Joel David Hamkins published a paper where he analyzes the "math tea" argument, namely, the argument that some real numbers are undefinable. He constructed a countable model of set theory where all sets are definable without parameters, and this apparently shows the flaw with the math tea argument. However, I still think some real numbers are undefinable. Here is why. Since we know that, "in the real universe", the real numbers are uncountable, they can't fit inside a countable model $M$, even if $M$ thinks the set of reals in that model are internally uncountable. So, basically, I think any countable model of set theory will never have all of the "real" real numbers, and this is why I think some real numbers are undefinable. I would like some clarification of this, and want to be corrected if I misunderstand something.
$\begingroup$
$\endgroup$
6
-
$\begingroup$ Maybe relevant: mathoverflow.net/questions/354201/… $\endgroup$– Qiaochu YuanCommented Jan 13, 2021 at 21:11
-
$\begingroup$ Definitely relevant: karagila.org/2015/name-that-number $\endgroup$– Asaf Karagila ♦Commented Jan 13, 2021 at 21:17
-
4$\begingroup$ The problem is that, if you are assuming a true ambient universe where the "real" real numbers live, in that universe, you cannot even state the math-tea argument due to Tarski undefinability. $\endgroup$– BurakCommented Jan 13, 2021 at 22:42
-
$\begingroup$ Also relevant: math.stackexchange.com/a/3954453/28111. $\endgroup$– Noah SchweberCommented Jan 13, 2021 at 23:33
-
$\begingroup$ @AsafKaragila In both this and in Hamkins' paper, I don't understand why the simple requirement of needing to use a finite number of symbols is immediately abandoned. The argument seems to me to be a kind of "in principle" argument, ignoring the fact that it is not in fact possible to uniquely specify every real number. It strikes me as misinterpreting the math tea argument. $\endgroup$– EraCommented Oct 10, 2021 at 21:50
|
Show 1 more comment