> 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.
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(?).
>In an infinite universe everything is in some sense the same, interchangeable, meaningless.
That something would be "meaningless" if it existed is not a valid or true argument against its existence.
>Physicists are naturally fairly conservative people when it comes to belief. They require evidence and experimentation before they say they believe something - so you can see why the theory of an infinite universe has not gained much support.
This is another one of those Christian apologists trying to use arguments against "actual infinity" to argue that the universe is finite and therefore had a beginning and therefore had a creator. You know, William Lane Craig-esque mental gymnastics. You know, like the Kalam Cosmological Argument or something.
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[ 3.0 ms ] story [ 22.6 ms ] threadThis 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.
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.
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(?).
That something would be "meaningless" if it existed is not a valid or true argument against its existence.
>Physicists are naturally fairly conservative people when it comes to belief. They require evidence and experimentation before they say they believe something - so you can see why the theory of an infinite universe has not gained much support.
Actually, it has quite a bit of support. The universe appears flat, which tends to connotate that it would also be infinite: https://books.google.com/books?id=KhTJZG-U3ssC&pg=PA161#v=on...
This is another one of those Christian apologists trying to use arguments against "actual infinity" to argue that the universe is finite and therefore had a beginning and therefore had a creator. You know, William Lane Craig-esque mental gymnastics. You know, like the Kalam Cosmological Argument or something.