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Is there a paper or other reference that explores the implications of a theory that denying the existence of a truth predicate in its own language (perhaps based on ZFC or PA)?

I know that a theory that proves itself inconsistent will prove that it has no truth predicate, but I'm looking more for examples like ZFC+there is no truth predicate for set theory (which is the same as ZFC + no model of set theory is sound).

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    $\begingroup$ How would you formulate it? There are several distinct things going under the name “truth predicate”, but generally they are described by an infinite schema, hence you can’t negate it. Anyway, the proof of Tarski’s undefinability theorem gives you a sense in which every theory denies the existence of its truth predicate. $\endgroup$
    – Emil Jeřábek
    Commented Jul 12, 2017 at 8:46
  • $\begingroup$ @EmilJeřábek mathoverflow.net/a/273121/65915 shows a finite construction of a truth predicate. Also, I'm thinking in terms of second of "there is no function that determines truth" (which is not implied by Tarski's theorem) as opposed to "there is no definable function that determines truth (which is). $\endgroup$
    – Christopher King
    Commented Jul 12, 2017 at 12:02
  • $\begingroup$ A truth predicate (in the linked sense) is a proper class. How do you intend to universally quantify in ZFC (or in PA) over proper classes that are not definable? The theory has no means of referring to them. $\endgroup$
    – Emil Jeřábek
    Commented Jul 12, 2017 at 12:29
  • $\begingroup$ @EmilJeřábek Oh hmm, I guess formulating such a theory is harder than I thought. I'm going to ask a separate question about it. $\endgroup$
    – Christopher King
    Commented Jul 12, 2017 at 12:33
  • $\begingroup$ @EmilJeřábek Here it is. $\endgroup$
    – Christopher King
    Commented Jul 12, 2017 at 12:47

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