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.

  • 2
    $\begingroup$ 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 $\endgroup$ – Neal Jul 16 '20 at 22:44
  • 1
    $\begingroup$ I suggest Set Theory: An Introduction To Independence Proofs, by Kenneth Kunen. $\endgroup$ – 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.


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