Timeline for Consistency strength needed for applied mathematics
Current License: CC BY-SA 2.5
6 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 10, 2011 at 3:23 | comment | added | Daniel Mehkeri | @Marcos: To your last question, see en.wikipedia.org/wiki/Ordinal_analysis for precise strengths of these five systems, and others. The first two theories, $RCA_0$ and $WKL_0$, have the same strength, namely $\omega^\omega$, and then $ACA_0$ is somewhat higher at $\epsilon_0$. And like Carl says it's unlikely you'll find anything physically applicable going beyond that. | |
| Feb 9, 2011 at 23:09 | comment | added | Carl Mummert | It seems unlikely to me that what people usually call "applied mathematics" will require more than ACAo. The proof theoretic ordinal of that theory is the same as Peano arithmetic, and so is not particularly "large" in the realm of proof theoretic ordinals. | |
| Feb 9, 2011 at 9:59 | comment | added | Tom Ellis | The Wikipedia page says $ACA_0$ is necessary for some results in analysis including Bolzano-Weierstrass and Arzela-Ascoli. Can you really do applied mathematics without these? | |
| Feb 8, 2011 at 23:24 | comment | added | Marcos Cramer | Thanks for the reference to reverse mathematics. From a first glance at the theorems listed for each theory in the Wikipedia article, it is obvious that WKL0 is needed for applied mathematics, but for the stronger systems this is at least not obvious at first sight. So I could now reframe my question as follows: Are any theorems from the stronger Reverse Mathematics theories needed in applied mathematics? Is there any need to go beyond these five systems to something as strong as ZFC? Do the five theories of reverse mathematics have different consistency strengths? | |
| Feb 8, 2011 at 22:30 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |