Infinity doesn't exist

> This is ultimately why I believe infinity should not be an axiom of mathematics. It cannot be imagined - and it is not right to declare something exists which cannot be imaginable - not even in mathematics. If you say you believe in infinity, say you understand it, say you can manipulate it and do mathematics with it - it isn't true. It can't be imagined, it can't be realized, it can't be used in mathematics - only finite approximations can.

This just shows a really odd understanding of what maths is, and how it's used. Firstly, there are no fundamental axioms of mathematics, only particular axiomatic systems. You shouldn't "believe" or "disbelieve" in infinity; rather you choose to use a mathematical model where it exists or where it does not.

But, more than that, this idea that you should only use things you can "imagine" is fundamentally nonsensical. You can't imagine a "5" any more than you can imagine an "∞" or a "3+2i"; they are all just mathematical objects that happen to map to useful physical properties. A "5" is not a physical thing. It's not a thing at all outside of its mathematical formulation, and it certainly isn't the nature of having five of a thing.

Mathematics uses infinities when it's useful, and discards it when it's not. We don't admit infinite computations on a Turing Machine, but not admitting the naturals to have an infinite size just makes maths harder. What's the point of having discontinuities at arbitrary places?

Your argument against "Keep Adding One" again shows a misunderstanding of how maths works.

> So when people say that infinity exists because they can keep adding one, what they really mean is that infinity exists, given infinite time or given infinite space or given an infinite counting speed.

But the issue doesn't show up only in such cases. Consider the successor function defined over the naturals.

    succ(x) = x + 1

Only, if there is a largest element k, then succ(k) is not defined, and succ is now a partial function. Handling this is just busywork, and one would be silly to dismiss infinity from your axioms at this cost just for some strange idea of purity.

There's a whole bunch of other wrong stuff here, like some false claims about an infinite universe and misrepresentations of Russell's Paradox and divergent summations, but I don't want to get into too much of a rant.

Likewise, the square root of 2 doesn't exist. (Write it out. How many sheets of paper will it take?)