Timeline for Axiom of dependent choice (up to $\omega_1$) and group rank
Current License: CC BY-SA 3.0
8 events
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| Aug 6, 2013 at 9:32 | vote | accept | user38200 | ||
| Aug 6, 2013 at 5:49 | comment | added | Asaf Karagila♦ | Joel, that is what I meant when I said weaker. I don't know th answer off-hand, nor I believe it is known generally. | |
| Aug 6, 2013 at 4:55 | comment | added | Joel David Hamkins | But perhaps if such a group has a minimal size set of generators, then it has a basis? | |
| Aug 6, 2013 at 4:52 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
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| Aug 6, 2013 at 4:52 | comment | added | Asaf Karagila♦ | @Joel: Ah. Silly me. I generally recall (although not in 100%) that free implying the existence of a basis is itself equivalent to the axiom of choice. This seems weaker, but probably unprovable as well. I'll remove it from the answer for now. | |
| Aug 6, 2013 at 4:47 | comment | added | Joel David Hamkins | Asaf, for the second question, you seem to be assuming that the free abelian groups are free over the set of generators that you happen to pick. But the OP defines "rank" not in terms of the size of such a basis, but in terms of the size of a smallest generating set. Can you prove in ZF that if a free abelian group is generated by set X, the it has a basis (over which it is free) of at most the same size? | |
| Aug 6, 2013 at 4:42 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
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| Aug 6, 2013 at 4:36 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |