
Infinity doesn't exist - firmgently
http://theorangeduck.com/page/infinity-doesnt-exist
======
Veedrac
> 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.

~~~
Recurecur
Yes, it is certainly confused thinking to state that if the universe is
infinite, every possible physical thing has to be inside it.

One can easily imagine a mostly empty infinite universe...

Personally, though, I think his description of that is more akin to the
multiverse, where there might be an infinity of universes with
aleph=infinity(?).

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

~~~
aisofteng
The square root of two doesn't exist any more than two does. The ideas of the
square root of two and of two itself, however, clearly do exist.

~~~
paulddraper
And so does infinity.

