Timeline for Cardinalities larger than the continuum in areas besides set theory
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 20, 2021 at 11:03 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Nov 4, 2010 at 3:05 | comment | added | Terry Tao | Hmm, that was a subtlety I was not aware of, though in retrospect it is clear that some weak form of choice is still used in the above cardinality argument. OK, I stand corrected! | |
| Nov 4, 2010 at 2:09 | comment | added | Andrés E. Caicedo | "(This issue arose at another MO question a few months back.)" I found it! mathoverflow.net/questions/32720/… | |
| Nov 3, 2010 at 23:20 | comment | added | Joel David Hamkins | Terry, you still need some AC (in the form of DC) to exhibit non-Borel sets of reals, because it is consistent with ZF that the reals are a countable union of countable sets, in which case every set of reals is Borel. (This issue arose at another MO question a few months back.) | |
| Nov 3, 2010 at 21:43 | comment | added | Hany | For the example of measurability I would say it is still within the realm of set theory (descriptive set theory). I think there are results on infinite dimensional topology that require a space of very large cardinality, but the ideas and techniques are borrowed from model theory. Actually it seems that whenever we study questions of infinite cardinals we require tools from model theory or set theory. | |
| Nov 3, 2010 at 18:45 | comment | added | Terry Tao | The axiom of choice is needed to exhibit sets that are not Lebesgue measurable. However, it is not required in order to exhibit sets that are Lebesgue measurable but not Borel measurable. (An explicit example of such a set is given for instance at en.wikipedia.org/wiki/Borel_set ). | |
| Nov 3, 2010 at 18:44 | history | edited | Terry Tao | CC BY-SA 2.5 |
added 106 characters in body
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| Nov 3, 2010 at 18:37 | comment | added | muad | I thought we needed axiom of choice to show that there are non-measurable sets? (unless the axiom was used in counting the size of the algebras and I missed it) | |
| Nov 3, 2010 at 18:30 | history | edited | Terry Tao | CC BY-SA 2.5 |
added 1468 characters in body; edited body; added 6 characters in body; added 2 characters in body
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| Nov 3, 2010 at 18:18 | history | answered | Terry Tao | CC BY-SA 2.5 |