Infinity Doesn't Exist

The article is internally inconsistent.

It's certainly true that you can choose not to have the axiom of infinity, and it's certainly true that most mathematics that does not directly invoke of use infinity is subsequently possible, and consistent with the usual mathematics.

However, the author having said that it's a choice, s/he then goes on to say ...

    So, although it is an axiom of mathematics, the belief in
    infinity is not a purely aesthetic choice. If you say you
    believe in infinity, say the 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.

    You cannot imagine infinity, use infinity, describe, or
    realized infinity. Not only does infinity not exist - it
    cannot exist - not even in your imagination.

The argument is that the author doesn't like some of the consequences, and so chooses to reject them. That's fair enough, although it feels to me a little like sticking your fingers in your ears and singing "La la la la ..." Calculus becomes really awkward without infinity, and personally, declaring that there's no such thing as a set that contains all natural numbers just seems unnatural and perverse.

But you can do that. What you can't do is have it both ways. You can't declare it to be a choice, and then declare that it's impossible.

Standard mathematics makes the choice to have the axiom of infinity, and part of standard mathematics is exploring the consequences of that choice. The author may choose to be a finitist[0] and that's fair enough. Just don't declare that the alternate choice is somehow wrong.

[0] https://en.wikipedia.org/wiki/Finitism