Logic - 2017-09-17

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5:02 PM

@LeakyNun We talk in the meta-system. Let M(N) denote what M thinks is the naturals N, and M(2^N) denote what M thinks is the powerset of N. For any countable model M of ZF, you have a bijection f between M(N) and M(2^N), so you can construct the set S = { x : x∈M(N) ∧ x∉f(x) }, and prove that S⊆M(N), but you cannot prove that S∈M(2^N), so you cannot obtain some c∈M(N) such that f(c) = S. If you could, then you get c∈S ⇔ c∉f(c) ⇔ c∉S. So in fact this proves that S∉M(2^N).

Namely, any countable model of ZF must think that 2^N does not have that set S that we can construct outside it.

> you cannot prove that S∈M(2^N)

should have put this in boldface; kind of the most important sentence :P

why can't you prove that?

Haha.

The rest of my comment shows why you can't prove it.

Because you can prove the negation.

And we believe that our meta-system is consistent, so if we can prove the negation of something then we can't prove it.

how do you prove that there is a bijection?

It's from the definition of "countable". If M is a countable model of ZF then M(2^N) must be countable.

hmm

alright

5:09 PM

If you want to see a more careful explanation of what "M thinks" means, then you need to look at the definition of interpretation.

"∈" will have to be interpreted by any model of ZF.

this is very paradoxical

let's say we biject n to {n}

wait, what

no, I can't do this at all

Cantor's theorem (diagonalization)
1. f bijects N to 2^N
2. S = {x∈N:x∉f(x)} is a set [schema of replacement]
3. S is a subset of N [definition of S]
4. S is an element of 2^N [definition of 2^N]
5. the sentence

the crucial step is 3->4

@LeakyNun This is a theorem proven by ZF. It is not a theorem about models of ZF.

@LeakyNun Sort of.

@user21820 @_@ doesn't model of ZF obey ZF

@LeakyNun It does, but as you noticed 3−>4 doesn't work outside the model.

we need to expand 3->4

5:15 PM

Just remember that as long as M satisfies "∃x ∀y ( ∀z ( z∈y ⇒ z∈N ) ⇒ y∈x )", then M thinks that N has a powerset.

It doesn't matter how small that powerset is. M does not care.

All it cares is that it has everything that it thinks is a subset of N.

M thinks that every element in S is in N, right?

@LeakyNun Yes, but I recommend you don't reuse variables that I've already used otherwise it's not clear which S you're talking about. I'll always use my definitions, so not your version of S.

so M thinks that every element in S is in M(N)?

M does not think anything about M.

using your notation

waaaat

5:18 PM

For every x∈S, M thinks that x∈N.

ha

my brain is imploding upon the universe

whatever that means

You have to be clear what is what. M(N) is an object in the meta-system MS. N is an object inside the model M.

M(N) may not even be the same as what MS thinks are the naturals.

wait, metasystem?

You need to work somewhere. You can't work in a vacuum.

oh, ok

so M(N) is outside, N is inside

5:21 PM

Yes.

and ∈ is just a binary function that returns true/false?

Inside M, it doesn't know that "∈" is a function. It even knows that it can't be a function otherwise Russell's paradox comes back.

I mean, outside

what can you do on the outside?

Outside M, since M is a model of ZF, by definition MS has a function that interprets "∈" for elements in M, yes.

are there any rules?

5:24 PM

Um.. of course MS itself is a formal system.

If you're curious, typically modern logicians use ZFC as the MS. But most results in logic actually hold in much weaker MS.

interpreting ZFC in ZFC.

MS can't talk about models of formal systems unless it has some notion of functions and collections.

but if you use ZFC as the MS, what is outside ZFC?

The question is not what is outside. If you are working inside MS, you can't see anything outside.

Intuitively remember that inside a model you cannot see outside.

Anyway I have to go soon.

ok, thanks for the time

5:27 PM

Since I said before that I don't buy ZFC as meaningful, here is a brief sketch of what I think should be meaningful:

9

A: Are sets and symbols the building blocks of mathematics?

The things you actually write on the paper or some other medium are not definable as any kind of mathematical objects. Mathematical structures can at most be used to model (or approximate) the real world structures. For example we might say that we can have strings of symbols of arbitrary length,...

so long @_@

Haha.

Ok see you!

so for every x in MS, x in S implies x in M(N)

[we're outside the model]

6:04 PM

1. there are only countably many objects x in M such that M(x) "∈" M(2^N).
2. cardinality inside the model no longer has any meaning once you go out of the model, since it's just another first-order logic formula.
3. let SM={ M(x) "∈" M(N) : M(x) "∉" f(M(x))}
4. SM doesn't need to correspond to anything inside M, since SM is a set defined outside M.

@user21820 resolution of Skolem's paradox

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