Timeline for Defining $\mathbb{Z}$ in $\mathbb{Q}$
Current License: CC BY-SA 3.0
14 events
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| Aug 19, 2020 at 18:58 | history | edited | user44143 |
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| Nov 10, 2016 at 13:16 | comment | added | Joel David Hamkins | For example, the integers are not definable in the real field, since the latter has a decidable theory. | |
| Nov 10, 2016 at 13:01 | comment | added | Todd Trimble | @MikhailKatz "then express natural numbers as those that are sums thereof" -- that's not a first-order characterization. | |
| Nov 10, 2016 at 12:59 | comment | added | Mikhail Katz | Can someone comment on the reason these results are considered difficult, and were published in the Annals? Naively speaking, apparently using the ring structure it should be easy to characterize the multiplicative unit, then express natural numbers as those that are sums thereof, and using the ring structure introduce the negative integers as well. Is the difficulty getting so few quantifiers? | |
| Nov 10, 2016 at 12:51 | history | edited | GH from MO |
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| Nov 10, 2016 at 12:41 | history | edited | Todd Trimble | CC BY-SA 3.0 |
specified the language
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| Nov 10, 2016 at 11:58 | comment | added | HeinrichD | Which language does this refer to? The language of ring theory? This should be added to the question, right? | |
| Nov 10, 2016 at 11:37 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
Updated links and gave titles of papers
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| Nov 10, 2016 at 11:26 | comment | added | David Roberts♦ | There is presumably some sort of categorical logic/topos-theoretic way to interpret what this says about integers (for instance, preservation by certain functors). And, I'm sure, classical model theoretic versions of said results. | |
| Jan 28, 2012 at 8:24 | history | edited | user16974 | CC BY-SA 3.0 |
added 302 characters in body; added 2 characters in body; added 2 characters in body
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| Jan 28, 2012 at 0:42 | comment | added | Joel David Hamkins | Probably it should be mentioned in the question that the first known definition of $\mathbb{Z}$ in $\mathbb{Q}$, very surprising at the time, was the 1948 result of Julia Robinson. Poonen's impressive result should be seen as a refinement of Robinson's theorem, lowering the complexity of the definition. | |
| Jan 27, 2012 at 23:22 | comment | added | Zack Wolske | See this question: mathoverflow.net/questions/19840/… @Guillaume: Here is a link to Poonen's paper lifted directly from above www-math.mit.edu/~poonen/papers/ae.pdf | |
| Jan 27, 2012 at 23:01 | comment | added | Guillaume Brunerie | Do you have a reference for these results? (without knowing how this is proved, it will be difficult to give a geometric interpretation) | |
| Jan 27, 2012 at 21:17 | history | asked | user16974 | CC BY-SA 3.0 |