Timeline for Is non-existence of the hyperreals consistent with ZF?
Current License: CC BY-SA 3.0
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| Jan 15, 2019 at 7:28 | comment | added | Timothy | the same as the ones defined for the real numbers. Are you asking whether it's consistent with ZF that it's the case that no system satisfies the additional requirement because for every system that satisfies the other requirements for a hyperreal number system, there does not exist a way to pick one member from each equivalence class of infinitesimally close hyperreal numbers such that it's closed under addition and subtraction because the axiom of choice is false? | |
| Jan 15, 2019 at 7:24 | comment | added | Timothy | What are you trying to ask? On possibile definition of a hyperreal number system is a system satisfying all the requirements for a complete ordered field but the requirement of completeness and replaces that requirement with the two requirements that for any Dedekind cut where you can take a number in each part to be arbitrarily close, the cut has a boundary number position, and there exists a number that exceeds all natural numbers. Another possible definition is the same definition with the additional requirement that it includes the real numbers and its operations on the real numbers are | |
| Dec 23, 2014 at 3:11 | vote | accept | Arseniy Sheydvasser | ||
| Dec 8, 2014 at 18:06 | history | edited | Mikhail Katz |
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| Dec 5, 2014 at 9:42 | comment | added | Asaf Karagila♦ | @katz: Just because you added a comment about the philosophical implications of not assuming the axiom of choice, and just because you added that as an answer instead of a comment, doesn't mean that the question itself is about mathematical philosophy. This is not a question for "What are the philosophical arguments in favor of assuming a hyperreal field exists" or something similar. This is just a simple question, can we prove in $\sf ZF$ that a hyperreal field exists with the transfer principle? The answer to which has nothing to do with philosophy, or mathematical philosophy in particular. | |
| Dec 5, 2014 at 9:40 | history | edited | Asaf Karagila♦ |
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| Dec 5, 2014 at 9:33 | history | edited | Mikhail Katz |
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| Dec 5, 2014 at 9:33 | comment | added | Mikhail Katz | @BenCrowell, I saw a remark of this sort in a paper by Keisler but I don't recall seeing it in Robinson. Do you have a source for this? | |
| Dec 4, 2014 at 16:33 | answer | added | Mikhail Katz | timeline score: 5 | |
| Dec 4, 2014 at 16:27 | history | edited | Mikhail Katz |
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| Dec 4, 2014 at 2:45 | comment | added | user21349 | Abraham Robinson suggested that ZF and ZFC were in some sense constructed exactly so as to allow us to do analysis on the reals. From that point of view, it's not surprising that in ZF(C) the reals exist and are unique, whereas ZF doesn't make the hyperreals exist, and ZFC doesn't make the hyperreals unique. | |
| Dec 4, 2014 at 2:17 | answer | added | Joel David Hamkins | timeline score: 31 | |
| Dec 4, 2014 at 2:00 | review | First posts | |||
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| Dec 4, 2014 at 1:58 | history | asked | Arseniy Sheydvasser | CC BY-SA 3.0 |