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which says that, at least one out of 2, 3, 5 is a primitive root modulo infinitely many primes: but we can't know which. I have thought for a while that a logician's commentary on this would be interesting, and less hackneyed than some of the other examples of non-constructive techniques and possible
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Herbrand's theorem, in its original form and for many classical theories, tells us something weaker than the existence property; namely it tells us that for some t1 ... tn, phi(t1) \/ ... \/ phi(tn) is a theorem, so any model must identify which of those ti is the (a) witness. I think
Herbrand's
285:, and also the argument that in a cut-free proof with last line proving a disjunction, the penultimate line must prove a disjunct (sorry if I'm putting this badly - my knowledge of the field is a bit amateurish). Primitive roots - I'm thinking of the approximation to the
331:, the candidates are {3, 5, 7}. It says the proof is non-constructive, fine... but you claim, "we can't know which", and I can't tell if you mean to say that Heath-Brown somehow proved that it's undecidable (?) or if it's simply unknown as yet. --
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What does it mean? If I understand it correctly, the language of IZF has no function symbols (including constants) in its signature. Thus, the only kind of terms are variables. Existence property declares, that for the theorem
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I don't know the conjecture (I've heard of the Artin conjecture, but I can't recall what it's about, should read more wikipedia, I guess), but it looks like this would be an excellent addition to the page ----
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In fact this isn't a theorem (I assumke you're talking about the sequent calculus): in LK the last rule applied can be a contraction, which is why the disjunction property can't be proven for LK;
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theorem does establish that the proof has content, in the sense you describe, for non-constructive theories. Oh, and I'd like to hear more about your number theoretic example. ----
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John Myhill (1973) showed that IZF with the axiom of
Replacement eliminated in favor of the axiom of Collection has the disjunction property, the numerical existence property, and
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In particular the existence property is fundamental to understanding in what sense proofs can be considered to have content: the essence of the discussion of existence theorems.
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at least one out of 2, 3, 5 is a primitive root modulo infinitely many primes: but we can't know which.
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The number theory example I'm thinking of is a quite striking one about primitive roots.
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There are quite concrete examples in number theory where this has a major effect.
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doen't imply that the formula is closed — I hope that I got this right).
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isn't closed: for example, (a version of) Heyting arithmetic proves
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So, it would be good to have discussion of the proof theory : both
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I think it would be helpful to give two or so examples of this.
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Make this page vaguely intelligible by a non-mathematician
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Discussion of
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