I don't have a particular interest in mathematics, but I have been introduced to some concepts and examples involving "infinity". I am told on good authority that it is not a number, real, rational, irrational, integer, whole or natural.
Despite this, paradoxes such as the Hilbert Hotel example essentially begin by saying "so imagine an infinite number of something interacting with another infinite number of something else".
Why is this tolerated when it isn't a number? Irrespective of the "its supposed to demonstrate the problem", it defies the sole axiom of its existence as a concept by quantifying it so. It isn't a number.
I'm more than happy have an explanation as to why I am wrong; like I say I am in no way a mathematician.
>>7791335
Those paradoxes do not invoke it as a number but in the context of sets. And infinite sets are rigorously defined and perfectly rational (in the sense that they make sense).
>>7791370
But surely the fact the a set contains within itself a non-finite condition, it is still applying such a concept in quantifiable terms.
If this "infinity" has no bearing on the set and how it is dealt with, why not use a finite condition instead?
>>7791405
>But surely the fact the a set contains within itself a non-finite condition, it is still applying such a concept in quantifiable terms.
We are still applying that but in terms of mathematics, that isn't breaking the rules. In the real world infinity may be out of our reach but in math you can play with infinity in the same way you play with the number one, for example.
>why not use a finite condition instead?
Because this would not be enough to explain mathematics.
The set of all natural numbers has infinitely many members. So do the integers, reals, rational, hyperreals, etc. And here it makes perfect sense.
The paradoxes you are talking about try to break the bounds of infinity and apply to the real world and they do so in order to prove a point. To show how logic and common sense immediately break down when you introduce infinity.
You have a full hotel with infinitely many rooms? Full implies that there is no room left but with infinity, that does not matter. That definition is made meaningless by infinity.
In other words, don't take the paradoxes as characteristics of infinity because infinity doesn't really exist in the real world.
>>7791405
>I am told on good authority that it is not a number, real, rational, irrational, integer, whole or natural.
That's wrong. Infinity is a cardinal number. In certain contexts this means that it cannot be used. But not in all contexts.
>If this "infinity" has no bearing on the set and how it is dealt with, why not use a finite condition instead?
Of course it has bearing on the set. An infinite set is very different from a finite set. The entire point of the "paradox" is to illustrate this.
>>7791370
>And infinite sets are rigorously defined and perfectly rational
to you
>>7791439
Thanks for taking the time with that, I think I get it now.
>"so imagine an infinite number of something interacting with another infinite number of something else"
If we were to say "so imagine an small number of something interacting with another small number of something else" the word small does not become a number.
>>7791484
An "infinite" number uses a descriptor which contradicts itself. A "small" number uses a descriptor which doesn't.
>>7791511
The word number is being used as a synonym for quantity. You could say 'unknown number' and this wouldn't imply the existence of some number that exists but has an undefined value, it simply states a property about a set of something.
>>7791335
was going to help you, but fuck you.
