[FOM] "Mathematician in the street" on AC

Daniel Méhkeri dmehkeri at yahoo.ca
Sun Aug 23 13:37:38 EDT 2009

> > (Indeed, in most cases in which the theorem about the countability
> > of the countable union of countable sets is applied in practice,
> > We have such X and F available, so we do not have to use AC).

> Exactly so.  Depriving the street mathematician of her witnesses is like 
> depriving a boxer of his fists.  (Hilbert didn't think to apply that 
> argument to AC, which he acknowledged as problematic, while applying it 
> to intuitionistic logic, where Goedel's translation shows that fists only 
> flatten theorems by erasing the distinction between P and ~~P, they do 
> not deprive the mathematician of any theorems.)  

Well, since you mentioned it, it works for arithmetic, but not set 
theory, which is what we are talking about. CZF is equiconsistent with 
a fragment of second-order arithmetic, but CZF + classical logic = ZF.

> Why should she care that foundationalists make things harder by 
> killing off her witnesses? You have to give her a situation she 
> cares about where she has a list of countable sets with no witnesses 
> to their countability.  Good luck with that, she's probably never 
> run across such a thing.

Well maybe not but what about using the countable union principle 
in the negative?

Is \aleph_1 regular? Basic counterexamples in topology use that. For 
that matter, could the real numbers be the union of a countable family 
of countable sets? 



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