Can set theory be non-extensional?

Question

Here is Juliet Floyd stating "Wittgenstein's non-extensionalism, like Russell's in Principia, precluded development of an extensional theory of the infinite (set theory). https://youtu.be/WGv8pqzHfoE?t=1094

So what has happened since? Is there still this view of Wittgenstein's that collections of infinite objects can't have extensional definitions?

With the term “extension” Wittgenstein has two things in mind. First, he will strictly distinguish between sequences of numbers that the extensionalist considers to be, in Cantor’s sense, “finished” [fertig] entities or sets – these are the “extensions” – from the techniques or rules by means of which such entities may be produced, assessed, or accessed. If there are such techniques, the extensionalist’s interest is ultimately only in their results, the produced sequences, and not the possible processes or conceptual motifs or definitions leading to them.

Wittgenstein’s criticism is directed at the common view of “modern” mathemati- cians that mathematics is about extensions. We call this the extensionalist point of view and the contrary view proffered by Wittgenstein the non-extensionalist one. We do not use the term “intensionalist” because it would suggest that words get their meaning via certain entities individually attached to them, entities called “intensions”, and this is a conception totally contradicting Wittgenstein’s stance. According to him, words (linguistic expressions, signs) get their meaning – their “life”, as he sometimes says – through the uses we make of them within language, and for him it is a sort of category mistake to base or ground this so-called meaning in immediate relations to certain entities accompanying the words.6

For the non-extensionalist, on the other hand, it is the processes and structured conceptual motifs, the grammar or logic of the notions, we should be concerned with. The most important cases discussed in Wittgenstein’s texts are given by the conception of a real number as a rule-governed calculational procedure for which we can see that for any given n it will generate n digits: for example, a recipe for generating more and more successive digits of √2. About the number√2 Wittgenstein explicitly says that “we [as extensionalists] have a tendency to think that there is one result produced by√2, viz., an infinite decimal fraction”. But on his view √2 produces a series of results, but no single result: “√2 is a rule for producing a fraction, not an extension” (AWL, p. 221) cover

The second context in which Wittgenstein speaks of “extensions” is the context of sets, paradigmatically sets of numbers: N, Q, R, and subsets of them. Considered extensionally, the laws or rules or techniques through which we may approach them are taken as irrelevant, as their identity is only determined by the elements of which they consist. From Wittgenstein’s non-extensionalist view, however, it is precisely these laws, rules or techniques we should take as primary.

Juliet Floyd, Felix Mühlhölzer Nordic Wittgenstein Studies 7 Springer International Publishing; Springer, Year: 2020 9783030484804

Doesn't this at least put people in a tough position? Reject the "greatest philosopher of the 20th century" or reject set theory.

Answer 1

For sets, extensionality is defined as follows.

∀S∀T(S=T ↔ ∀x(x ∈ S ↔ x ∈ T))

All modern set theories have this as an axiom or theorem, so they are all extensional.

Russel did not reject extensionality, but (at least at one point) he thought that extensionality is too limited. In general, this is a disagreement that is characteristic of the different concerns of philosophers vs. mathematicians. Mathematicians can do pretty much all they want to do extensionally, but philosophers want to do more. For one thing, they want a logically sound foundation for mathematics (something mathematicians have never been all that concerned about).

The essential problem with extensionality can be seen in a problem first pointed out by Frege. The problem is, "Why are mathematical statements like 1+1=2 useful?" Now, it may be obvious to you why such statements are useful, but you aren't a philosopher who is interested in understanding the meanings of sentences as a separate abstraction.

So, let's examine what this statement means. It has four distinct symbols. To distinguish the symbols being discussed from the semantics, I'll quote the symbols. Here are the meanings:

A. "1" means 1
B. "2" means 2
C. "+" means λxy.x+y
D. "=" means λxy.x=y

(the λ-expressions represent functions).

So far, so good. But now, let's give meanings to the clauses based on the meanings of their parts:

E. "1+1" means (λxy.x+y)(1,1) which means 2
F. "1+1=2" means (λxy.x=y)(2,2) which means True

The problem is that "1+1=2" means the same thing as "2=2", and "2=2" doesn't really provide any information. So how does "1+1=2" provide information when it means the same things as a phrase that does not provide information?

The problem occurs in step E where the meaning of "1+1" is the same as the meaning of "2". This is the step of extensionality. We replace "1+1" with 2 strictly on the basis of its value, throwing away the sense of the expression, and then we end up with a trivial statement at the end.

You may have noticed that the definition for set extensionality doesn't really apply here; that's because "extensional" and "intensional" have more general definitions, and the definitions for sets is just a specific application.

To get to the general definition, we need to discuss categories a bit. A category is any form of division or classification. It can refer to classes such as "man" or "banana" or properties such as "red" or "immortal". A category is said to "apply to" the objects that it categorizes or describes. If you are holding a ripe banana in your hand, then the categories "ripe", "yellow", and "banana" apply to that object.

Actually, there are many different kinds of banana, not all of which are yellow when ripe. The ones you normally buy in the grocery store are Cavendish bananas. You can often also buy plantains which are not as sweet but are great when deep fried and dipped in sour cream. We say that the categories "Cavendish" and "plantain" imply the category "banana". Similarly, "banana" implies the categories of "fruit" and "grows on a tree".

So, with that background we can define:

intension of a category P: all of the categories that P implies.

extension of a category P: all of the objects that P applies to.

You can see how this applies to sets. A set is the extension of a category. If you have a category "even numbers less than 5", the extension of that category is the set {2,4}.

So our solution to the problem is to say that the meaning of "1+1" is not 2, it's a category C which picks out 2 as the only thing in its extension. We call C the meaning or sense of the phrase, and we call 2 the reference or denotation. C has an intension which includes things like "refers to 1", "refers to addition". By contrast, the phrase "2" denotes the same thing, 2, but the intension does not include "refers to 1" or "refers to addition". It is the difference in intensions which explains how the sentence is useful and provides knowledge.

Can you have an intensional set theory? Technically, no, because in modern language "set" implies "extensional". However, you can have an intentional theory of categories (categories in the sense I defined above, not in the sense of today's category theory), which is essentially the same thing. It would have much the same rules as set theory, but without the axiom of extensionality. In such a theory, {x: x is even and x < 5} is not the same as the set {2,4}, which makes it rather an odd theory, and I'm not sure it would be useful for anything. However, set theory is really just higher-order logic restricted to unary predicates, and there are intensional logics.

Answer 2

The style of mathematics expressed extensionally as set theory, would be a type or perhaps category theory if intensional. IIRC, one set theorist (I forget his name) has even said that the axiom of extensionality is the most "analytic" of all the axioms, meaning (if taken in a Gödelian way) the most "intrinsically justified by the iterative conception of sets."

Also, it is difficult for me to accept that:

If there are such techniques, the extensionalist’s interest is ultimately only in their results, the produced sequences, and not the possible processes or conceptual motifs or definitions leading to them.

Either there were no such "extensionalists" in the first place, or by now it looks like there aren't very many, if any, because the entire process of characterizing larger and larger weird cardinals is highly interwoven with obsessions over the processes involved (i.e., in general, embedding theory) and the conceptual motifs at play (arguments about possible multiverses, with attendant talk of "buttons" and "switches" and "set-theoretic geology" delving towards some kind of "Ultimate Mantle"; or see the vast musings of Penelope Maddy and the history she covers in her famous "Believing the Axioms" work; or the whimsical talk of mice and weasels and "zero pistol and zero hand grenade"). Or reconsider this part of the quote:

For the non-extensionalist, on the other hand, it is the processes and structured conceptual motifs, the grammar or logic of the notions [emphasis added], we should be concerned with.

So, contrary to this seeming mythology of extensionalism/non-extensionalism, contemporary transfinite set theory involves overwhelmingly explicit and massive reliance on logical details in its (hyper)construction of new large-cardinal typologies. For example, indescribable cardinals depend on variations over structures jointly known (I think) as the Lévy hierarchy, which has to do with the existential and universal quantifiers, which are logical matters, after all. Then there are weakly and strongly compact cardinals, consideration of which emerged from reflection on infinitary logic. Embedding theory is a subset of model theory, as far as I can tell, and model theory "metaphysically" complements proof theory, which is again a very logically obsessive realm of analysis (proof-theoretic ordinal analysis being just about as cool as seeking new large-cardinal types); and anyway, part of model theory, esp. with respect to large cardinals, is the basic, generic definition of a cardinal as "large" relative to a given set theory T when and only when that cardinal is large enough to be a model of T, i.e. prove T (vs. relative consistency in the background: yet another definition in terms of logical functions).

The SEP article on Wittgenstein's philosophy of mathematics quotes him as asserting:

There’s no such thing as ‘all numbers’, simply because [emphasis added] there are infinitely many.

So Wittgenstein seems to have had a difference of opinion with set theorists on a level independent of the alleged extensionalism issue, unless his very concept of extension was itself entangled with this rejection of finished infinities. But then I would call into question his concept of extension. The intensional/extensional dichotomy has gone by many names over time: sense/reference, connotation/denotation, or in Kant discursion/intuition. Now as Kant notes pretty quickly in the first Critique, "Thoughts without content are empty, intuitions without concepts blind," so sets and types (or categories) might be interpreted as different sides of the same coin, or two eyes each of which is needed in order for us to clearly see. At any rate, mathematics occupies that weird threshold between the discursive austerity of pure logic on the one hand, and the intuitive "creativity" (effluent plurality) of imaginative-perceptual space on the other. The concept of extensionality has an intension to its name, and the concept of intensionality has endlessly many extensions (quantifying over everything whatsoever "in the end"). In this deep unity, it appears that mathematics can be framed as either/both intuitive knowledge of the intuition/discursion distinction or/and discursive knowledge of the discursion/intuition distinction. Never just an extensional knowledge, then.