Talk:Post's theorem

The article has been sitting in a half-finished state for a while. I expanded it and set it up for further improvement.

• This article needs to be at an undergraduate level at the highest
• More references, especially at the undergrad level, are needed
• The notation should be kept as simple as possible (but not more simple than that).
• I think that a proof would be nice, and the proof is elementary anyway.
• More corollaries would be good.

CMummert 13:42, 25 October 2006 (UTC)

I believe the 'alternative' definition I added using quantifier blocks and bounded quantifiers should probably be the primary definition. However, I don't remember and need to get back to working on my thesis.Logicnazi (talk) 02:46, 24 November 2007 (UTC)

The definition of $\backslash Sigma^\{0\}\_m$ differs from the traditional one for $m=0$ since it would not allow bounded quantifiers. Since Post theorem doesn't say anything about $\backslash Sigma^0\_0$ relation this doesn't affect the theorem, but it seems misleading. Catrincm (talk) 13:34, 30 September 2014 (UTC)

What does it mean: "at most if and only if"? Is it equivalence or one-direction consequence?

Eugepros (talk) 10:08, 6 October 2018 (UTC)

:I think they mean to say that $\backslash varphi(n,n\_1)$ is true if and only if $T$ halts at or prior to time $n\_1$. Related to that, the assertion that $\backslash varphi$ can be a $\backslash Delta\_0$ formula (only having bounded quantifiers) is almost certainly incorrect, or else it needs a citation. I've never seen any formalization of turing machines where the halting time function is $\backslash Delta\_0$. Obviously $\backslash varphi$ is $\backslash Delta\_1$, which is sufficient for Post's theorem, and it's even primitive recursive, but the idea that it could be $\backslash Delta\_0$ seems implausible. Jade Vanadium (talk) 20:13, 8 July 2024 (UTC)