Wednesday, June 10, 2009

Sets and Truth

In this post, I will explain a way to create a theory of sets, given a theory of truth. These two foundational issues are typically treated as separate matters, so asking for a particular relationship to hold between a theory of truth and a set theory adds some interesting constraints to the situation.

(Part of this post is an edited version of an email I sent originally to Randall Holmes.)

Claim: A set is essentially a sentence with a hole in it. Set membership is characterized by the truth/falsehood of the sentences when we fill the holes.

The major justification for this way of understanding sets is the way we use the comprehension principle to talk about sets. The comprehension principle is what allows us to say, in a set theory, that if we can describe the membership requirements for a set then that set exists. For example, I can describe the requirement "it must be a prime number that is over 5 digits long", so the set of prime numbers over five digits long exists. (The comprehension principle with no restrictions leads to naive set theory, however, which is paradoxical.)

This view is not far from the way Frege explained sets, as I understand it. However, he distinguished the set as the extension of the sentence-with-hole; meaning, the things for which it is true.

So, suppose we've got a logic with enough machinery to represent computable functions, and furthermore we've got a description of the language in itself (ie, Godel-numbering-or-something-equivalent). Furthermore, we've got some theory of truth. Then, via the claim, we can already talk about sets, even though they haven't been purposefully introduced. In particular, "x is an element of y" is
interpreted as:

"When "x" is used to fill in the partial sentence Y, the result is true"

where "x" is the name of the term x (Godel-numbering-or-equivalent, again, for those who are familiar with such things), and Y is the sentence-with-hole corresponding to the set y.

The truth predicate is needed here in order to assert the result of the substitution. With the naive theory of truth, it is always meaningful to apply the truth predicate to a sentence. So, the naive theory of truth gives us the naive theory of sets, in which set-membership is meaningful for any set we can describe. Of course, this is inconsistent under classical logic.

So, what I'm saying is: if the claim is accepted, then the set theory is pinned down completely by the theory of truth. The naive theory of truth gives us the naive theory of sets. A tarski-hierarchy theory of truth gives us something vaguely resembling type theory. Kripke's theory of truth gives us a theory in which all sets exist, but not all membership evaluations are meaningful. In particular, Russel's set "all sets that do not contain themselves" exists. We can meaningfully say that any set of integers is in Russel's set, and that the set of all sets (which exists) is not. The paradoxical situation, in which we ask if Russel's set is a member of itself, is simply meaningless.

So good so far. But, there is the issue of extensionality to deal with. The axiom of extensionality is taken as a very basic fact of set theory, one that not even nonstandard set theories consider rejecting. Given the above discussion, however, the axiom of extentionality would be false. Two different sentences-with-holes can be logically equivalent, and so have the same extension. For example, "_ is an even prime number" and "_ added to itself equals 4" are the same set, but they are different sentences.

My solution here is to interpret the notion of set-equality as being a notion of logical equivalence between sentences-with-holes, rather than one of syntactic equivalence. In other words, "x=y" for two sets x and y needs to be interpreted as saying that X and Y mutually imply each other given any slot-filler, rather than just as saying X=Y. But this is doable within the language, since all we need to do is quantify over the slot-filler-names.

This can be thought of as my way of interpreting Frege's concept of the "extension" of a sentence-with-hole. Rather than being a seperate entity, the extension is a "view" of the sentence: the sentence up-to-equivalence.

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