Against Set Theory

Questions

Recursive Definition of Addition Base case: x + 0 = x Recursive step:x + s(y) = s(x + y)

Prove that + exists! Ordered pairs Ordered triples Relations Functions The Recursion Theorem

Frege-Analyticity Analytic = “true in virtue of meaning” Frege-Analytic = provable from logic + definitions Same thing?

Set Theory: the Pros If there are some things, then there’s a collection of those things. Set theory is obvious. Mathematics is mysterious. It’s mysterious what numbers are, and how we know about them– but, if set theory is true, then we can explain mathematical truths and mathematical knowledge in terms of truths about sets and knowledge about sets.

Benacerraf’s Problem

Paul Benacerraf American philosopher Born in 1931 Teaching at Princeton since 1960 Argues against identifying numbers with sets

Ernie and Johnny Ernie and Johnny are each raised by parents who believe that numbers are sets. Each child is taught set theory first, before they learn to count. Then Ernie is taught von Neumann’s construction of arithmetic and Johnny is taught Zermelo’s construction.

Ernie and Johnny All of the “pure set theory” each boy can prove will be the same, for example: There is only one null set. All of the “pure arithmetic” each boy can prove will be the same, for example: = 4

But… For Ernie, 3 ϵ 17, but not for Johnny. For Ernie, every set with 3 members has the same number of members as the number 3, but not for Johnny. For Ernie, 3 = { Ø, { Ø }, { Ø, { Ø } } }, but for Johnny, 3 = { { { Ø } } }

From “On What Numbers Could Not Be” If “Is 3 = { { { Ø } } }?” “has an answer, there are arguments supporting it, and if there are no such arguments, then there is no ‘correct’ account that discriminates among [the different constructions of arithmetic in set theory]”

Frege’s View Frege actually didn’t use either Zermelo’s or von Neumann’s construction. For Frege 3 = the set of all 3- membered sets.

Frege’s View 3 = { x | Ǝy 1 Ǝy 2 Ǝy 3 y 1 ≠ y 2 & y 2 ≠ y 3 & y 3 ≠ y 1 y 1 ϵ x & y 2 ϵ x & y 3 ϵ x & ~Ǝz (z ≠ y 1 & z ≠ y 2 & z ≠ y 3 & z ϵ x) }

Argument for Frege’s View When we say “the sky is blue”, this is true iff the sky ϵ { x | x is blue } When we say “Michael is smart”, this is true iff Michael ϵ { x | x is smart } So when we say “These lions are 3 (in number)” it should be true that these lions (namely, the set of the lions) ϵ { x | x has 3 members } And that’s Frege’s view!

Weak Argument Benacerraf doesn’t think this is very plausible. Numbers in language function more like what we call quantifiers: All lions are in the zoo. Some lions are in the zoo. Five lions are in the zoo. Finally, the set of all three membered sets leads to a paradox.

From “On What Numbers Could Not Be” “There is no way connected with the reference of number words that will allow us to choose among them, for the accounts differ at places where there is no connection whatever between features of the accounts and our uses of the words in question.”

From “On What Numbers Could Not Be” “[A]ny system of objects, whether sets or not, that forms a recursive progression must be adequate. But… any recursive set can be arranged in a recursive progression.”

From “On What Numbers Could Not Be” [This] suggests that what is important is not the individuality of each element but the structure which they jointly exhibit… ‘Objects’ do not do the job of numbers singly; the whole system performs the job or nothing does.”

From “On What Numbers Could Not Be” “I therefore argue, extending the argument that led to the conclusion that numbers could not be sets, that numbers could not be objects at all; for there is no more reason to identify any individual number with any one particular object than with any other (not already known to be a number).”

From “On What Numbers Could Not Be” “To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by… Any object can play the role of 3; that is, any object can be the third element in some progression… [3 represents] the relation that any third member of a progression bears to the rest of the progression.”

From “On What Numbers Could Not Be” “Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particular objects – the numbers.”

Discussion If this is right, what’s left of Kant’s claim that arithmetic is synthetic a priori? If this is right, what’s left of logicism? If this is right, should we say the same thing about sets? Is set theory not the science of some objects – the sets?

The Banach-Tarski Paradox

The Axiom of Choice If you have a set S And all of S’s members are sets, And S has infinitely many members, And all of S’s members are non-empty Then there exists another set X Where X has exactly one member from each of S’s members.

Constructive vs. Non-Constructive Axioms

Video Time!

Nominalism

Quine & Goodman

Quine & Goodman’s Nominalism Reject abstract objects Say ‘x is a dog’ without admitting abstract thing: being a dog Say ‘P and Q’ without admitting ‘and’ represents anything (syncategorematic) Can’t say ‘x is a species’ or ‘x is extinct’ Motivation: paradoxes vs. arbitrary axioms Reject infinite numbers, infinite expressions if there are only finitely many concrete things

Quine on What There Is “To be is to be the value of a bound variable.” Ǝx x = Fido & x has the property of being a dog (Committed to Fido, but not properties.) Ǝx ƎP x = Fido & P = the property of being a dog & x has P (Committed to both.)

Reductions Instead of “the set of all A’s is a subset of the set of all B’s” we can say: “Everything that is an A is also a B.” Instead of “the set of A’s has 3 members” we can say: “X, Y, and Z are all A’s & X ≠ Y & Y ≠ Z & X ≠ Z & anything that is an A is either X, Y, or Z.”

Frege’s Ancestral Recursive definition of A = { x : x is an ancestor of Michael } a.Michael’s parents are ϵ A b.If x ϵ A, then x’s parents are ϵ A

Fusions “A broken dish is no less concrete than a whole one.”

Q&G’s Ancestral “Z is an ancestor of Michael” a.Michael’s parents are part of A b.If x is part of A, then x’s parents are part of A c.Z is a parent of some part of A

Size Comparisons “There are more dogs than cats” Set theory way: { x : x is a cat } < { x : x is a dog } Why can’t we say “the fusion of all the dogs is (physically) bigger than the fusion of all the cats”?

Size Comparisons “There are more dogs than cats” First, find the smallest dog and the smallest cat. Then take whatever one is smaller and call its size a “bit”. Consider the object that is made from a bit-sized portion of all the cats and the one made from a bit-sized portion of all the dogs. The latter is bigger.

What about Math? Math is the rule-governed writing down of symbols. We don’t need mathematical entities to understand what we’re doing when we do math, we just need the symbols and the rules for manipulating them.