[marcus_holmes]

I think you missed the point.

So, firstly, you have split the particle 5 times. That's not infinite times. You can split it more, so that would be 6 times. And more. Even if you could split it 1000 times, that's not infinity.

The standard argument for infinity is that "you can always add 1 to any number, so there must be an infinity of them", and the refutation is that no matter how many times you add 1 to a number, all you've done is create a larger number. You never reach the point of actual infinity, no matter how long you keep doing this. You need to have infinite time in order to create an infinity by adding 1 to each number, so you're starting with the axiom that infinity exists (because you need an infinite number of operations to actually create an infinity). If you don't start with that axiom, then you can never reach infinity by addition (or any operation).

[alex_young]

Time has nothing to do with it. There are an infinite number of ways to divide anything. You don’t need time to prove that. Whatever number you think of you can divide by a larger number.

[marcus_holmes]

Yes, and that gets you to another number. Not infinity. You need an infinity of operations to create an infinity.

[alex_young]

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.

[lostmsu]

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)

[alex_young]

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.

[lostmsu]

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.