[umanwizard]

"the study of how to organize abstract collections of objects" is not really a great explanation of set theory. But it is true that the usual way (surely not the only way) to formalize mathematics is starting with set-theoretic axioms and then defining everything in terms of sets.

[k_bx]

"Usual", "most common by far" etc. are all great phrases, but not "all of mathematics", esp when we talk about math related to computer science

[umanwizard]

Math related to CS is typically formalized starting with set theory, just like other branches of math.

[A_D_E_P_T]

What's important to note is that this is just a matter of convention. An historical accident. It is by no means a law of nature that math need be formalized with ZFC or any other set theory derivative, and there are usually other options.

As a matter of fact, ZFC fits CS quite poorly.

In ZFC, everything is a set. The number 2 is a set. A function is a set of ordered pairs. An ordered pair is a set of sets.

In ZFC: It is a valid mathematical question to ask, "Is the number 3 an element of the number 5?" (In the standard definition of ordinals, the answer is yes).

In CS: This is a "type error." A programmer necessarily thinks of an integer as distinct from a string or a list. Asking if an integer is "inside" another integer is nonsense in the context of writing software.

For a computer scientist, Type Theory is a much more natural foundation than Set Theory. Type Theory enforces boundaries between different kinds of objects, just like a compiler does.

But, in any case, that ZFC is "typical" is an accident of history, and whether or not it's appropriate at all is debatable.

[jltsiren]

In CS, types are usually a higher level of abstraction built on top of more fundamental layers. If you choose to break the abstraction, you can definitely use an integer as a string, a list, or a function. The outcome is unlikely to be useful, unless your construct was designed with such hacks in mind.

When I did a PhD in theoretical computer science, type theory felt like one niche topic among many. It was certainly of interest to some subfield, but most people didn't find it particularly relevant to the kind of TCS they were doing.

[umanwizard]

Sure, but it fits the rest of mathematics "poorly" for exactly the same reasons. No working mathematician is thinking about 3 as an element of 5.

The reason ZFC is used isn't because it's a particularly pedagogical way of describing any branch of math (whether CS or otherwise), but because the axioms are elegantly minimal and parsimonious.

[k_bx]

Sorry but if you try to prove 2+2=4 in ZFC versus Type Theory (HoTT-style) – nothing looks elegant about ZFC afterwards.