I already defined infinity several times. A set is infinite if it can be bijected with a proper subset of itself. That definition is due to Dedekind in the 1880s.
But nothing could be further from the truth. There are lots of bounded infinite sets. I gave the unit interval as one example. Another is the ordinal (\omega + 1). That’s the usual set of natural numbers 0, 1, 2, 3, … with an extra element at the end, like this: 0, 1, 2, 3, 4, …, (\omega). That’s an infinite set that contains a largest element. It’s bounded.
You keep insisting that infinite sets must be unbounded, but there are so many counterexamples that you’re simply wrong, absent a compelling argument. I have no problem with you holding different opinions than standard math. I do have a problem that you can’t intelligently defend your position, or frankly even articulate it.
A set is infinite if it may be placed into bijection with at least one of its proper subsets. Galileo noted this fact about the natural numbers and there are references in Indian mathematics hundreds of years earlier. en.wikipedia.org/wiki/Galileo%27s_paradox. This is not a new idea.
I see. So earlier when you wrote …
… you were just kidding? You have contradicted yourself. You reject infinity and accept infinity.
Easy. We know its cardinality is uncountable, (2^{\aleph_0}) in fact. And its powerset has cardinality (2^{2^{\aleph_0}}), which is strictly greater.
You are not the first person to have felt that way. If you wish to reject all of modern mathematics from 1874 onward, that’s perfectly ok. Back in those days many people rejected these ideas. Today they’re almost universally accepted. But I did say “almost.” There are finitists who reject the axiom of infinity. The only problem with that doctrine is that it’s difficult and presently impossible to found physical science! You can’t do relativity and quantum physics without infinitary math. Now that is in fact a genuine philosophical problem. I don’t think you appreciate it though. You wave your hands and say, “absurd,” but you haven’t proposed how to salvage physics; and you are decidedly out of step with modern math. Nothing wrong with that, but you have the burden of justifying your position. Everyone else accepts infinitary math because it is (1) interesting; and (2) useful. Whether it’s “true” in any meaningful sense, I cannot say.
So you see I am not disagreeing with you. I am only pointing out some of the subtle issues with your position that you may not have considered.
I make no claim that any mathematical objects “exist.” Existence is a technical term. If we can prove something exists by our axioms, it has mathematical existence. I make no further ontological claims. You’re arguing with a strawman.
Do I claim the unit interval of real numbers exists in the physical universe? Of course not. Do I claim it exists in some sort of abstract Platonic sense? No not even that!
I only claim it has mathematical existence. I make the same claim for the number 3. That has mathematical existence. I have no idea if it has Platonic existence. You’re fighting a strawman argument that you’ve made up yourself.
No I make no claims of existence at all. Only of mathematical existence within infinitary set theory. That’s a very limited claim and it is all I claim. Just as when I play chess, I don’t claim the knight “really” moves that way. It only moves that way as long as I accept the rules of the game.
Mathematical infinity only works if we accept the rules of the game of set theory. But this game is (1) interesting; and (2) useful. So we play the game.
I believe this ontological point is the heart of our disagreement. I make no claims whatever for the “truth” or “existence” of mathematics. I only claim interestingness, usefulness, and the agreement of almost all modern mathematicians on these matters.
I make no claim about truth or Platonic or physical existence.
No it’s not. I have no “belief.” I only enjoy playing the game. If I accept the rules of chess, I may move my knight in such-and-so a manner when it’s my turn to move. If I accept the rules of set theory, I may prove various facts about infinite sets.
That’s all I’m claiming. You are the one fighting against a position I don’t hold.
Without infinitary set theory you lose quantum physics, which is based on a mathematical discipline called functional analysis. You simply lose QM. Relativity too. A century of physics, gone.
Ah. Well calculus can not be placed on a logically sound basis without infinitary set theory. That was the great advance of the 1880’s, what’s called in math history the arithmetization of analysis.
You don’t need set theory to do freshman calculus. You do need it to do higher mathematical analysis. That was the lesson of the 1800’s and 1900’s.
You are confusing the terminal point of your own education with the terminal point of human knowledge.
I have no problem with that. But why are you on a philosophy discussion forum talking about things you don’t understand?
From an engineering perspective? You’re several disciplines removed. From engineering to applied physics to theoretical physics to mathematics to philosophy.
You can not hope to elucidate the infinite from an engineering perspective.
But no. You could write a play about pink elephants and if your play was entertaining, people would enjoy it. And it would be useful in the sense that it brings joy to children. Again these two criteria: Interestingness and usefulness.
I appeal to the interestingness and usefulness of math. I make no claims about truth or existence.
Posting that hoary old quote from Gauss is the last refuge of someone with no argument to make. A lot has happened since the 1820’s when Gauss did his work.
Likewise Kronecker. Kronecker was wrong, and Hilbert was right when Hilbert said, “No one shall expel us from Cantor’s paradise.” And quoting historical figures out of context really doesn’t make you look clever. It makes you look like someone who can surf Wiki but hasn’t got any background or knowledge of the historical developments.
You don’t even know what the word bounded means. Any infinite cardinal is bounded by that of its powerset.
LOLOLOLOL. I stopped reading here. Wildberger’s ideas on infinity are universally regarded as cranky.
All the best. I get that you like to surf Wiki. You would be better off learning some math or philosophy.