The very fabric of space time is stretching. It does not need anywhere to expand into; time and space don’t even exist beyond the boundaries

What's infinite is the waste of time where ignorance debates dogma without benefit of good definitions - or at least agreed definitions. The question of the OP is to "actual infinities." Here's one, and once this one is comprehended you will see that you almost cannot get out of bed in the morning, or get into it at night, without blundering into any number of them.

The surface of a sphere is a finite quantity. It is also unbounded. It's reasonable to plot a path on the surface of a sphere. We do it all the time. What would you say the sum of the distances of the possible paths on the surface of a sphere is? And the surface of the sphere is just exactly a collection of those paths. I guess it's aleph-c and maybe greater, but not less.

You do not have a truely continuous sphere on which to make such a construction.

You also have no geometrical equipment with precision 1/oo to make the construction

That's an appeal to authority. — Metaphysician Undercover

By this argument you destroy argument. Or can you produce an argument not at some level based on authority?

modern mathematics is full of contradiction. — Metaphysician Undercover

Please cite some. I always did like a good contradiction, and if you're right then very likely there are not just a lot of them, but an infinite number of them.

You do not have a truly continuous sphere on which to make such a construction. — Devans99

Huh?!

I guess you cannot read, too.

It's reasonable to plot a path on the surface of a sphere. We do it all the time. — tim wood

Did you miss, "We do it all the time"? And what do you imagine a path is?

You are assuming space is continuous to get smooth sphere on which to plot. IE you are assuming what you want to prove.

It's not true that the "normal operations" can be performed with transfinite numbers. Analogous operations can be defined, but the are not the SAME operation. The fact that transfinite numbers have mathematical properties has no bearing on whether or not they have a referent in the real world - mathematics deals with lots of things that are pure abstraction with no actual referent (look into abstract algebra). — Relativist

I did not claim that because transfinite numbers have mathematical properties they can have real world referents. I don't see where in the part you quoted of me indicates that, the part you quoted was my response to a user claiming infinite numbers "act non-numerically". Yes, I've studied abstract algebra, I never claimed all mathematics was applied math.

That said, addition and multiplication can still be done with transfinite numbers. Cantor himself showed this, so it's old hat. If you mean that it's not literally the same operation, I'm just questioning the relevance. Transfinite arithmetic is arithmetic for infinite numbers. Is it a bit different? Yeah, but I never said otherwise. My point was that the results are odd because you're not in a finite domain anymore, e.g. ℵ0 + 8 = ℵ0.

It would be like looking at negation in paraconsistent logic and saying "Hey that's not negation because it isn't explosive".

I just think it's misleading to say, "addition and multiplication can still be done with transfinite numbers" , and that's because (as you say)- it's not the same operation. But sorry if I misunderstood where you were coming from. I thought you were claiming the mathematical relations involving infinities implied they had real world referrents.

Nah it's fine. It's technically not the exact same operation but since it behaves similarly enough I don't know what else to call it (additive operation?) My initial points were that infinity isn't inherently off the table when talking about reality, as the OP and another user were arguing that infinity is a contradictory concept (which is just flatly untrue); so if anything in reality is infinite or not is an empirical matter, there's no strictly logical argument against it being instantiated. Anyway, sorry if I was unclear!

How else could you explain 1/3 = 0.33reapeating if not with infinity?

A decimal representation has a fixed number of digits; always will have a fixed number of digits. No need for actual infinity.

How else could you explain 1/3 = 0.33reapeating if not with infinity? [/quote]
That's indeed how you explain the mathematical relation between thirds (which have real world referrents) and the abstract mathematical process of dividing 3 into 1 - which does not have a real world referrent. As Devans99 alluded, any real world representation of the result of this division (such as in a computer) will be an approximation.

" if anything in reality is infinite or not is an empirical matter, there's no strictly logical argument against it being instantiated."
I don't see how an instantiated infinity could ever be established empirically since we can't count to infinity. On the other hand, I think in some cases, infinity can be ruled out. For example: the past cannot be infinite. Here's my argument:

1. It is not possible for a series formed by successive addition to be both infinite and completed.
2. The temporal series of (past) events is formed by successive addition.
3. The temporal series of past events is completed (by the present).
4. (Hence) It is not possible for the temporal series of past events to be infinite.
5. (Hence) The temporal series of past events is finite.

I'm assuming a sphere. Or is a sphere not possible?

I myself believe Absolute Infinity as an mathematical entity exists. It's just a personal hunch that it is so.

Cantor reserved the knowledge of Absolute Infinity to God. As a deeply religious person this sidenote shouldn't be overlooked as Cantor obviously viewed it as something very important. I think it's the cornerstone that is simply yet missing from our basic knowledge of mathematics.

The reason why it would be so:

a) mathematicians have yet to solve the Continuum Hypothesis. Hence our understanding of infinity is still lacking.
b) Usually if some mathematics is useful in physics, the math is right. I don't know of applications in physics of using the cascading system of larger and larger infinities.
c) All the discussions we have of infinity.

I'm assuming a sphere. Or is a sphere not possible? — tim wood

We were looking for examples of actual infinity in nature
You said a sphere with infinite segments
But it’s not proven that nature is continuous
So your sphere can’t have infinite line segments
For the purposes of this proof

I think Devans99 is a troll.

I don't see how an instantiated infinity could ever be established empirically since we can't count to infinity. — Relativist

The same way we can empirically establish anything at all. We don't necessarily need to count to infinity for that, just as we don't need to write out all the digits of pi in order to empirically establish the harmonic oscillator solution. If a model that makes use of infinities provides a good fit for many observations, is parsimonious, productive, fits in with other successful models, etc. then we consider it to be empirically established, infinities and all.

On the other hand, I think in some cases, infinity can be ruled out. For example: the past cannot be infinite. Here's my argument:

1. It is not possible for a series formed by successive addition to be both infinite and completed.
2. The temporal series of (past) events is formed by successive addition.
3. The temporal series of past events is completed (by the present).
4. (Hence) It is not possible for the temporal series of past events to be infinite.
5. (Hence) The temporal series of past events is finite. — Relativist

"Successive addition" implies a starting point, which obviously precludes an infinite past. Your argument simply begs the question. An infinite past is a past that does not have a starting point.

I myself believe Absolute Infinity as an mathematical entity exists. It's just a personal hunch that it is so. — ssu

You don't need any hunches in order to believe that a mathematical entity exists: all you need is a mathematical theory that says that such and such entity is infinite - and such mathematics exists, there is no question about that.

You don't need any hunches in order to believe that a mathematical entity exists: all you need is a mathematical theory that says that such and such entity is infinite - and such mathematics exists, there is no question about that. — SophistiCat

Is there a theory of Absolute infinity? Please tell me if there is!!!

Cantors' system of larger and larger infinities, his transfinite set theory, where by using Cantor's theorem one can generate an infinite sequence of infinite sets whose infinite sizes are larger and larger infinities basically collides with the notion of Absolute infinity. Now Cantor didn't know how to deal with it, so I guess he left it to God to know.

With Absoluty Infinity we have right in our face basically Russel's Paradox, the 'set of all sets', or Cantor's Paradox or Burali-Forti Paradox, you name it. I think the problem here is that we start mathematics from counting. For a theory of the Absolute to exist you need to show just how you cannot have anything larger or basically the paradoxes of the infinity aren't something to be solved, but answers to be understood. Or something like that (and hence the talk of hunches).

"'Successive addition' implies a starting point, which obviously precludes an infinite past. Your argument simply begs the question. An infinite past is a past that does not have a starting point."

But an infinite past still entails an infinite series that has been completed; that is the dilemma. Consider how we conceive an infinite future: it is an unending process of one day moving to the next: it is the incomplete process that is the potential infinity. The past entails a completed process, and it's inconceivable how an infinity can be completed.

"You don't need any hunches in order to believe that a mathematical entity exists: all you need is a mathematical theory that says that such and such entity is infinite - and such mathematics exists, there is no question about that."
Mathematical entities are abstractions, they have only hypothetical existence.

" If a model that makes use of infinities provides a good fit for many observations, is parsimonious, productive, fits in with other successful models, etc. then we consider it to be empirically established, infinities and all."
How is this different from the infinity of mathematical operation of dividing 3 into 1? Just because it equates to an infinity of 3's after the decimal doesn't imply infinity exists in the world. The real world dividing of a thing into 3 equal parts entails no infinity, the infinity just arises in the math. Mathematics is descriptive (or purely hypothetical), not ontic.

[addition and multiplication of transfinite cardinals is] technically not the exact same operation [as addition and multiplication of the natural numbers] — MindForged

Interestingly, addition and multiplication of real numbers, of rational numbers, and of integers, are also all different from the addition and multiplication of integers:

Starting with the natural numbers, every time we enlarge the set of numbers, the algebraic properties change. There's no reason for us to be surprised when it changes yet again when we move from the reals to the cardinals (including transfinite cardinals).

FWIW, the cardinals form a commutative monoid under addition and a commutative monoid under multiplication, and multiplication is distributive with respect to addition. Like all other sets of numbers, the set of cardinals is totally ordered.

I don't see how an instantiated infinity could ever be established empirically since we can't count to infinity. On the other hand, I think in some cases, infinity can be ruled out. — Relativist

Sure but that's not really how one gets to infinity in math. It's not like when someone says the natural numbers are infinite they mean they've counted to some point called infinity. As you know, in modern mathematics it means the set can be put into a one-to-one correspondence with a proper subset of itself. If making the assumption that something in the universe (space, time, something else) is infinite is a part of a very good theory, that's perfectly reasonable a basis to think reality is infinite in that respect, even if in other respects it might not be possible. The Everettian/Many-Worlds Interpretation of QM seems really solid to a lot of physicists, and it seems to make such an assumption about the number of worlds, for example.

Edit: SophistiCat has already put it better than I:

If a model that makes use of infinities provides a good fit for many observations, is parsimonious, productive, fits in with other successful models, etc. then we consider it to be empirically established, infinities and all. — SophistiCat

Starting with the natural numbers, every time we enlarge the set of numbers, the algebraic properties change. There's no reason for us to be surprised when it changes yet again when we move from the reals to the cardinals (including transfinite cardinals). — andrewk

Yeah that's why I didn't see it as a relevant objection even if it's technically correct. Your points were well made though. *thumbs up*

The surface of a sphere is a finite quantity. It is also unbounded. It's reasonable to plot a path on the surface of a sphere. We do it all the time. What would you say the sum of the distances of the possible paths on the surface of a sphere is? And the surface of the sphere is just exactly a collection of those paths. I guess it's aleph-c and maybe greater, but not less. — tim wood

The problem is that spheres are only conceptual, just like infinities. So the question is, does a concept, like "infinity", have actual existence.

Please cite some. I always did like a good contradiction, and if you're right then very likely there are not just a lot of them, but an infinite number of them. — tim wood

I'm starting with "infinite set" which is very obviously contradictory. A "set" is limited, restricted, by the defining terms of the set. "Infinite" means unrestricted, unbounded, or unlimited. Therefore "infinite set" is very clearly contradictory. Once you grasp this obvious contradiction, then I might be able to show you some other, more complex contradictions within mathematics, but if you cannot see the contradiction here, in this very simple example, I don't see any point in giving any other examples.

My initial points were that infinity isn't inherently off the table when talking about reality, as the OP and another user were arguing that infinity is a contradictory concept (which is just flatly untrue); so if anything in reality is infinite or not is an empirical matter, there's no strictly logical argument against it being instantiated. Anyway, sorry if I was unclear! — MindForged

If you're referring to me, saying that someone is arguing that infinity is a contradictory concept, then this is wrong, it's not what I've been arguing. What I have been arguing is that "infinite set" is a contradictory concept.

"infinite set" which is very obviously contradictory.

It can't be all that obvious, since so many mathematicians and scientists have failed to observe the contradiction, and some of them have been reputed to be quite bright.

We must all be grateful that this thread has finally come to light, so that the said mathematicians and scientists can be freed from the delusion under which they have been labouring.

The problem is that spheres are only conceptual, — Metaphysician Undercover

Really MU? There's no such thing as a sphere? That's just stupid!

Infinite sets very obviously contradictory? How about the set of numbers greater than two? The set of irrational numbers between zero and one?

Or the set of possible paths on the surface of a sphere. Actually, the set of paths on any surface?

Is there a theory of Absolute infinity? Please tell me if there is!!! — ssu

OK, so you make a distinction between something you call "Absolute" infinity and any other sort of infinity. I don't know what that difference is, and it doesn't look like you have a very definite idea either. When you want to find out whether something exists, you don't start by giving it a name, you start by giving it an operational definition, laying down requirements that need to be satisfied for anything to be recognized as that thing. It's no use just saying: "Well, it's Absolute, you know..."

But an infinite past still entails an infinite series that has been completed; that is the dilemma. Consider how we conceive an infinite future: it is an unending process of one day moving to the next: it is the incomplete process that is the potential infinity. The past entails a completed process, and it's inconceivable how an infinity can be completed. — Relativist

Well, inconceivable is a subjective assessment, it's a far cry from being provably impossible. If you just want to say that you don't believe the past can be infinite because an infinity of elapsed time seems inconceivable to you, you are welcome to it. Does an absolute beginning of time, such that right at the beginning there is no before, seem more conceivable to you?

Mathematical entities are abstractions, they have only hypothetical existence. — Relativist

That's neither here nor there, because this is true for all our thoughts, concepts, imaginings. When you think of a dog, even when the thought is prompted by looking at one, your thought is not the dog - it's an idea in your head, an abstraction of a dog.

How is this different from the infinity of mathematical operation of dividing 3 into 1? Just because it equates to an infinity of 3's after the decimal doesn't imply infinity exists in the world. — Relativist

You mean dividing 1 into 3, right? Exactly, very good example. You don't say that for there to be thirds we need to be able to write out all the decimal digits of 1/3, right? That would be an arbitrary, unjustified requirement. So why do you maintain that for there to be a "completed" infinite sequence we need to be able count out each individual element of the sequence? Does its existence somehow depend on us speaking or thinking it into existence, one element at a time? Bottom line, you can't just throw out such arbitrary requirements, you need to justify them.

OK, so you make a distinction between something you call "Absolute" infinity and any other sort of infinity. I don't know what that difference is, and it doesn't look like you have a very definite idea either. — SophistiCat

Ah, you didn't know the issue. It's basically about what Georg Cantor proposed. See here.

Yeah, that was before we realized the naive comprehension scheme resulted in Russell's Paradox and trivializes the math if you keep the classical logic. There is no "set of all ordinal numbers" so Cantor's notion of an "absolute infinity" cannot be expressed in ZF set theory.