Serendipper wrote:How did you make the determination that the set can be placed into bijection with the subset without having prior knowledge that the set was unending/unlimited/unbounded?

There is no way. You must know the set is unending before you can show the bijection.

I take this objection seriously. I've asked it of myself from time to time. I'll outline my thoughts.

* First, if we accept the axiom of infinity, which says that there's an infinite set, we can establish the bijection with no trouble. But the axiom of infinity bakes infinity into mathematics, so it's too strong for our purposes.

The next step back from the axiom of infinity is the Peano axioms. Here we have each of the natural numbers 0, 1, 2, 3, ..., but we do not have a completed "set" of them; where a set is defined as something that obeys the particular axioms you're assuming.

It was in this context that Galileo made his famous observation that the counting numbers may be placed into bijection with a proper subset of themselves, in particular the squares. That is not a "appeal to authority" as you put it, but rather a statement of historical record. 1638, in his final work,

Two New Sciences. You can find a copy online these days.

More generally if I assert that the planets revolve around the sun, I am not making an "appeal to authority" when I accept this universally agreed upon conclusion of the greatest scientific minds in history. "Oh yeah, Newton and Einstein, you're just appealing to authority." Is that the argument YOU are making here? That Galileo was some jerk and YOU know better?

If you take a step back and view your position with objectivity you will agree that the burden of proof is on you.

Before analyzing the case of Peano, let me first cover the third alternative: You deny that all of 0, 1, 2, 3, ... exist. You claim that at some point, there aren't any more. You deny not only infinite sets, but mathematical induction too.

That position is called

Ultrafinitism.

I completely agree with you that from an ultrafinitist position that it's meaningless to talk about a bijection. There's no map that inputs n and outputs 2n. At some point you put in a big n and it says, "Sorry Dave, I can't do that."

Ultrafinitism is a really interesting idea. There have been a couple of serious ultrafinitists, though many more adherents are cranks. That said, ultrafinitism is useless. Even if it's true it's useless. You can't do math with it and if you can't do math you can't do science and then we're back to living in caves and throwing rocks. If you reject mathematical induction you lose all of finite math, combinatorics, everything. First you threw out calculus, and now basic probability theory?

So I reject ultrafinitism for the same reason I reject solipsism. Even if it's right it's worthless. It's an essentially nihilistic philosophy.

* Ok. Back to Peano. We have 0, 1, 2, 3, ... and each one of them "eventually" exists. We have the law of induction; which says that if 0 exists; and if whenever n exists, n+1 exists; then all natural numbers exist. Of course "exist" just means mathematical existence.

Now what sense can we make of a notion such as the function, or mapping, denoted as f(n) = 2n, which inputs a natural number n, and outputs an even number 2n?

If we are in the realm of ZF set theory, this is no problem. Given the SET of natural numbers we can form another set that serves the role as a function; namely, the set of ordered pairs (n, 2n).

But without the axiom of infinity, we can't get that off the ground.

What, then, is a function?

I think that this is the core question you are asking. I have a strong answer.

* The map n -> 2n is a

computable function on the natural numbers. That is: There is a Turing machine that, when given the number n, eventually outputs the number 2n,

in a finite number of clearly defined steps.

Therefore even if someone rejects the axiom of infinity; if they are a constructivist who demands that all mathematical concepts and objects should be computable; then the bijection between the naturals and the evens exists.

Do you see my point? In order to define the map that sends n to 2n, I do not need any "a priori" unbounded or infinite sets or collections. All I need is a FINITE string of symbols that represents the operation of a Turing machine (or a Python or Java or Javascript program, same thing) that inputs the number n, and outputs the number 2n. And such a thing exists.

A computer program is a purely finitistic object. It's a finite string of symbols. A finite string of bits if you like. And there's a computer program that doubles any natural number.

So: I say to you:

If you are a constructivist, you must accept the bijection.

If you are an ultrafinitist, you are a nihilist. I can't defeat your argument with logic. Only with practicality. So in the end my position is based on pragmatics. Usefulness.

Ok thanks for reading I think there's a good anwer there, I hope you'll give some thought to what I said.

Serendipper wrote:I just copied the dictionary as a starting point, but it turned-out that particular definition cannot be a definition which is something I didn't know at the time.

Haha. I'll resist the temptation to point out that not only don't you seem to read what I write; you don't even seem to read what YOU write. I'll keep that in mind. No just kidding. I think you asked a very good question and I gave you my best answer.

** tl;dr: If you are an ultrafinitist, you are right. I can't defeat your idea with logic. Only with a claim to practicality. And by practicality I mean ALL of known math and science. My gosh, you even reject induction? That's a lot worse than just rejecting the axiom of infinity.

On the other hand if you understand that the map that sends n to 2n is expressible in a finite number of symbols; and is a

computable function in the sense of Turing; then that mapping exists; without any need to invoke boundless collections. Unless you are ready to throw out all of computer science too?

To me your argument is irrefutable; yet nihilistic. It denies everything.