Timeline for Can something infinite be absolute?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 6, 2011 at 14:11 | comment | added | Niel de Beaudrap | I don't see how this is an acceptable answer. The constant π is not "undefined as a ratio of integers"; it is simply not definable (nor expressible) as a ratio of integers, nor as the root of any polynomial over the integers. This does not make it undefined! Similarly, 1/3 is not definable or expressible as a finite decimal expansion: does that make it undefined? Furthermore, just because one must occasionally explore alternative axiomatic systems to consider new ideas, does not make the new ideas "undefined" or "less well defined". | |
| Jun 18, 2011 at 20:38 | comment | added | Dan Brumleve | New axioms are needed to justify new ideas (e.g. negative numbers or inaccessible cardinals) and the acceptance of new axioms requires a widely-held belief that they won't lead to a contradiction. e.g. the Axiom of Choice has passed this belief threshhold. The Riemann Hypothesis has not, although it is believable enough that there are interesting theorems contingent on it and some have applications in cryptography. So I think "defined" depends on language (and it is not as simple as having a distinguished name), and "existence" depends on beliefs. | |
| Jun 17, 2011 at 2:48 | comment | added | Mitch | Forgetting what 'absolute' is intended to mean...what do you mean by 'undefined' and 'exist'? As inscrutable as 'inaccessible cardinals' is to most people (even most mathematicians), the can be proven to exist just as easily as, say, negative numbers. | |
| Jun 16, 2011 at 19:00 | vote | accept | RolandiXor | ||
| Jun 16, 2011 at 5:40 | history | answered | Dan Brumleve | CC BY-SA 3.0 |