Timeline for Uncountable family of infinite subsets with pairwise finite intersections
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 28, 2012 at 12:46 | comment | added | Andrej Bauer | Right, "bigger" does not make as much sense constructively as it does classically, at least not if injections or surjections are used for comparison of size. | |
| Feb 27, 2012 at 18:00 | comment | added | Aaron Meyerowitz | OK, so (from a constructive point of view) countable means having an enumeration and uncountable means having a proof that there could never be an enumeration. However it does not mean "bigger than countable". | |
| Feb 27, 2012 at 17:22 | comment | added | Andrej Bauer | No, I am not saying that from a constructive point of view countable is the same as recursively enumerable. There is one particular model of constructive mathematics, namely Russian constructivism or the effective topos, in which "countable" happens to coincide with "recursively enumerable". But that aside, yes it is consistent to assume that a subset of a countable set is not countable. For a striking example, see "Embedding the Baire space into natural numbers" at math.andrej.com/2011/12/06/… | |
| Feb 27, 2012 at 8:09 | comment | added | Aaron Meyerowitz | Are you saying that countable=recursively enumerable (from a constructive viewpoint)? So we could have a countable set $P$ (say all finite sequences of of characters from some alphabet) and yet have a subset which is not countable (say the sequences which code a valid algorithm outputting a binary sequence.) | |
| Feb 26, 2012 at 23:13 | history | answered | Andrej Bauer | CC BY-SA 3.0 |