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 Mon, 30 Jan 2012 07:14:48 +0000 hourly 1 http://wordpress.org/?v=3.2-beta2-18055 By: Lew http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/comment-page-1/#comment-17287 Lew Thu, 20 Oct 2011 22:06:46 +0000 http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-17287 It appears that Hardy and Littlewood's conjecture that there are infinitely many primes of the form x^2+1 is still unsolved. This would give a sentence that is smaller by one symbol. It appears that Hardy and Littlewood’s conjecture that there are infinitely many primes of the form x^2+1 is still unsolved. This would give a sentence that is smaller by one symbol.

]]> By: Andrej Bauer http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/comment-page-1/#comment-12111 Andrej Bauer Sat, 20 Jun 2009 07:10:38 +0000 http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-12111 Yes, you misunderstand <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem" rel="nofollow">Gödel's Completeness Theorem</a>. The theorem guarantees that <b>valid</b> formulae are provable, not <b>true</b> ones. The difference is perhaps a bit subtle: a formula is valid when it holds in every model. A formula is considered true when it holds in a preferred model (referred to as the standard model). Yes, you misunderstand Gödel’s Completeness Theorem. The theorem guarantees that valid formulae are provable, not true ones. The difference is perhaps a bit subtle: a formula is valid when it holds in every model. A formula is considered true when it holds in a preferred model (referred to as the standard model).

]]> By: Richard http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/comment-page-1/#comment-12103 Richard Fri, 19 Jun 2009 18:39:53 +0000 http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-12103 “All true small sentences of PA are provable.” Aren't all true sentences provable in general? Or, did I misunderstand the Completeness Theorem? Interesting idea though. “All true small sentences of PA are provable.”

Aren’t all true sentences provable in general? Or, did I misunderstand the Completeness Theorem?

Interesting idea though.

]]> By: Andrej Bauer http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/comment-page-1/#comment-11609 Andrej Bauer Wed, 22 Apr 2009 19:33:39 +0000 http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-11609 Oops, thanks, I fixed the axiom so that it reads `x+0=x`. Oops, thanks, I fixed the axiom so that it reads `x+0=x`.

]]> By: David http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/comment-page-1/#comment-11607 David Wed, 22 Apr 2009 16:09:08 +0000 http://math.andrej.com/2006/11/04/are-small-sentences-of-peano-arithmetic-decidable/#comment-11607 `x+0=0` is a rather unusual axiom. `x+0=0` is a rather unusual axiom.

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