Bourbaki and foundations

[Timothy Y. Chow]

Monroe Eskew wrote:

>> On 09.05.2022, at 02:01, Michael Sheard <msheard at stlawu.edu> wrote:
>>
>> Earlier, Aczel had already said that Bourbaki's "greatest error" was 
>> "letting Grothendieck go and disagreeing with his vision for the future 
>> of mathematics" -- specifically, its unwillingness to revise its 
>> earlier volumes in order to replace set theory, a "discipline full of 
>> theoretical holes," with category theory, which "does not suffer from 
>> the inherent limitations of set theory," as the foundation of 
>> mathematics.
>
> It sounds like there is a belief that categorical foundations can avoid 
> G?delian incompleteness.  Is that a fair reading?

I don't think that is a fair reading.

More likely, an example of an (allegedly) "inherent limitation of set 
theory" would be the fact that in ZFC, everything is a set.  So if you 
firmly believe that (say) topological spaces and groups are fundamentally 
different types of mathematical objects, then you may chafe at the 
requirement to define both topological spaces and groups in terms of sets. 
You may feel that doing so creates the misleading impression that 
topological spaces and groups are fundamentally "made of" the same kind of 
stuff, just packaged slightly differently.

Tim