is there some statement equivalent to consistency of $ZFC$ in $ZFC$?

Question

In my studies I learn the strengthened finite Ramsey theorem is equivalent to consistency of peano arithmetic and is a sentence of the language of arithmetic. I wanna a example of set theory

Answer 1

Constistency statements for effective formal systems like PA and ZFC can be expressed in arithmetic by Godel numbering, and logicians usually think of consistency statements as sentences of arithmetic by default. In fact, in the example you cite, what is happening formally is that, within some weak arithmetical theory, the statement of the combinatorial theorem in the formal language of arithmetic is shown to be equivalent to the arithmetical statement of Con(PA). The language of set theory can express arithmetic (and much more), so Con(PA) and Con(ZFC) can also be viewed as set theoretical statements.

But the question that you want to ask is not trivial, though it is somewhat vague. What's special about the combinatorial theorem you mention that is equivalent to Con(PA) is not that it is arithmetical, but that it is 'natural' in some sense. While Con(PA) is an arithmetical statement cooked up precisely to correspond to the idea of PA being consistent, the combinatorial theorem has nothing to do with logic on its face, and it is a statement a mathematician might reasonably consider even if they have no interest in logic. So it is noteworthy that such a theorem exists, and the fact that it cannot be proved in PA (except in the unlikely event that PA proves PA is inconsistent) is interesting.

There are a number of general purpose results that show that these consistency statements can always be made into more mathematical questions. For instance, there is a diophantine equation that has solutions in the integers if and only if PA is inconsistent (and another one for Con(ZFC). Also one can readily show there is a turing machine that halts if and only if the theory is inconsistent. (If I recall correctly, somebody has explicitly constructed one for ZFC with a few thousand states). But these things aren't really natural since they would seem arbitrary to a mathematician (why that Turing machine?). But they can be a useful place to start.

While there are several well-known 'natural' equivalents to Con(PA) it is a good deal harder to find something for ZFC. Harvey Friedman has had a program to find natural combinatorial problems whose solution prove Con(ZFC) and the consistency of stronger systems with large cardinal axioms. I'm not all that familiar with his work on this front, but this answer seems to have some leads.

If you expand the search radius from 'simple combinatorial theorems' to 'set theoretical statements', then there are quite a few very natural statements that imply Con(ZFC). The most prominent is probably the statement that inaccessible cardinals exist. That, and the whole hierarchy of large cardinal stronger than it, imply that ZFC is consistent. But these are all strictly stronger... I don't know of a natural statement that is equivalent off-hand. (The only new thing that springs to mind now that we have sets is 'there is a model of ZFC', but that's hardly natural in the prescribed sense.)