Godel's incompleteness theorems
in [Logic]
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From: Marc Alcobé García on 1 Apr 2010 13:52 On 30 mar, 19:01, zuhair <zaljo...(a)gmail.com> wrote: > Suppose we have a set theory T, which cannot define > the notion of "natural number", would that make T > escape the incompleteness theorems of Godel's. > > Zuhair I know of a partial result due to H. M. Löb quoted in Boolos' "The logic of provability" Chapter 2 (page 16). The reference given there for that result is "Solution of a problem of Leon Henkin" Journal of Symbolic Logic, 20 (1955), 115-18. The result in question is this one: If Z is a theory in which a few simple facts about natural numbers can be proved (namely the first six axioms of PA and "every non-zero number is the sucessor of some number", and there is a formula B(x) such that for all S, T sentences of PA: 1. If Z proves S then Z proves B([S]), where [S] is the numeral that stands for the Gödel number of S (Boolos' uses Quine's corner notation, but I don't know how to type it). 2. Z proves that B([S->T]) -> (B([S]) -> B([T])) 3. Z proves that B([S]) -> B([B([S])]) then Z cannot prove its own consistency. These three numbered conditions over Z are called derivability conditions. Usually a theory that proves things about natural numbers is one for which in some extension by definitions (these are always conservative) one defines 0, s(x), + and ·, and proves some fragment of PA (usually varying only in how much induction you take into account). I have not read all contributions, so I hope not having repeated anything already said. Sorry if I have and I hope this helps. |