Timeline for Independence of the countable axiom of choice
Current License: CC BY-SA 3.0
8 events
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| Mar 15, 2015 at 7:46 | comment | added | Asaf Karagila♦ | @Andreas: Yes, that is a possibility. (I am familiar with the basics of Eric Hall's dissertation, by the way.) | |
| Mar 15, 2015 at 4:30 | comment | added | Andreas Blass | In the present situation, you can get from an urelement proof (the basic Fraenkel model) to a pure-set proof more easily than by the Jech-Sochor construction. Adjoin to the basic Fraenkel model an $A$-indexed family of Cohen reals (force with Fin($A\times\omega$,2)), and then take the pure part of the resulting model. You get the basic Cohen model, which, of course, violates countable choice. | |
| Mar 14, 2015 at 21:16 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
added 43 characters in body
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| Mar 14, 2015 at 20:54 | comment | added | Asaf Karagila♦ | And I don't like them exactly because the approach is "Shove it all into one cardinal", rather than "reiterate the construction carefully, and you can't lose!" :-) | |
| Mar 14, 2015 at 20:54 | comment | added | Noah Schweber | That's a very good point - I like the embedding theorems partly because they lead to a nice combinatorial question, "What's the lowest rank where we can shove this construction in?" | |
| Mar 14, 2015 at 20:53 | comment | added | Asaf Karagila♦ | Noah, it is true, and it didn't end there. I prefer to look at the meta-meta-arguments, rather than embedding theorems. Meaning under what circumstances we can "essentially repeat" the same construction with forcing and obtain the same results in terms of truth values of the statements we wish to investigate; and thus discard all the use of atoms from the get go. | |
| Mar 14, 2015 at 20:51 | comment | added | Noah Schweber | To elaborate on (2), the issue of how to transfer those core ideas from the context of urelements to genuine ZF results was seriously investigated by Jech and Sochor, and later by Pincus, who developed powerful machinery to say: "Oh hey, we can do this thing with urelements, so we must be able to do it without them." | |
| Mar 14, 2015 at 20:44 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |