most recent 30 from http://mathoverflow.net 2013-05-22T10:45:40Z http://mathoverflow.net/feeds/user/1496 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12426/is-there-a-computable-model-of-zfc/12431#12431 Anonymous 2010-01-20T17:44:19Z 2010-01-20T18:54:43Z

<p>No, it cannot be computable. It's a theorem of Tennenbaum from the 50's that there is no computable, non-standard model of Peano arithmetic. If there were a computable model of ZFC, then it would give a computable, non-standard model of Peano arithmetic. Specifically, it would contain some non-standard model for PA, specified by a set of elements and two relations, and the computability of the model of ZFC would imply computability there. (If, say, you want the sum of elements a and b in the model of PA, search by brute force for an element c of the model such that (a,b,c) is in the sum relation. This may be terribly slow, but it will eventually terminate.)</p> <p>Edit: That's a good point about the computability of (a,b,c), and I'm not sure how to compute that. Fortunately, it turns out not to be needed here. Specifically, if we define (a,b,c) = ((a,b),c) and define (u,v) = {{u}, {u,v}}, then even though it's not clear whether you can computably produce (a,b,c) given a, b, and c, given something you know a priori is a triple, you can try to check whether it comes from a,b,c by finding appropriate elements. (In the simpler case of pairs, to check whether w = (u,v) given that it is some ordered pair, we just need to find x and y such that both are in w, u is in x, and u and v are in y.) Now we just do major brute force: not only searching over all possible choices of c, but also over the auxiliary elements that would establish that (a,b,c) is in the relation. I hope there's a nicer way to do this, but I think it works.</p>

http://mathoverflow.net/questions/4172/where-does-a-math-person-go-to-learn-statistical-mechanics/4212#4212 Anonymous 2009-11-05T04:27:57Z 2009-11-05T04:27:57Z

<p>"Statistical Mechanics: Entropy, Order Parameters, and Complexity" by James Sethna (my favorite) and "Statistical Mechanics" by Kerson Huang are both really good books. Sethna's book is very readable and engaging; Huang's is more of a syatematic textbook.</p> <p>Ruelle's book is mathematically rigorous but is aimed at people who already know something about the field. Baxter's book is incredibly valuable for what it covers, but it is highly specialized and provides no motivation for people who aren't already comfortable with the basic formalism.</p>