From: israel@math.ubc.ca (Robert Israel) Subject: Re: I am not a crook Date: 26 Jul 1999 22:27:52 GMT Newsgroups: sci.logic,sci.math Keywords: Loeb's (Lob's) Theorem: "Is provable" is provable In article <7nfprp\$u3k\$1@iradj.is-bremen.de>, "Marco Manfredini" writes: > I somewhat curious about self-confirming statements: > > (1) 'This statement is false' is the basic form of well known and > popularized antinomies > > (2) 'This statement is true' is what? It's not necessarily either: in any system of logic that allows "this statement is false", there is no consistent way of assigning "true" or "false" to all statements. However, the similar statement "This statement is provable" is indeed provable. The formal version of this is called L\"ob's Theorem. See G. Boolos, "The Logic of Provability", American Mathematical Monthly 91 (Oct. 1984) 470. Here's an informal proof of "This statement is provable". Call that statement S (i.e. S asserts "S is provable"). Let T be the statement that asserts "if T is provable then S". If T is provable, there is a proof of "If T is provable then S". But by combining this with the proof of T and modus ponens, you have a proof of S. Thus if T is provable then S is provable. But since S asserts "S is provable", we see that if T is provable then S. Now the above paragraph is a proof of T. Thus T is indeed provable. Therefore S is true (and thus provable). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2