Comments on: Are small sentences of Peano arithmetic decidable? http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/ Mathematics for computers Wed, 20 Aug 2008 12:03:16 +0000 http://wordpress.org/?v=2.3.3 By: Andrej Bauer http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-5740 Andrej Bauer Sun, 22 Jul 2007 22:57:19 +0000 http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-5740 Primality is diophantine, but as far as I am aware the equation(s) expressing it involve many variables, which gives us many existential quantifiers, not just one. And even if somehow we could express "`n` is prime" with a single existential "`EE m. p(n,m)=0`", would that not still give us a `AA EE EE` sentence "for all `a` there exists `b` and exists `m` such that `p((a+b)^2 - 2, m)=0`"? So I am afraid your comment does not convince me. (Note that Tung's result is about a single universal followed by a single existential.) Primality is diophantine, but as far as I am aware the equation(s) expressing it involve many variables, which gives us many existential quantifiers, not just one. And even if somehow we could express “`n` is prime” with a single existential “`EE m. p(n,m)=0`”, would that not still give us a `AA EE EE` sentence “for all `a` there exists `b` and exists `m` such that `p((a+b)^2 - 2, m)=0`”? So I am afraid your comment does not convince me. (Note that Tung’s result is about a single universal followed by a single existential.)

]]> By: jonathan http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-5729 jonathan Mon, 09 Jul 2007 17:05:55 +0000 http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-5729 I believe that the "near-square" conjecture you state above is decidable. Since primality is diophantine, it can be determined with one existential quantifier. Therefore, the "near-square" conjecture is a universal-existential sentence, which is decidable by the result of Tung referenced here: http://cs.nyu.edu/pipermail/fom/2006-October/011050.html Jonathan I believe that the “near-square” conjecture you state above is decidable. Since primality is diophantine, it can be determined with one existential quantifier. Therefore, the “near-square” conjecture is a universal-existential sentence, which is decidable by the result of Tung referenced here:

http://cs.nyu.edu/pipermail/fom/2006-October/011050.html

Jonathan

]]> By: Andrej Bauer http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-415 Andrej Bauer Wed, 27 Dec 2006 10:41:33 +0000 http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-415 I have to make a small correction. The sentence `forall b, c \in NN: a != (S(S b))*(S(S c))` does not mean "`a` is a prime" but rather "`a` is 0, 1 or a prime". Luckily, the short sentence with unknown status is not affected by this mistake. I have to make a small correction. The sentence `forall b, c \in NN: a != (S(S b))*(S(S c))` does not mean “`a` is a prime” but rather “`a` is 0, 1 or a prime”. Luckily, the short sentence with unknown status is not affected by this mistake.

]]>