind, like, "I wonder what it was that caused him to consider it important to include it? Was it perhaps some paradox discovered, or maybe just a specific type of problem that couldn't be solved without it?" and then also, "What was it about the time of roughly 1908 and/or of 1930 that had sparked interest in axiomatic views like Zermelo's" and also, "Who else was exploring such axiomatic ideas, and what were their systems like, and why did we end up converging on Zermelo-Fraenkel, and not some others' systems?" Personally, I find such background information often helps me connect-the-dots between different areas of maths and form a 'bigger picture' of the role and purpose of various fields of math, and how they often actually do have a sort of 'logical' progression from one thing to the next (even if the 'logic' of that progression may sometimes just be one of historical accident; more often than not, though, it can be the case that one development inspires a future development).

@1:50 Why does it conflict?

@SeanEberhard

2 жыл бұрын

Here a is "minimal for \in" in A means there is no b \in A such that b \in a (including b = a), or put another way a \cap A = emptyset. So if a \in a then {a} has no \in-minimal element, and if a \in b \in a then {a, b} has no \in-minimal element.

Why are 4 of the 7 videos in this playlist hidden? Please make them public! :-)

@oh3991

2 жыл бұрын

Probably he is revising it or editing some minor mistakes

Why is set C not in M?

Robinson = Abraham Robinson, not Julia Robinson.

yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

@blairkilszombies

2 жыл бұрын

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Borcherds, you are a field's medalist! Why do you spend your time making youtube videos if you have already conquered math?!

@him21016

2 жыл бұрын

Why do you have disdain for his teaching? He’s decided to spend his time offering invaluable university education to everyone, for free. Teaching is a very impressive thing to do!

@zapazap

2 жыл бұрын

@@him21016 Asking the question does not _strictly_ entail disdain.

@CasualGraph

2 жыл бұрын

Not sure I see the mismatch between those two things, people still go on hikes after summiting Everest after all.

@franciscodanieldiazgonzale2096

2 жыл бұрын

Easy: Teaching keeps you sharp. It is also an activity NOT related with his current work, that he is doing in a difficult problem he is not ready to share yet. One of the classic advices for mathematicians to get the ball rolling when doing mathematics research is to get engaged in a couple of other activities. It will free your mind of the problem and it provides stimulus to other unconscious deep processes. He already answered these points in one of the Q&A videos in this channel!

@martinepstein9826

2 жыл бұрын

Nobody has conquered math.

The Axiom of Foundation is sometimes called the Axiom of Regularity, but I dislike that, since Regularity means something else in Set Theory (a cardinal is Regular if any sequence of numbers less than it of length less than it, also has its supremum less than it). Foundation says that you cannot have an infinite series: ...a3 ∈ a2 ∈ a1 ∈ a0 You can however, have a series: a0 ∈ a1 ∈ a2 ∈ a3... Stated formally, "If A is a nonempty set, we can find a∈A which is minimal for the "∈" relation." (I have no idea what that means. If "minimal" means just a leaf of the set tree, an empty set, then this axiom would mean that every set must have ∅ as its leaf. But that would be weird, since it said "a∈A", and ∅ is not an element of every set.) "In particular, ¬∃a(a∈a), nor exist chains of sets such that a∈b, b∈c, c∈a." 1:45 Applications of Foundation. In a Von Neumann universe, all sets have a rank. Rank(A) is the smallest ordinal greater than the ranks of all elements of A. Rank(A) also turns out to be the smallest α such that A∈Vα. (Borcherds casually roasting Foundation for being useful only in making models of set theory, and so meritless for all the rest of mathematics, that Zermelo didn't even include it in his first published axioms.) 3:45 What happens when you omit Foundation: Obviously, you can get a={a}, and Borcherds said you could build a Von Neumann universe using {a} instead of ∅ as the basic element (because by Extensionality, there is only one such {a}), BUT HE IS WRONG! {a} will reduce itself from any Neumanning! E.g. {{a},a} = {a,a} = {a} = a (Also, Pow(a) = {∅,a}, so we will not escape using ∅ anyway.) From 6:45 to 10:45 things are nonsensical to me. Then, Borcherds considers ZFC + a new axiom X="ZFC is Inconsistent" And I understand that if ZFC is Consistent it cannot prove it, so ZFCX would be Consistent, but what if ZFC is Inconsistent?? Then its Inconsistency will be provable within it, and if ZFC is Inconsistent, then ZFCX is also Inconsistent! (If he meant that X="ZFCX is Inconsistent" nothing changes. Yes, while ZFCX is Consistent it cannot prove its Inconsistency, but if ZFC has an Inconsistency somewhere out there, it will still make ZFCX Inconsistent. (The fact that one of ZFCX's axioms confirms its Inconsistency does nothing to prevent that Inconsistency.)) 13:45 Well-foundedness cannot be defined by First Order statements. So the Axiom of Foundation (which is in FOL) is only doing its best to describe it. And weird things sprout from it, things I do not understand. I best just ignore it. 19:00 "Foundation together with Extensionality more-or-less describes what a 'set' is: A well-founded (ugh, let us approximate it to mean all branches are finite), rooted, rigid (satisfying Extensionality) tree." (But wait, I though we define sets by Von Neumanning?) 20:45 The limit of finite branches does not mean the rank of a set must also be finite. The rank can be any ordinal, shown visually by Borcherds here.