Bourbaki and foundations

Timothy Y. Chow tchow at
Mon May 16 19:28:55 EDT 2022

Monroe Eskew wrote:

> I don't see how concerns about the "true nature" of mathematical objects 
> leads one to assertions that set theory is "full of theoretical holes."
> What are the holes? I can only think of incompleteness as the thing 
> being referred to here, which is of course a very naive reason to prefer 
> an alternative foundation.

Aczel was writing for a popular audience, so I don't think that his words 
can be taken too literally.  I'm basing my interpretation on the kinds of 
criticisms of set theory that I have heard over the years, and assuming 
that Aczel heard some of the same criticisms.  Other than in one notorious 
lecture by Voevodsky, I don't recall incompleteness being touted as a 
reason to reject "traditional foundations" in favor of "new foundations." 
On the other hand, I have certainly heard many complaints that set theory 
is ill-suited for this purpose or that purpose (generally having to do 
with the perceived mismatch between how mathematics is encoded using sets 
and how mathematicians actually think about or work with mathematical 
entities), and it is not hard to imagine that a writer for a popular 
audience might come up with the turn of phrase "full of theoretical holes" 
to convey the gist of such complaints.

Sadly, Aczel died in 2015, so we'll probably never be able to find out 
exactly what he meant by that phrase.


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