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by alex_young 42 days ago
Create an infinity? What does that mean? Why would you need to do that?

Is there a limit to how many times something can be logically divided? If not, then there’s your infinity. It doesn’t require you to continue brute forcing it, just reason about it.
1 comments
lostmsu 42 days ago
Maybe? Can you prove there's no limit? The default proof by induction requires postulate of infinity. (this statement is potentially incorrect, but takes across the point)
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alex_young 42 days ago
Does half of something have a limit? Not by its definition. Same thing with addition or multiplication. All of these only work with some concept of infinity.

We could redefine "half" to mean "half of whatever you're talking about until you get to some arbitrary limit", but doing that to all of arithmetic is going to wind up in a very odd place.

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lostmsu 42 days ago
Half of something has a value, and that value is not infinity. You need to be more specific about how exactly do you get infinity from the fact that half of something has a value.
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drdeca 42 days ago
Not from “that half of something had a value”, but from “that half of any thing has a value”.

If you accept that every natural number has a successor which is a natural number, and no two natural numbers have the same successor, and that there’s no loops (e.g. by saying that there’s a total order on natural numbers and that any natural number is less than its successor), then there can’t be a finite collection which is all the natural numbers.

You could say “there’s no collection which has all the natural numbers”, which, ok, how do you want to talk about things true of all natural numbers then?

Formulating descriptions of physics without the axiom of infinity (or, without something to play the role of the real numbers) is super icky. You, in practice, can’t do any significant mathematical physics in an ultrafinitistic approach.

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lostmsu 42 days ago
> how do you want to talk about things true of all natural numbers then

There's an entire branch of math for that: https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...

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drdeca 40 days ago
I’m aware of constructive math. You still have the type of natural numbers in that?
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