# Reference Request - What texts must I go through to be able to understand the second half of this answer?

I've just encountered this gem over on mathoverflow. I understand up to half of the answer. Somewhat.

My troubles begin about halfway down:

The point is that the concept of definability is a second-order concept...

Mainly I have no idea of:

• First-order, second-order "concepts"
• I am reading up on $$ZFC$$, but I have no idea what $$VOD$$, Skolem functions, Condensation... even are

It's safe to say that anything beyond the first half is unknown to me.

The answer links a paper that touches on the subject but I'm not well equipped I believe.

Any references would be helpful even if they're articles, thank you.

• I'm pretty sure "first-order" and "second-order" refer to first- and second-order logic. So I'd start by reading up on second-order logic. plato.stanford.edu/entries/logic-higher-order – Neal Jul 16 '20 at 22:44
• I suggest Set Theory: An Introduction To Independence Proofs, by Kenneth Kunen. – DanielWainfleet Jul 17 '20 at 2:18

You should probably read this post. Note that it does not contradict the MO post because it is about a different kind of "definable real". To make things clear, I will repeat the key definitions here:

A definable object over a theory $$S$$ is given by some 1-parameter sentence $$φ$$ such that $$S$$ proves $$∃!x(φ(x))$$. An element $$c$$ in a structure $$M$$ is definable over $$M$$ (without parameters) iff there is some 1-parameter sentence $$φ$$ over the language of $$M$$ such that $$c$$ is the unique element in $$M$$ that satisfies $$φ(c)$$. For the more general notion of "definable from parameters" see wikipedia.

First note that an isomorphic copy of the reals is definable over ZFC (i.e. there is a $$1$$-parameter sentence $$ρ$$ over ZFC such that ZFC proves ( there is a unique $$R$$ such that $$ρ(R)$$, and furthermore this $$R$$ is a model of the real axioms ). So we can conservatively extend ZFC to ZFC' by adding a constant-symbol $$ℝ$$ and axioms stating $$ρ(ℝ)$$ and ( $$ℝ$$ is a model of the real axioms ). Then (working within a meta-system that can construct the reals) the two key theorems from my post and the MO post are:

(1) Every model $$M$$ of ZFC' with a copy of the true reals (i.e. there is an isomorphism from the reals to $$ℝ^M$$) has uncountably many members of $$ℝ$$ (i.e. $$ℝ^M$$ is uncountable) but only countably many of them are first-order definable over ZFC (since there are countably many 1-parameter sentences over ZFC).

(2) If there is a model of ZFC, then there is a model $$M$$ of ZFC in which every element is definable over $$M$$. In other words, $$M$$ extends directly to a model $$M'$$ of ZFC' in which every element of $$ℝ^{M'}$$ is definable over $$M'$$. Obviously, such an $$M$$ does not have the true reals (since only countably many elements in $$M'$$ are definable over $$M'$$).

The point of the MO post is that not every model of ZFC' has undefinable elements of $$ℝ$$. After all, if ZFC has a model at all then it has a countable model, so internal uncountability within a model says nothing about definability.

The point of my post is that if we believe ZFC to be truly foundational then we ought to also believe that it has a model with the true reals, and such a model will of course have members of $$ℝ$$ that are undefinable over ZFC.

For the time being, you can ignore terms like "second-order concept" and "outside the universe" as they are intuitive but not really precise. For instance, NBG could be considered like a second-order axiomatization of ZFC, where classes are the second-order sort, since each class is some subset of the 'entire universe' (hence second-order). Since a definable element in a structure is defined by a 1-parameter sentence $$φ$$ over the language, and $$φ$$ corresponds to the class $$\{ x : φ(x) \}$$, in that sense definability is a second-order concept.

Anyway, $$V$$ is the class $$\{ x : x=x \}$$ and $$HOD$$ is the class of hereditarily ordinal definable sets.