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In Frederic Schuller's lecture series Lectures on Geometrical Anatomy of Theoretical Physics, he declares (here) the first axiom of ZFC set theory to be the Axiom on $\in$-Relation, stated as follows:

Informal:

"$x\in y$" is a proposition if and only if $x$ and $y$ are both sets.

Formal:

$\forall x:\forall y:(x\in y)\veebar \neg(x\in y)$.

The "formal" version above (where $\veebar$ means "exclusive or") is not given in the lecture, but is stated in Simon Rea's transcription of the lectures, found here (p. 8).

Schuller does not include the traditional Axiom of Extensionality that I have seen in every other ZFC book, which states that two sets are equal if and only if their elements are identical.

Two questions:

  1. Does the "formal" statement above actually encode the "informal" version given by Schuller? I can maybe see this, if the implication is that our axioms simply don't discuss variables $x$ and $y$ unless they are sets, but no other formal encodings are of the same nature, so I'm not convinced.

  2. Is Schuller's axiom equivalent to the Axiom of Extensionality, all other axioms being equal? On the one hand, Schuller is a very smart man whose lectures on physics tend to be carefully thought out and steeped in mathematical rigor. On the other hand, I can't possibly see how the two are equivalent and can find absolutely no other references to Schuller's axiom online.

tryst with freedom
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WillG
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    I am no logician but this feels fishy to me. This axiom can be deduced from the law of excluded middle. On the other hand, the axiom of extensionality actually says something about set theory: it relates the $=$ and $\in$ relations. – diracdeltafunk Jan 14 '20 at 04:40
  • @Buraian: While it is helpful to add links to Wikipedia for technical terms, it is not at all necessary to bump old posts just for that. – Asaf Karagila May 07 '21 at 21:03
  • Ok! Will keep it in mind @AsafKaragila – tryst with freedom May 07 '21 at 21:06

1 Answers1

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No, that's absolutely bonkers: Schuller's "axiom" is just an instance of LEM, which is built directly into the underlying logic itself. Extensionality is much more interesting than that: it says that the elementhood relation completely determines the identity of a set. One direction of this is trivial - certainly two equal sets have the same elements - but the other direction is much less trivial than it may first appear.

I think that what's happened is that Schuller has internalized Extensionality as something so basic that it's on the level of logical rules as opposed to axioms, and so he's managed to conflate it with an (instance of an) actual logical rule. But in fact we can quite easily work with failures of Extensionality without breaking logic!

The most important kind of failure of extensionality in my opinion is given by urelements. Urelements are things other than the emptyset $\emptyset$ which don't have any elements; the only way we can have them is if Extensionality isn't present. Generally, when we switch from Extensionality to (the option of having) urelements things change in quite interesting ways:

  • It's relatively easy to prove that the axiom of choice is not provable in ZF - Extensionality, whereas proving its independence from ZF is much harder. That said, the two approaches turn out after the fact to be closely related.

  • Looking at weak subtheories of ZF in the context of (higher) computability theory, Barwise showed that urelements are extremely useful in the context of admissibility theory; see his book.

  • For an even more extreme if more technical example, consider the difference between Quine's set theories NF and NFU (= NF without Extensionality). On the one hand, NF disproves Choice, consequently proves Infinity, and is not known to be consistent even relative to large cardinals (Holmes has a claimed consistency proof, but I don't think it's been fully vetted yet). On the other hand, NFU is consistent relative to PA, is consistent with Choice, and is consistent with the negation of Infinity. So here in fact we have a situation where adding Extensionality to a reasonably-well-behaved theory results in something which at least according to our current understanding is quite wild!

Noah Schweber
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    Looking at the video, it appears that Schuller treats equality as defined by extensionality, rather than as a logical symbol. He seems to neglect to then add any substitution axiom for equality. – Eric Wofsey Jan 14 '20 at 05:03
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    @EricWofsey Ugh. – Noah Schweber Jan 14 '20 at 05:05
  • Sorry, I'm only now realizing that he does indeed treat extension as a definition of set equality (see 5:43 in video). @NoahSchweber does that make Schuller's approach more valid? – WillG Jan 14 '20 at 05:26
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    @WillG Not really: if he's going to change what equality means he should explicitly present the whole logical apparatus around equality (see Eric Wofsey's comment above). It's just a sloppy way of doing things; it's fixable, but why do it in the first place? – Noah Schweber Jan 14 '20 at 05:29
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    Meanwhile, if he's doing that then why include the "axiom on $\in$-relation" at all? It's just an instance of LEM, so either he's not assuming LEM in general (in which case things suddenly get a lot more finicky, and he should say that explicitly) or - more likely - it's just redundant. – Noah Schweber Jan 14 '20 at 05:31
  • @NoahSchweber so let's say the note transcriber just didn't correctly encode the axiom in logic (maybe he was jumping at the "$x\in y$ is a proposition" part and appealed to LEM as a best effort). Does it hold together logically using the "informal" version above? – WillG Jan 14 '20 at 05:36
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    @WillG I honestly don't even get what the "informal" version means. What a proposition is is dictated by the syntax; what's being gained there? – Noah Schweber Jan 14 '20 at 05:40
  • That makes sense. I don't know much about formal logic, but I'm wondering if Schuller uses nonstandard conventions for his logic? He defines a proposition in the first video with "A proposition p is a variable that can take on the values 'true' or 'false,' and no others." Is this not the usual approach? – WillG Jan 14 '20 at 05:51
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    @WillG, so $R\lor Q$ isn't a proposition for him because it's not a variable? I don't think he meant to say that or he and I don't have the same definitions of variable. – Mark S. Jan 14 '20 at 13:14
  • Could you explain " level of logical rules as opposed to axioms, and so he's managed to conflate it with an (instance of an) actual logical rule." more? Does this mean that schullers axiom set is missing an axiom relative to the standard ZFC? @NoahSchweber – tryst with freedom May 07 '21 at 20:05
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    @Buraian In modern mathematical logic, we distinguish between a logic and a theory in that logic. For example, ZFC is a theory in first-order logic; its syntax and the relevant proof rules are part of the logic, not of ZFC itself. (There's a vague analogy here with the difference between a programming language and a specific program in that language.) Now amongst the rules for first-order logic are the laws of excluded middle and non-contradiction, and Schuller's expression "$\forall x:\forall y:(x\in y)\underline{\vee}\neg (x\in y)$" is a consequence of these. (cont'd) – Noah Schweber May 07 '21 at 20:12
  • That would mean, he didn't have to define those! That should have been implied by the programming language he was using – tryst with freedom May 07 '21 at 20:14
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    @Buraian Well, "logic," not "programming language" - like I said, that's only an analogy. But yes, what Schuller wrote as a ZFC axiom is totally redundant. And in particular, it's not the axiom of extensionality; per WillG's comment above, it looks like Schuller is folding extensionality into the underlying logic (which you really shouldn't do). – Noah Schweber May 07 '21 at 20:14
  • The way you said it, it suggests to me that there maybe other 'programming' language/ logics which you could write set theory in , is that correct? – tryst with freedom May 07 '21 at 20:15
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    @Buraian Yes, and there's nothing special about set theory here. The general study of arbitrary theories in first-order logic (which is by far the most common logical system used) is called model theory, and the broader study of general logics is called abstract model theory. For example, besides first-order logic we have second-order logic and infinitary logic(s). – Noah Schweber May 07 '21 at 20:16
  • My mind is expanding just like the universe – tryst with freedom May 07 '21 at 20:18
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    Meanwhile, note that in ZFC everything is a set - so "$x\in y$ is a proposition if and only if $x$ and $y$ are both sets" is a weirdly redundant way of stating the (specific case of the) law of the excluded middle here. Schuller's presentation is highly ideosyncratic; even if it works for his specific purposes (and I don't know if it does), I really wouldn't recommend it as an introduction to set theory or logic in general. – Noah Schweber May 07 '21 at 20:19