Timeline for Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Aug 12, 2013 at 20:59 | comment | added | GEdgar | Consistency of the first-order theory of the reals (a result of Tarski) is MUCH EASIER than consistency of set theory (which is not known). | |
| Aug 12, 2013 at 20:21 | comment | added | Brusko651 | @GEdgar But the real numbers contain the integers. As far as my understanding goes, the real numbers are only proved to be consistent under the assumption that set theory is consistent (and this is impossible to prove.) | |
| Aug 11, 2013 at 17:45 | comment | added | Nate Eldredge | But as Henning's answer points out, the fact that we can construct the complex numbers using ZFC means that if we find a contradiction in the theory of complex numbers, then ZFC was already inconsistent. | |
| Aug 10, 2013 at 13:24 | comment | added | Denis | Yes, I didn't go into these details because I assumed ZFC was the framework of all this. | |
| Aug 10, 2013 at 13:23 | comment | added | GEdgar | Gödel's result is only for sufficiently expressive systems. The (first-order) theory of the reals has, indeed, been proved consistent. That theory is far simpler than the theory of the integers... | |
| Aug 10, 2013 at 13:06 | history | answered | Denis | CC BY-SA 3.0 |