Bourbaki and foundations

[Timothy Y. Chow]

On Mon, 9 May 2022, JOSEPH SHIPMAN wrote:
> So far I have not seen anyone give an explanation that makes sense of 
> what is WRONG with set theory.
[...]
> WHO is forcing anyone to use set theory in a way that requires any 
> significant and unnecessary effort?

To be clear, I don't think there's anything "wrong with set theory" in any 
absolute sense.  When Aczel speaks of "myriad problems and inadequacies" 
and a "flawed system," I think he's overstating the case for the sake of 
telling a good story.

On the other hand, I think I understand what's behind the complaints.  If 
you are a practitioner of some subfield of mathematics, then one thing you 
typically want from a "foundation" is an axiomatization that accurately 
captures the fundamental nature of the structures you are studying. 
Faced with various logically equivalent alternatives, you are going to 
pick the one that you feel best reflects the "essence" of the 
subject---without any artificial or extraneous features---and that lets 
you develop the basic theory with ease and elegance.  From this point of 
view, what's "wrong" with sets is that defining objects in terms of sets 
often comes across as an artificial encoding procedure.

Here's an analogy that may help.  If I'm studying sorting algorithms, I 
will probably want to write my algorithms in some high-level programming 
language.  I don't want to have to program a Turing machine.  What's 
"wrong" with the Turing machine is that the process of encoding an 
algorithm for a Turing machine comes with a lot of extraneous baggage that 
obscures the essential differences between the sorting algorithms.

I don't think that any of the critics of set theory are saying that you 
*can't* use sets as a foundation, nor are they denying that set theory is 
well suited for certain tasks.  What they're objecting to is the 
suggestion that the only viable way---or at least the uniquely best 
way---to develop a foundation is by means of set theory, even if what you 
are seeking is Essential Guidance and not Risk Assessment or Generous 
Arena.

Tim