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There is a natural analogy between the theories PA and ZFC. See the linked question by Gro-Tsen here.

Peano arithmetic (PA) is a first order approximation to the natural numbers. As is well known, there are true facts about the natural numbers that we cannot prove in PA, but we can prove them in stronger systems that seem to be entailed by philosophical assumptions about the natural numbers.

Informally, my question is: Does this aspect of the analogy extend to ZFC?

In a trivial way, the answer to my question is yes. Since we (supposedly) believe the axioms of ZFC, we accept the statement Con(ZFC) that asserts ZFC is consistent, and also Con(ZFC+Con(ZFC)), and so forth. My question then is whether or not there are non-trivial examples of higher order considerations giving us theorems about sets that are not provable in ZFC, just as the Paris-Harrington theorem is true but not provable in PA.

To attempt motivating an example, consider the following: The "axiom of union" in ZFC is a first order approximation of the philosophical assertion that putting collections together results in a new collection. (The classic paradoxes show that we have to be careful what we mean by a collection, but let's ignore that issue for now.) In particular, if we have some collection $S$ of subsets of the natural numbers, then we have no problem asserting the existence of $\bigcup S$ as a subset of $\mathbb{N}$.

If we have a first order, recursively enumerable theory $T$, extending ZFC, we can define a truth predicate $P_n$ for the formulas cut off at level $\Sigma_n$. Philosophically, we thus expect the existence of a full truth predicate---not one definable in any first order way in $T$ (that would be impossible by Tarski's theorem), but one that "exists" nonetheless. Can we leverage this fact, or other facts like it, to make philosophical advances in understanding "true" set theory?

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  • $\begingroup$ I suspect the truth predicate is outside the arithmetical hierarchy as well. However, we need an expert or someone with more than a cell phone to research this properly. Definability of truth might be a good search phrase. Gerhard "Not Ready For Big Screen" Paseman, 2020.01.30. $\endgroup$ – Gerhard Paseman Jan 30 '20 at 23:08
  • $\begingroup$ Do the usual arguments pro- or anti-large-cardinals fit into this? Or are you looking only for "unobjectionable" claims? $\endgroup$ – Noah Schweber Jan 31 '20 at 0:04
  • $\begingroup$ @NoahSchweber If you have an argument that by considering the usual axioms of ZFC from a second order perspective, large cardinals are affected, that would be very interesting. $\endgroup$ – Pace Nielsen Jan 31 '20 at 2:07
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    $\begingroup$ Reflection principles ("for any property of the universe of all sets we can find a set with the same property") are second order, and often seen as "entailed by philosophical assumptions" about sets. Cantor asserts something to this effect, but one has to be careful with "properties" to avoid inconsistency. You may like Maddy, Believing the Axioms. $\endgroup$ – Conifold Jan 31 '20 at 8:40
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    $\begingroup$ Does this mean something provable in second order ZFC? See also Williams, The Structure of Models of Second-order Set Theories. $\endgroup$ – Conifold Feb 1 '20 at 8:00

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