most recent 30 from http://mathoverflow.net 2013-05-24T23:15:57Z http://mathoverflow.net/feeds/user/6997 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9864/presburger-arithmetic/29145#29145 Doukas 2010-06-22T20:27:05Z 2010-06-22T20:27:05Z

<p>I like the discussion. (I am currently working on similar themes.) I have come to the following conclusions. In order for a theory to be provably consistent, it needs either (i) to be provably consistent but not by itself, like in Presburger or in Peano, or (ii) to be provably consistent by itself but in that case the theory has to disallow the inference from the statement that it is consistent to the <em>I am improvable</em> sentence, like in Willard. The introduction of the reflection principle is possible there exactly because of what you say Adam. Notice that in case the fixed point theorem did work in Willard, this, in combination with the reflection principle, would suffice for a contradiction, and no matter what the status of the first incompleteness theorem would be. </p>