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We know that given a set of axioms in a formal system, if the system is consistent, then some true statements cannot be proven, but isn't truth based on the proof? If we can't prove a statement, then what asserts the truth?

Nathan Kaufmann
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    True things are true regardless of whether or not that truth is asserted (or indeed even known) by anyone. –  Feb 07 '24 at 13:24
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    @jwhite So how do we know if it is true? – Nathan Kaufmann Feb 07 '24 at 13:42
  • @MauroALLEGRANZA How is the "strength" of a theory measured? – Nathan Kaufmann Feb 07 '24 at 13:43
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    The idea isn't that we run across some proposition $P$ and know that $P$ is true without being able to prove it. The idea is that, in the world of all of the unprovable propositions, some of them are true (but we don't know which ones). –  Feb 07 '24 at 13:45
  • Analogous to how the intermediate value theorem will tell us that a point of intersection exists (but not what that point of intersection is), we can know that a true, unprovable proposition exists without knowing which propositions one(s) it may be. –  Feb 07 '24 at 13:47
  • And there might be some statements like "this statement can't be proven" that some may find obviously true. – Henrik supports the community Feb 07 '24 at 13:47
  • I don't think jwhite's comment is apposite. Gödel's theorem takes a formal system and provides an explicit construction of a specific statement that is true but not provable in that system. – MJD Feb 07 '24 at 13:54
  • I think the answer here is that we have a concept of the arithmetic of the natural numbers (the “standard model”) that precedes the axiomatic definition, and a concept of the truth of the standard model of the natural numbers that is separate from what is provable in any formal system about it. I'm sure I have read a discussion of this before on this site. But I have to take the car for service so I won't have time to look for it. Meanwhile Nathan Kaufmann should check out the "Related" links in the sidebar. – MJD Feb 07 '24 at 13:57
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    Aha,, I found it. Check out Henning Makholm's and Asam Karagila's answers here: https://math.stackexchange.com/a/213264/25554 (I am voting to close as a duplicate of that post.) – MJD Feb 07 '24 at 14:06

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