Timeline for Are there natural examples of mathematical statements which follow from consistency statements?
Current License: CC BY-SA 2.5
11 events
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| Sep 5, 2017 at 23:16 | comment | added | Christopher King | "Thus, there are interesting consistent theories TT, such as the one I just gave, such that T+Con(T)T+Con(T) proves any statement at all!" +1 | |
| Jul 23, 2010 at 10:39 | comment | added | Joel David Hamkins | That is fine, and the Vitali statement is obviously close to a consistency statement---I had meant merely that most mathematicians can understand directly the question of whether or not we can make a Vitali construction without AC, without needing to express it as a consistency statement. Of course, you are asking for a statement provable in T+Con(T) but not in T, so any example will be close to a consistency statement. The diophantine examples, however, are just polynomials. | |
| Jul 23, 2010 at 0:28 | comment | added | Kaveh | @Joel: What I had in my mind when I asked the question was some theorem $\varphi$ that an ordinary mathematician (not a logician) would appreciate. I am not sure that "We can perform a Vitali construction without AC" satisfies it. By the way, out of curiosity, could you please explain how one would express it not as a consistency statement? I mean how can we say that one can not perform a Vitali construction other than stating that the negation of its consequence is consistent with the theory? | |
| Jul 21, 2010 at 23:46 | comment | added | Kaveh | Thank you Joel. The first half is really nice, but I have to think about it a little more. The second part refers to the theory since the polynomial from the MRDP theorem will depend on the encoding of the theory. This will make it metamathematical. | |
| Jul 20, 2010 at 2:44 | comment | added | Timothy Chow | This is a nice result that I was not aware of. However, Kaveh might disqualify it as a "metamathematical" statement. | |
| Jul 17, 2010 at 1:03 | comment | added | Sergei Tropanets | It is really great answer! | |
| Jul 16, 2010 at 2:35 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Jul 16, 2010 at 1:17 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Jul 16, 2010 at 0:45 | comment | added | Joel David Hamkins | My statement $\psi$ was "our inability to carry out a Vitali construction without using AC." While it is true that this statement can be thought of as a consistency statement, it needn't be. | |
| Jul 16, 2010 at 0:34 | comment | added | Andrés E. Caicedo | Joel, I believe the question is about $T+Con(T+\varphi)$ proving a statement $\psi$, not about $Con(T+\varphi)\leftrightarrow Con(T+\psi)$ being provable in $T$ (or in a weaker system). Your example is of the second kind. | |
| Jul 16, 2010 at 0:26 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |