User michael o'connor - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:38:41Z http://mathoverflow.net/feeds/user/1574 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44208/is-there-any-formal-foundation-to-ultrafinitism Is there any formal foundation to ultrafinitism? Michael O'Connor 2010-10-30T02:27:45Z 2013-01-25T05:35:23Z <p>Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to <a href="http://en.wikipedia.org/wiki/Ultrafinitism" rel="nofollow">wikipedia</a>, it has been primarily studied by Alexander Esenin-Volpin. On his <a href="http://www.math.rutgers.edu/~zeilberg/OPINIONS.html" rel="nofollow">opinions page</a>, Doron Zeilberger has often expressed similar opinions.</p> <p>Wikipedia also says that Troelstra said in 1988 that there were no satisfactory foundations for ultrafinitism. Is this still true? Even if so, are there any aspects of ultrafinitism that you can get your hands on coming from a purely classical perspective?</p> <p>Edit: Neel Krishnaswami in his answer gave a link to a paper by Vladimir Sazonov (<a href="http://www.csc.liv.ac.uk/~sazonov/papers/lcc.ps" rel="nofollow">non-Springer link</a>) that seems to go a ways towards giving a formal foundation to ultrafinitism. </p> <p>First, Sazonov references a result of Parikh's which says that Peano Arithmetic can be consistently extended with a set variable $F$ and axioms $0\in F$, $1\in F$, $F$ is closed under $+$ and $\times$, and $N\notin F$, where $N$ is an exponential tower of $2^{1000}$ twos.</p> <p>Then, he gives his own theory, wherein there is no cut rule and proofs that are too long are disallowed, and shows that the axiom $\forall x\ \log \log x &lt; 10$ is consistent.</p> http://mathoverflow.net/questions/44208/is-there-any-formal-foundation-to-ultrafinitism/44232#44232 Comment by Michael O'Connor Michael O'Connor 2010-10-30T17:53:06Z 2010-10-30T17:53:06Z Thanks! That Sazonov paper (which is available not through Springer here: <a href="http://www.csc.liv.ac.uk/~sazonov/papers/lcc.ps" rel="nofollow">csc.liv.ac.uk/~sazonov/papers/lcc.ps</a>) seems quite interesting. It makes me think the correct answer to my original question might be &quot;yes&quot;! http://mathoverflow.net/questions/44208/is-there-any-formal-foundation-to-ultrafinitism Comment by Michael O'Connor Michael O'Connor 2010-10-30T17:20:13Z 2010-10-30T17:20:13Z Thanks, you guys! http://mathoverflow.net/questions/44208/is-there-any-formal-foundation-to-ultrafinitism Comment by Michael O'Connor Michael O'Connor 2010-10-30T03:26:55Z 2010-10-30T03:26:55Z Thanks, I'll take a look! I'd looked a bit at his version of nonstandard analysis a while ago, but I didn't read any of his ultrafinitist writings.