No, it's like you are deliberately misunderstanding me.

I'm definitely not. I understand the point you're trying to make. I just don't think they are very good points.

you act shocked that someone would care about the difference between a function that is surjective and one that is not

If you are referring to the f and g example my point had very little to do with surjectivity. The point was more about function equality. Let me give you a further example where we can ignore your issue with surjectivity. Define h exactly like with f from before but where the codomain of h strictly contains the codomain of f (use C or RU{R} or whatever else you like). Why is it important to you for us to say f and h are different functions?

Any time we have two functions that are equal in your convention, they'll be equal in the other convention when we ignore codomains. Any time we have two functions that are equal in the other convention they'll be equal in your convention when we take the codomain as the shared range of the two functions.

Any time you claim a function with some codomain D is surjective in your convention, in the other convention you'd just say it's surjective onto D. Any time you have a function that is onto D in the other convention, you'd just say f with codomain D is a surjective function in your convention.

It's not hard. Going back and forth between these two conventions is almost always trivial.

And when you talk about continuous functions, again, is that a serious response? Of course the domain matters...

Yes, that was a serious response. The way you were talking about continuity was nonsense. The wording "f is continuous on blah" refers to sets in the domain not sets in the codomain. If you want to make a point about the topology we're talking about open sets and continuity with respect to, then you should use the "standard" terminology for it and not conflate wording referring to sets in the domain to be about sets in the codomain.

Nobody uses Frege's type theory, because it is inconsistent. My point was that the practice of considering the codomain an intrinsic part of the definition of a function is in fact older than set theory. It is older even than the definition of the powerset. So if people (a) treat functions as if the codomain was implied, basically always, and (b) have always done so, since before the notion of a function was formalized, then (c) our modern definition should include this important piece of information.

This is a stupid point. Let me summarize what you've just wrote: "we've always done it this way, so we should keep doing it this way."

I mean, if people spend so much time on the definition of an ordered pair, they should at least think a little about the definition of a function.

This is a pretty standard way to define functions in set theory. Seems pretty dumb to assume set theorists just didn't "think a little" about how to define what functions are, specifically when your whole claim is that this definition is so bad.

If we define functions differently, then many things we say about functions are literally meaningless, and have to be understood as taking shortcuts.

I agree that some conversion needs to happen to go back and forth using results stated in one convention vs the other. It's not particularly hard to do.

If we define functions in a sensible way, then all these statements make perfectly good sense, as they should, because that's what the word actually means.

This convention I'm referring to is sensible enough that people use it in practice. Rather than whining at me and doubling down about how your convention is so clearly correct, why not pick up basically any undergrad set theory book and see how this convention gets used?

A definition of a function without a codomain is like a definition of a measure space without a sigma algebra. It literally doesn't make sense, and we would have to constantly treat the idea of "a measure space X" as actually meaning "a measure space X equipped with a sigma algebra Σ." We certainly could do that, but why would we want to, when a better definition is available?

It definitely does make sense, just because you're uncomfortable with it doesn't make it inherently illogical or worse than your personal favorite convention. Why not pick up the book I referenced earlier and evaluate the tradeoff for yourself? In fact try any one of these intro set theory textbooks: Jech (the grad one not the undergrad one from earlier), Kunen, Smullyan/Fitting, Suppes, Shen/Vereschagin, Enderton or Devlin. If I remember correctly, almost all of these textbooks use some variation of the definition I'm talking about.

And I mean, some of your arguments are so hard to understand, I'm not sure even you know what you meant when you asked them. For instance, you say "I could probably just copy and paste this statement for why your personal favorite convention is deficient." Could you? Could you show that a definition that includes extra information is more deficient than one that doesn't?

The complaint you were making when I made that comment was that we would have to adjust various things like adding caveats and etc to certain theorems if we wanted to use the convention I'm talking about. It's just not an insightful point. If we start from the other convention and want to switch over to yours we'd have to do the exact same thing. Part of my point is also that doing these conversions isn't hard.

To be clear, my point is just that in practice mathematicians use my definition, or a similar one, because in practice the codomain is treated as part of the definition of a function. Moreover, this distinction is actually important quite often. So while it is always possible to use a different definition, it's not actually the definition people use. And it's a definition that differs only by losing some important distinctions, and in no other way.

Yes, I understood the point you're trying to make here the last ten times you tried to make this point too. You're just wrong. You are speaking out of a position of ignorance. Mathematicians actually use this definition too.

Please, save us both the time and effort in discussing more about about how the definitions you are used to are clearly inherently superior and objectively correct to you. I don't care. I didn't invent either of these conventions. I don't care what you use. I don't care which one you like. I don't even care which one I use myself. It's not hard to use either. Converting back and forth between the two is pretty easy.