Timeline for Did Brouwer evade uncountability?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 12, 2015 at 1:02 | vote | accept | Frode Alfson Bjørdal | ||
| Jan 12, 2015 at 0:59 | comment | added | Frode Alfson Bjørdal | (I have been adviced by lawyers that only those who have the copyright may make themselves available at en.bookfi.org/s, so that such books are quotable.) I will have some go at these things. | |
| Jan 12, 2015 at 0:58 | comment | added | Andreas Blass | My impression is that Brouwer would probably not have accepted exponentiation as we view it in, for example, topoi. His reals and his sequences of natural numbers would have been some sort of choice sequences. And "some sort" here does not refer only to my ignorance; I think Brouwer changed his mind on what is allowed in a choice sequence. Free choice sequences are pretty clear, but general choice sequences allow one to choose restrictions on future choices. Can one choose restrictions on future restrictions? I think Brouwer vacillated on that. | |
| Jan 12, 2015 at 0:54 | comment | added | Andrej Bauer | (I am not sure about making public links to books that look like they are copyrighted.) As I said, I am not an expert on the history. It seems to me you can answer your own question better than I can. | |
| Jan 12, 2015 at 0:52 | comment | added | Frode Alfson Bjørdal | The real numbers $\mathbb{R}$ would be denumerably $unfinished$ according to the Brouwerian jargon in his dissertation. Or so it seems from a quick glance on these things. | |
| Jan 12, 2015 at 0:51 | comment | added | Frode Alfson Bjørdal | Also Dirk van Dalen's first chapter confirms so much. | |
| Jan 12, 2015 at 0:48 | comment | added | Frode Alfson Bjørdal | Here is the complete book I linked to above. Chapter 2 seems to confirm that Brouwer indeed did attempt to avoid the uncountable. dropbox.com/s/qigvod7tx6dpe0w/… | |
| Jan 12, 2015 at 0:45 | comment | added | Andrej Bauer | I am pretty sure that Brouwer takes $\mathbb{R}$ as a totality, i.e., a space which exists as an object. After all, there is a lot of discussion on the nature of $\mathbb{R}$, how it cannot be decomposed into parts, there is talk about its points, etc. Countable choise suffices to show that $\mathbb{R}$ is not countable. But it is an open problem whether $\mathbb{R}$ is uncountable in pure intuitionistic logic. | |
| Jan 12, 2015 at 0:44 | comment | added | Andrej Bauer | "Rejecting higher infinities" can mean something subtler than "I don't believe there are any infinities past $\aleph_0$", especially since intuitionistically infinities cannot be shown linearly ordered and so "past $\aleph_0$" does not mean "everything but countable". There can be subsets of countable sets that are not countable. | |
| Jan 12, 2015 at 0:32 | comment | added | Frode Alfson Bjørdal | (I have encountered constructivists of the Martin Löf type who have this conviction that there is no set of the real numbers.) | |
| Jan 12, 2015 at 0:29 | comment | added | Frode Alfson Bjørdal | How do we e.g. account for the text on the top of page 26 here? May it be that Brouwer rejected such totalities as $\mathbb{N}^\mathbb{N}$? books.google.com.br/… | |
| Jan 12, 2015 at 0:22 | history | edited | Andrej Bauer | CC BY-SA 3.0 |
added 528 characters in body
|
| Jan 12, 2015 at 0:16 | history | edited | Andrej Bauer | CC BY-SA 3.0 |
added 528 characters in body
|
| Jan 12, 2015 at 0:05 | history | answered | Andrej Bauer | CC BY-SA 3.0 |