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Comment by alex_young | Hacker News Reader

And no discussion of Zeno? Pish.

The idea that nothing is demonstrative of infinity is clearly incorrect.

Take the screen you're reading this on. One pixel is composed of a bunch of different atoms, and once you get down to one of them, that atom subdivides into a bunch of subatomic particles, some of which even have mass. Let's take one of those for argument's sake. Split that, and you get some quarks.

Now let's imagine that's the smallest you can go. We can still talk about half of a down quark, or half of that, etc. Say, uh, infinitely so. There you go, everything is infinite. That wasn't so hard was it?

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The paradoxes of Zeno are caused by his lack of understanding of the symmetry between zero and infinity. It is also possible that he actually understood more than is apparent from his paradoxes, but those were intended only to troll the other philosophers.

Zeno understood things like zero multiplied by a number being zero and a number multiplied by infinity being infinity, but he did not understood that neither of zero and infinity is stronger than the other, so that the product of zero and infinity may be any finite number, i.e. the limit of a sequence of products where one factor decreases towards zero and the other increases indefinitely can be any number.

While Zeno either ignored or faked ignorance about the existence of limits of infinite sequences, other later Ancient Greek mathematicians, like Eudoxus and Archimedes, computed several limits, so they had an intuitive understanding of their behavior, even if they did not have a comprehensive theory.

You can't split a quark, partial quarks doesn't exist. In fact, singular quarks can't exist, if you try to pull quark out of nucleus, it produces another quark to pair with. Quarks can be destroyed in particle accelerators collisions but those aren't components.

Also, all of the components of an atom, electrons and nucleus, have mass.

top quarks exist as lone quarks and decay before they can pair

Huh? I thought color confinement prevented this?

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).

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.

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

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.

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)

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.

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.

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.

> 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_...

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

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