Can Infinity Truly Be Considered a Number?

[CrankFan]

Normal arithmetic isn't provably consistent. Of course it isn't known to be inconsistent, and we don't have any reason (right now) to believe that it isn't consistent. But we can't prove it.

Right.

Perhaps you misinterpreted what I was saying: I meant "not provably consistent" but it kinds of sounds like I'm saying "not consistent".

I did misunderstand your meaning; but now it's clear to me.

 

[JonF]

Matt I am well aware of what mathematicians think of '1' but it does not apply in the real world -- it's quite impossible to confirm that any formed identity is the same as the original -- except by assumption as in maths. .

Just because we can not measure the difference between two object in a perfectly accurate way it doesn’t mean that two objects can not be similar. It simply means in practicality we can not make (or find) similar objects. All of this is kinda moot though, because if you needed use mathematic for some application to the level of accuracy you describe, you would include a rang of error in your calculation. Relating to your example with sticks: You may call some arbitrary amount of “stick mass” 1. To keep your data accurate in measurement error the next time you measure a stick don’t call that value 1, call it constant that is c which satisfy: .999<c<1.001. You may not be able to find exactly 1 but you sure can find a value it is in-between.

Maths can be reduced to grouping of related "identical" quantities per addition multiplication etc. etc. the real world does not allow such simple occurences.

I beg to differ. That is exactly what the real world allows. In the real world these simple occurrences just interact with each other and there are a infinite amount of them.

it's quite impossible to confirm that any formed identity is the same as the original … The mathematical identity '1' is a concept just as '0' and the rules and the results of maths follow

This is a silly argument, that can be said of anything describe in symbolism, which is just about everything. The thing is with math its rules are consistent with reality.

Also you have a small point with 1 (but see my previous response to one of your quotes) but you got no ground with zero. In reality comparing two “ones” may require an inequality but you can compare two zero’s with no such tricks.

 

[Philocrat]

The article is merely confirming the existence of infinity in a non-existent way. In fact, the clearer picture is that infinity is quantitatively finite. That is, infinite and finite quantities are mathematically interchangeable. I personally think that there is no need to formalize this relation as everything is describable in both ways.

 

[quartodeciman]

It is unfortunate that we still tend to speak of "infinity" as a number. Suppose we spoke of its contrary as "finity". Then we would get stuff like:

finity + finity = finity
finity - finity = finity
finity*finity = finity

. Phooey on that! With discrimination between +ve infinity and -ve infinity, the situation only gets worse:

Is (+ve finity) + (-ve finity) equal to -ve finity, 0 or +ve finity?

Instead of infinities, I would prefer something like "transfinite numbers" as a class, but very individual and well-discriminated. Maybe there is even a better name (one lecturer at MIT tried "macho numbers", versus "wimps").

Alas, it is hard to undo centuries of habit!

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A comment on non-standard numbers: they are great for providing proofs of theorems, returning at the end to the original domain with only ordinary finite numbers. But trying to use non-standard numbers to produce results in an extended domain as the final result leaves us with a problem: there are pairs of non-standard numbers for which we can't tell their order:
In other words, we can't tell for some non-standard numbers x and y whether x < y, x=y or x > y. That is OK in the non-standard analysis usage, because those numbers don't occur in the final result. But if we want results in the extended system, then we are stuck.

But maybe someone has already found a constructible free hyperfilter or Lindstrom measure function and I don't know it. Happy thought!