most recent 30 from http://mathoverflow.net 2013-05-18T12:45:32Z http://mathoverflow.net/feeds/user/23660 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96510/have-we-ever-lost-any-mathematics/96804#96804 none 2012-05-12T22:37:49Z 2012-05-12T22:37:49Z

<p>I don't know if this is an example of what you're asking. In mathematical logic, the Hilbert Program of the 1920's intended to come up with a finitary consistency proof and a decision procedure for analysis and set theory. Many luminaries including Hilbert himself, Bernays, Ackermann, von Neumann, etc. gathered in Göttingen for this purpose. Ackermann in 1925 published a consistency proof for analysis (that turned out to be incorrect) and many other promising results emerged. Then in 1931, Gödel's incompleteness theorem shut the whole thing down. Some valid theorems came out of it, but the program as a whole had to be (in some interpretations) completely abandoned.</p> <p><a href="http://en.wikipedia.org/wiki/Hilbert_program" rel="nofollow">http://en.wikipedia.org/wiki/Hilbert_program</a></p>

http://mathoverflow.net/questions/96661/the-richardson-theorem-and-the-base-identities-problem/96801#96801 none 2012-05-12T21:45:19Z 2012-05-12T21:45:19Z

<p>Sergei, this is a reply to your comment asking about enumerating formulas in $\mathcal R$. Sorry to post it as a separate answer but I no longer have the browser cookie to post it as a followup comment.</p> <p>You don't need a particular standardized enumeration, but just some computable mapping between formulas and natural numbers so that each formula gets a unique number. Such a numbering scheme is traditionally called a "Gödel numbering" and the numbers are called "Gödel numbers" because the idea was (I think) introduced in Gödel's landmark paper (1931) about the incompleteness theorem.</p> <p>A simple Gödel numbering scheme (similar to the one Gödel used) is like this: say the formulas are written in an "alphabet" whose "letters" are $\{\sigma_1,\sigma_2,\ldots\}$. Treat those as natural numbers the obvious way (i.e. $\sigma_k\mapsto k$). So a formula F might be written as $(F_1,F_2,\ldots F_n)$ where the $F_i$ are natural numbers. Then let</p> <p>$$N_F=2^{F_1}\cdot 3^{F_2} \cdot 5^{F_3} \cdots p_n^{F_n}$$ </p> <p>where $p_i$ is the $i$'th prime number. That is the Gödel number for F (under this particular scheme). It's pretty easy to see how to convert a formula to a number and back. Some numbers won't correspond to valid formulas so treat them as identically zero, for example.</p> <p>Maybe you should read an introductory book on logic, if you want more clarity about this stuff. There are some other threads suggesting them.</p>