Pretend you’re talking to a 5-year old kid who doesn’t know what infinity means. If you say “we have an axiom of infinity”, the kid will look at you stupid. If you say, “we can align the set with a subset of itself”, the kid will look at you stupid. If you say, “the infinite is the unbounded, unlimited, unending” then the kid will say “Ooooh!” You cannot do any of these acrobatics (axioms, bijections) until you make it clear what infinity means.
There is no way I could perform a bijection of a set with a subset without already understanding the set is unending. So I would need to know what infinity means before I could understand the definition.
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?
I accept that planets revolve around the sun not because of authority, but because it’s the most sensible scenario.
If you take a step back and view your position with objectivity you will agree that the burden of proof is on you.
The infinite is something that cannot be observed (much like god) and there is no evidence for (much like god), yet I am required to prove that it doesn’t exist? There is also no evidence of a teapot orbiting the earth, so do I have to prove that doesn’t exist too?
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.
What’s the biggest number you can think of? Now make it bigger: square it, factorial, define new symbols to reduce the size and continue on and on until you run out of room on the forum to write that number down. Regardless what you devise, you will find a biggest number, but you won’t be able to do anything with it other than bask in its glory.
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.”
The issue isn’t whether the bijection is possible, but whether it defines infinity.
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?
I can integrate an area over a height to yield a volume without using infinity. I can add my grocery bill without infinity. What do you need infinity for?
What’s the limit of 1/x where x → 23+10^10^10^10^10^10^10^10^10^10? Zero right? If it’s not zero, then how far am I off? No machine that could ever exist will be capable of discerning the difference, so from the perspective of practicality, we don’t need infinity.
- 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.
Induction isn’t the same as deduction and if n exists, you cannot say with certainty that n+1 exists. However, if n+1 exists, you can say with certainty that n exists (deduction).
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.
Yes I understand your point, but we still must have a priori understanding that the turing machine never ends. Once again, infinity must be understood before it can be defined with the turing machine.