Does infinity exist?

Main Question or Discussion Point

Is infinity merely a mental concept, or is it entirely meaningful and common?

Infinity does not exist in the context of a number system. Example: what would infinity minus 1 be? It's not infinite since no finite number plus one equals infinity. We see that the rules of arithmetic are violated, because we can't subtract infinity from both sides to conclude that -1 = 0, which isn't true.

But at the same time there are for example, infinitely many real numbers between 0 and 1 (there is actually even an infinite number of *types* of infinities; this kind is called Aleph-1).

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Although there is no real number called infinity, infinity exists as a type of limit. In mathematics you can have sequences that diverge and calculus limits that approach infinity. I thought there were only two types of infinity...countable and uncountable. Isn't there even a proof showing that there are no "degrees" of infinity...just the two types (countable and uncountable) with nothing in between?

You can have finite limits right?...like pi, or -2 or 7, so imagine that boundless limits approach something...lets give that thing a name/symbol, infinity. Since I've defined it, it exists...even if its only an existence as a concept. Arguing that there is no infinity is like arguing that there is no such thing as a complex number or an imaginary number...whether or not such a thing exists in nature, or is useful, is irrelevant,...I have a definition so it exists as a concept. Mathematics isn't constrained by reality.

That argument doesn't sit right with me - boundless limits don't approach anything, that's the whole point. Anyway the thinking behind the finitists (and constructivists) is that you can't claim something to exist until you have constructed it. I'm not personally denying or affirming the existence of infinity or the validity of these philosophies, I just linked them to show that point of view.

Fredrik
Staff Emeritus
Gold Member
You could ask the same thing about the number 2 (or pretty much any mathematical concept). What's relevant is that they can all be constructed from the ZFC axioms of set theory, and that many of the mathematical structures obtained this way are useful in physics.

apeiron
Gold Member
Infinity in maths is modelling, so it can be both an artificial concept and meaningful, or rather useful.

But you are asking more for an ontological view, this being a philosophy forum.

I would look at it this way. When counting is defined as a non-terminating process, then it seems ridiculous that you could ever reach a termination. It seems patently self-contradictory. So any cut-off imposed as a limit would be arbitrary and untrue.

However, this is the local view. Now if we take the global view, we can see that change reaches a stage where there is no real or visible change going on.

To count from 1 to 2, that's a big change (twice as much). But to count 1 more from some tremendously huge number, well that hardly makes any difference from the global perspective of what has ontologically changed. Or same if you are counting down in infinitesimals.

So this is an equilibrium view - a state where there is free change at the microstate level but no apparent change anymore at the macrostate. So the system has clearly hit some limit to actual change (even if change remains as free and non-terminating as ever).

Infinity is ment to be beyound the scope of a single slice of cheese. It takes a whole sandwich to understand infinity.

I'm gonna say I agree with what Fredrik wrote.

But what's this about 'subtract infinity from both sides to get -1=0??? I surely hope you didn't mean to say that in terms of your 'what's infinity -1'.
If you(which I think you are) then let's look at it shall we? Using arithmetic rules you described, no other rules from other number systems apply...

infinity-1=infinity
infinity-infinity-1=infinity-infinity
-1=0
I think that's what you were thinking about... I hope you can see the error in your thinking, if you weren't thinking this then please explain cause that part confused the heck out of me.

It would be like saying
2-1=2
2-2-1=2-2
-1=0

It was wrong from the very beginning.

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Pythagorean
Gold Member
As far as I know, we've never observed infinity in nature.

Hurkyl
Staff Emeritus
Gold Member
Infinity does not exist in the context of a number system.
That's inaccurate. Sure, the natural numbers, real numbers, and complex numbers do not have an element called infinity. However:
• The projective complex numbers have an element named $\infty$
• The extended real numbers has two elements named $+\infty$ and $-\infty$
• The cardinal numbers do not have any elements named infinity, but they do have many elements that are infinite, such as $\aleph_0$. (aleph-null)

Example: what would infinity minus 1 be?
• In the projective complexes, $\infty - 1 = \infty$
• In the extended real numbers, $(+\infty) - 1 = +\infty$ and $(-\infty) - 1 = -\infty$
• In the cardinal numbers subtraction isn't well-defined. However, $\aleph_0 + 1 = \aleph_0$

We see that the rules of arithmetic are violated, because we can't subtract infinity from both sides to conclude that -1 = 0, which isn't true.
Each number system has its own rules. While they often have lots of rules in common, none of the ones I mentioned above satisfy the cancellation property that you just tried to use.

Oh good I was afraid to ask this in the astronomy forum relating to the infinite density of a black hole.

Perhaps philosophy is the proper place?

This is the question:

When infinities are mentioned in physics, such as in the density of a black hole,
are these static infinities? Or are they approaches towards infinity?

I will ask a different way.

When a massive star starts collapsing into a black hole this is a process. Is a black hole still continually completing this process.. becoming more dense.. approaching infinite density in an asymptotic line? Or is it simply at one point BAM infinitely dense?

I mean the difference between forever aproaching infinity and being at infinity.
Black Hole density which is it?

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Pythagorean
Gold Member
Is infinity merely a mental concept, or is it entirely meaningful and common?
by the way, meaningful and common are traits of mental concepts.

apeiron
Gold Member
When a massive star starts collapsing into a black hole this is a process. Is a black hole still continually completing this process.. becoming more dense.. approaching infinite density in an asymptotic line? Or is it simply at one point BAM infinitely dense?
Someone else can probably answer this better but energy density is limited by the planck scale so there is a "finite" cut-off. You don't get infinite density at a point but a black hole with an event horizon that is proportional to the mass contained. Our sun would reduce to about 3km as a black hole for example.

You may be thinking of something else? Like the time taken for an observer to fall into a black hole?

Anyway, a black hole small enough in mass to be about planck scale in extent (so looking as close to a point as possible) would also evaporate almost immediately. If one could be created in an accelerator, it would be BAM finitely undense.

SixNein
Gold Member
Although there is no real number called infinity, infinity exists as a type of limit. In mathematics you can have sequences that diverge and calculus limits that approach infinity. I thought there were only two types of infinity...countable and uncountable. Isn't there even a proof showing that there are no "degrees" of infinity...just the two types (countable and uncountable) with nothing in between?
Infinity does not exist as a limit in calculus; instead, it is used to communicate the way the limit does not exist.

Cantor created a proof showing that infinities can come in different sizes. For example, there are more real numbers than natural numbers. In addition, he created the continuum hypothesis that basically states that no set can come between the real and natural sets. But this hypothesis remains unsolved. Many mathematicians think its false. Several have went mad trying to prove it.

SixNein
Gold Member
Is infinity merely a mental concept, or is it entirely meaningful and common?

Infinity does not exist in the context of a number system. Example: what would infinity minus 1 be? It's not infinite since no finite number plus one equals infinity. We see that the rules of arithmetic are violated, because we can't subtract infinity from both sides to conclude that -1 = 0, which isn't true.

But at the same time there are for example, infinitely many real numbers between 0 and 1 (there is actually even an infinite number of *types* of infinities; this kind is called Aleph-1).
Infinity is meaningful and common, but it is very poorly understood. For example, an uncountable infinite amount of measurements can be taken from 0 inches to 1 inches. And there is nothing special about the inch unit (the same is true for smaller units). When infinity is brought into a subject, things can get weird quickly.

Hurkyl
Staff Emeritus
Gold Member
Infinity does not exist as a limit in calculus
I mentioned the projective reals and the extended reals; try looking them up.

But this hypothesis remains unsolved.
The status of the continuum hypothesis is completely known -- it is independent of the axioms of ZFC.

SixNein
Gold Member
I mentioned the projective reals and the extended reals; try looking them up.

The status of the continuum hypothesis is completely known -- it is independent of the axioms of ZFC.
See Woodin.

Observing infinty is impossible because you will never see the whole thing. But proving infinity should be easy because everything we have ever looked at goes on forever. All things can be broken down more and more. The thing is that infinity is always over looked as a concept or an idea. When you stop and start to think about infinty you realise that if infinty is real then ALL things become infinite. Time, space, mass. There is no end to anything. Every number is infinite. The value of 2 is infinite. lol, when you talk about infinity one tends to say infinity a lot.