Timeline for Can every nonempty set carry abelian group structure? [duplicate]
Current License: CC BY-SA 3.0
14 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 31, 2015 at 3:33 | comment | added | Boaz Tsaban | @AsafKaragila I believe that such questions (as do essentially all of mathematics outside set theory) assume AC implicitly. | |
| Mar 1, 2012 at 5:08 | history | edited | CommunityBot |
insert duplicate link
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| Mar 1, 2012 at 5:08 | history | closed |
Steven Gubkin Theo Johnson-Freyd Andreas Blass Emil Jeřábek Ryan Budney |
exact duplicate | |
| Feb 29, 2012 at 18:10 | comment | added | Emil Jeřábek | (Arturo’s and Steven’s examples are one and the same.) | |
| Feb 29, 2012 at 17:02 | comment | added | Asaf Karagila♦ | @Arturo: Indeed if you take $\bigoplus_X\mathbb Z$ then $X$ indeed embeds into it as you said. However without the axiom of choice it is possible that the group generated by this embedding has a strictly larger cardinality. | |
| Feb 29, 2012 at 16:56 | comment | added | Arturo Magidin | P.S. I believe that you only need the axiom of choice in order to prove that the resulting group is bijectable to $X$ (and thus use transfer of structure to endow $X$ with an abelian group structure); the fact that the group exists given a set $X$, and that you can embed $X$ into the group (via mapping $x\in X$ to the characteristic function of $\{x\}$), does not seem to me to require AC, but if I'm wrong I'm sure the experts on this will correct me. | |
| Feb 29, 2012 at 16:43 | comment | added | Arturo Magidin | You can take the direct sum of $|X|$ copies of $\mathbb{Z}$ (assuming the axiom of choice, at any rate). This is the set of all functions $f\colon X\to\mathbb{Z}$ of finite support, with pointwise addition. The cardinality is $|X|$. | |
| Feb 29, 2012 at 16:34 | vote | accept | Chris Heunen | ||
| Feb 29, 2012 at 16:29 | comment | added | Asaf Karagila♦ | You can read several answers on math.SE: math.stackexchange.com/q/105433/622 | |
| Feb 29, 2012 at 16:20 | answer | added | Joel David Hamkins | timeline score: 14 | |
| Feb 29, 2012 at 16:19 | comment | added | Steven Gubkin | The free abelian group on an infinite set X has the same cardinality as X | |
| Feb 29, 2012 at 16:16 | comment | added | Joel David Hamkins | See related question mathoverflow.net/questions/12973/…, concerning the use of the axiom of choice in imposing a group structure. | |
| Feb 29, 2012 at 16:12 | history | asked | Chris Heunen | CC BY-SA 3.0 |