Wednesday, November 10, 2021

"The whole is bigger than the part"

Some people don’t like Cantorian ways of comparing the sizes of sets because they want to have a “whole is bigger than the (proper) part” principle, denying which they consider to be counterintuitive.

Suppose that there is a relation ≤ which provides a way of comparing the sizes of sets of real numbers (or just the sizes of countable sets of real numbers) such that:

  1. the comparison satisfies the “the whole is bigger than the part” principle, so that if A is a proper subset of B, then A < B

  2. there are no incommensurable sets: given any A and B, at least one of A ≤ B and B ≤ A holds

  3. the relation ≤ is transitive and reflexive.

Then the Banach-Tarski paradox follows from (a)–(c) without any use of the Axiom of Choice: there is a way to decompose a ball into a finite number of pieces and move them around to form two balls of the same size as the original. And Banach-Tarski feels like a direct violation of the "whole is bigger" principle!

Thus, intuitive as the “whole is bigger” principle is, the price of being able to compare the sizes of sets of real numbers in conformity with the principle is quite high. I suspect that most people who think that denying the “whole is bigger” principle also think Banach-Tarski is super problematic.

For our next observation, let’s add one more highly plausible condition:

  1. the relation ≤ is weakly invariant under reflections of the real line: for any reflection ρ, we have A ≤ B if and only if ρA ≤ ρB.

Proposition: Conditions (a)–(d) are contradictory.

So, I think we should deny that, in the context of comparing the number of elements of a set, the whole is bigger than the proper part.

Proof of Proposition: Write A ∼ B iff A ≤ B and B ≤ A. Then I claim we have A ∼ ρA for any reflection ρ. For otherwise we either have A < ρA or ρA < A by (b). If we have A < ρA, then we also have ρA < ρ2A by (d), and since ρ2A = A, we have ρA < A, a contradiction. If we have ρA < A, then we have ρ2A < ρA by (d), and hence A < ρA, again a contradiction.

Since any translation τ can be made out of two reflections, it follows that A ∼ τA as well. Let τ be translation by one unit to the right. Then {0, 1, 2, ...} ∼ τ{0, 1, 2, ...} = {1, 2, 3, ...}, which contradicts (a).


Wesley C. said...

Isn't it still the case that the whole is greater than the part even for higher infinities and large cardinals, if one interprets greater as not just being limited to quantity or cardinality?

The whole would still be greater than the part conceptually, but not numerically, which isn't a necessary part of the truism.

Wesley C. said...

Another way the principle could be made to work here is that a proper subset B of A is still lesser than A conceptually, even though they may be equal numerically (say, B is the even integers and A is all the integers). Same for comparing the number of elements of sets - the set still conceptually contains the elements, even though they may be numerically identical.

Alexander R Pruss said...

I think my argument shows that there is no reasonable sense of "greater than" that makes the maxim true. For, any reasonable sense of "greater than" would satisfy (b)-(d).

Maybe, though I am a little worried about (b).

Alexander R Pruss said...

Another take on the argument here would be a finitist one: The "whole is greater" principle is true, and since there is no reasonable sense of "greater than" that makes the principle true if are actual infinites, there are no actual infinites. I am not happy with that.

Wesley C. said...

Couldn't one then say that it makes no sense to speak of a proper subset of an infinite set, since the very idea of subset already presupposes the "whole is greater than proper part" principle? Or to also say that the subset isn't actually a part of the infinite whole?

Maybe one could put a quasi-theological spin on it and say that infinity also reflects God in not having proper parts?

Alexander R Pruss said...

I do agree that "whole" and "part" are understood analogically in the case of sets. But there are definitely proper subsets: a proper subsets is just a subset that is not the whole. The set of even integers is a proper subset of the set of integers.