Actual Infinity in Reality?

[GrandizerII]

There is no end, but no matter how far you go through the set, there is already an element to observe.

---

Just use your imagination and assume all elements actually already exist in the set of positive integers:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

The elements are all already there, including 11 and so on. Can you add more positive integers to the set that aren't already there yet? No! No matter how far you go through the set, any last integer you reach would have already been an element of the set. It didn't need your observation to bring it into being. Even if it was Graham's number, or TREE(3), they're already there (assuming they're integers, of course; if not, ignore this last sentence).

What are you saying that the term “actual” means in the term.  
I take it to mean that it is completed or actualized.  Yet the term infinite means without limit or end... or in other words never completed. It is similar to a square circle.
There also would not be a total quantity of the set.

Yes, "actual" means "complete" ... as opposed to "potential". It doesn't mean that, necessarily, an actual infinity must have ends. A complete set means that it has all the required elements in it (which is rather redundant to say, but this is what I have to deal with in order to explain this to you).

[GrandizerII]

Regarding Zeno's paradoxes, I found this interesting:

Surely we can forgive Zeno for not seeing that his arguments can only be satisfactorily answered - from the standpoint of physics - by assuming Lorentzian invariance and the relativity of space and time. According to this view, with it's rejection of absolute simultaneity, we're inevitably led from a dynamical model in which a single slice of space progresses "evenly and equably" through time, to a purely static representation in which the entire history of each worldline already exists as a completed entity in the plenum of spacetime. This static representation, according to which our perceptions of change and motion are simply the product of our advancing awareness, is strikingly harmonious with the teachings of Parmenides, whose intelligibility Zeno's arguments were designed to defend.

http://www.mathpages.com/rr/s3-07/3-07.htm

So, mathematically, Zeno's paradoxes are easily resolved. And from a physics standpoint, they've also been resolved (B-theory all the way). And various other solutions (e.g., finite universe in an infinite wider cosmos). So no issues there with Zeno. Pick whatever solution you're happy with.

[robvalue]

I answered yes. I see no reason why an infinite number of things can't logically exist. Whether there actually are an infinite number of things, I have no idea at all.

[pocaracas]

I answered yes. I see no reason why an infinite number of things can't logically exist. Whether there actually are an infinite number of things, I have no idea at all.

Rob?
Is that you?
Are you back?!!!

[robvalue]

Hiya Poe-carcass

Yes on both counts!

[pocaracas]

Hiya Poe-carcass

Yes on both counts!

You've been missed.
What are you up to?
What *have* you been up to?

[robvalue]

Hiya Poe-carcass

Yes on both counts!

You've been missed.
What are you up to?
What *have* you been up to?

Thank you

I've been up to quite a lot! Please check out this thread

[polymath257]

What are you saying that the term “actual” means in the term.  
I take it to mean that it is completed or actualized.  Yet the term infinite means without limit or end... or in other words never completed. It is similar to a square circle.
There also would not be a total quantity of the set.

OK, so part of your problem is your definition of the term 'infinite' as being without limit or end. That is NOT the best definition of the term and is not the one used for infinite sets.

For collections of objects (sets), the term infinite means 'not finite'. In turn, 'finite' means that it can be put into correspondence with a positive integer.

So, the set {1,5,8} can be put into correspondence with the positive integer 3, so it is a finite set of size 3. The set {1,4,9,16,25} can be put into correspondence with the number 5 and so it is a set of size 5.

Being infinite just means there is no positive integer that can be put into correspondence with that set. So, for example, the set of ALL positive integers, call it N, cannot. So it is an infinite set. It is a *completed* set because we know exactly which things are in it and which are not. So there is no ambiguity in the set, no process going on that is generating the set. It is just there.

Now, we often write this set in the following way,

N={1,2,3,4,5,...}

but this is a convenience based on our understanding that the ellipses represent a pattern we can use to determine what is in and what is not in that set. It isn't a process, but a pattern.

Now, quantities are a bit strange when it comes to infinite sets. We say two sets have the same 'cardinality' when they can be put into correspondence with each other. A classical example due to Galileo (mentioned by Steve) is to look at the collection of perfect squares

S={1,4,9,16,25,36,....}

and compare it to the positive integers

N={1,2,3,4,5,6,....}

Every square can be paired off with a positive integer:

1 <-> 1
4 <-> 2
9 <-< 3
16 <-> 4
..
..

Again, both sets are *complete*; we know exactly which things are in both. The pairing is simple, but is even easier backwards: each integer x is paired with its square x*x. This gives a correspondence between the two sets. Because of this, we say the two sets have the same cardinality (loosely, they have the same size).

Now, you may have noticed that S is a subset of N: everything in S is also in N. And it is a proper subset: there are things in N that are not in S. For finite sets, a situation like this would force the size of S to be smaller than the size of N. For infinite sets, all we get is that S is no larger than N. And, in this case, they have the same 'size' in the sense of cardinality. That isn't a contradiction, it is simply a way in which finite and infinite sets act in different ways. In fact Dedekind defined an infinite set to be one that can be put into correspondence with a proper subset. For him, finite means no such correspondence is possible.

So the set S can be put into correspondence with N, the set of positive integers. Any infinite set for which this can be done is called countably infinite. Yes, I know you can't count all the elements, but the idea of the terminology is that we can still use 'numbers' to 'count' them in some way. In any case, it is now standard terminology.

Now, it turns out that there are infinite sets that *cannot* be put into correspondence with N. We say those sets are uncountable. The collection of all real (i.e, decimal) numbers is an uncountable set. So is the collection of all subsets of N.

In practice, physicists use the collection of real numbers all the time. Itis the realm of calculus. It is the realm of continuous functions and integrals, of differential equations and motion. The uncountably infinite set of real numbers is used as a matter of course in all of physics. So, all modern physics is based on the usage of not just actual infinities, but actual uncountably infinite sets.

Since calculus is foundational in physics and uncountable sets are foundational for calculus, the whole notion of objecting to actual infinities simply is silly. The issue literally never arises.

In fact, anyone using the terms 'potential infinity' today is showing they are basing their ideas on concepts that have been discarded. NOBODY uses these Aristotelian concepts *except* philosophers. Everyone else has progressed past them. NO modern math or physics even uses the term.

[SteveII]

Since there are new people joining the discussion, here is a summary of problems as I see it. 

INFINITE SET THEORY
Some have suggested that the area of infinite set theory in mathematics is an indication that an actual infinite is possible. So this is how that argument goes.

1. An actual infinite consists of real (not abstract) objects.
2. The Axiom of Infinity assumes an abstract object that is an actual infinite set
3. Paradoxes, contradictions, and absurdities (such as below) can be dismissed because we can apply infinite set theory parameters/restrictions to the actual objects
     Hilbert's Hotel
     Galileo's paradox
     Ross–Littlewood paradox
     Thomson's Lamp paradox
     Zeno's Paradoxes
4. Therefore actual infinities are possible.

Since (2) is of the class of axioms that are not self-evident, they are assumptions on which further mathematical equations can be developed (useful in calculus for example). To be clear, this axiom is not reasoned into--it is just assumed as a foundation for the subset of infinite set theory in mathematics. See this earlier post. 

This is CLEARLY a question-begging argument and therefore invalid. 

B THEORY OF TIME
Another argument that has been made is that if the B Theory of Time is correct, spacetime is infinite in extent. But there is nothing in the theory that says our spacetime is infinite in the past. To get that, you must also posit an infinite cosmology model. But such models are not thought to be the best candidates for our universe, so, while possible (broadly speaking), there are not good reasons to believe this to be the case. But, such a combination of theories seems possible, so then doesn't that show that an actual infinity is possible. No, not at all.

Under any theory of time there is some sequence that is countable whether you call it causes/connection/light cones/changes in entropy/states of affairs/or whatever. I'll call it causal connections (but insert whatever you want).  Any timeline would show that the causal connections that created the present were preceded by causal connections which were preceded by causal connections for an infinite series in the prior-to direction. If you posit an infinite number of these causal connection going back, you have a problem. How could we have traversed through an infinite number of sequential causal connections to get to the one that caused the present (causal connection 5, 4, 3, 2, 1, 0)? There will always have to be infinite more causal connections that still need to happen. We will never arrive at the present.

To illustrate it with a thought experiment, imagine a being who is counting down from eternity past to the present: 5, 4, 3, 2, 1, now. How is that possible? Wouldn't he have an infinite amount more numbers to get through to get down to 3, 2, 1? If you insist that this could be done, why didn't he get done 1000 years earlier or for that matter, an infinite time ago? 


AGAINST AN ACTUAL INFINITE
A positive argument against an actual infinite is:

5. An actual infinite consists of real (not abstract) objects.
6. In 100% of our experiences and 100% of our scientific inquiries, quantities of real objects can have all the operations of addition, subtraction, multiplication and division applied to them. 
7. As the following paradoxes show, these operations cannot be applied to the concept of an actual infinite without creating contradictions and absurdities
     Hilbert's Hotel
     Galileo's paradox
     Ross–Littlewood paradox
     Thomson's Lamp paradox
     Zeno's Paradoxes
8. Therefore an actual infinite of real objects is metaphysically impossible. 

Infinite set theory is not a defeater for (6) because infinite set theory is not itself a conclusion derived from a logical process. To defeat (6) you have to give logical reasons why we should expect an infinite quantity of objects to behave fundamentally different than a finite quantity of objects.

[polymath257]

Since there are new people joining the discussion, here is a summary of problems as I see it. 

INFINITE SET THEORY
Some have suggested that the area of infinite set theory in mathematics is an indication that an actual infinite is possible. So this is how that argument goes.

1. An actual infinite consists of real (not abstract) objects.
2. The Axiom of Infinity assumes an abstract object that is an actual infinite set
3. Paradoxes, contradictions, and absurdities (such as below) can be dismissed because we can apply infinite set theory parameters/restrictions to the actual objects
     Hilbert's Hotel
     Galileo's paradox
     Ross–Littlewood paradox
     Thomson's Lamp paradox
     Zeno's Paradoxes
4. Therefore actual infinities are possible.

Since (2) is of the class of axioms that are not self-evident, they are assumptions on which further mathematical equations can be developed (useful in calculus for example). To be clear, this axiom is not reasoned into--it is just assumed as a foundation for the subset of infinite set theory in mathematics. See this earlier post. 

This is CLEARLY a question-begging argument and therefore invalid. 

On the contrary, the fact that 1) these ideas are useful (and required) for calculus, 2) that calculus has served as a fundamental tool to help us understand reality, and 3) the fact that no contradiction is produced from these assumptions
is quite sufficient to show the *possibility* of an actual infinity.

Most importantly, the lack of an actual contradiction (as opposed to simply having different rules) is quite enough to make these ideas a possibility. That they are actually used and helpful is what makes them true in the real world.

But, let's go back a bit. What are the 'self-evident' axioms? Previous posts have suggested 1. Existence 2.Consciousness 3. Law of Identity 4. Law of Non-Contradiction, and 5. Law of Excluded Middle.

The problem with this list is that it is *very* restrictive and it is even redundant. For example, the Law of Identity can mean either 'For all x, x=x' OR it can mean 'for all propositions, p<=>p'. Both are true, but the second is from propositional logic and the first from the logic of equality. But, from these, you cannot even show equality satisfies x=y & y=z => x=z. You can't show that x=y => y=x. Those are additional assumptions about equality that need to be made.

Next, in propositional logic, the term 'p implies q' is logically equivalent to '(not p) or q'. So, the claim that p=>p is logically identical to (not p) or p, in other words, the law of excluded middle! It is a definitional thing. And the Law of Non-Contradiction is simply the statement ' not (p and (not p)). But guess what? (p and q) is *defined* to be not ((not p) or (not q)), so once again, the LNC is exactly the same statement as the LEM (as long as you know that not(not p) =>p, i.e, if it is false that p is false, then p is true---hmmm....that's another logical statement that was left out).

Next, nothing was assumed as a rule of inference. Typically, Modus Tolens is assumed: if we know p and we know p=>q, then we can conclude q.

Nothing was assumed about quantifiers: 'there exists' and 'for every'. Typically, logic assumes that the negation of an existence statement is the same as saying 'for all x, the statement is false'. This is yet another basic logic axiom that is left out of the list.

Finally, with this list, it is impossible to do math at all because no method is given for producing collections. That is yet another collection of assumptions concerning how collections can be constructed. if you want the natural numbers, you have to allow *some* sort of set theory. But that goes *way* beyond the list of 'basic' assumptions made, even for finite sets.

So, I would strongly suggest you take a *modern* logic course that covers propositional and quantifier logic and *then* look at the whole host of axioms needed to construct even finite set theory. Your claim that the axiom of Infinity is suspect is very strange given the other axioms that are required to even *start* talking about addition and multiplication.

And let's face it, division is NOT always defined: 5/0 is not a well defined object. Even ordinary division requires an extension into fractions, which were quite far from being 'self-evident' to many people in history (Even Aristotle). The point is what we might consider to be 'absurd' today is likely not to be so once we get over our biases and realize there are no actual contradictions involved.

Finally, there seems to be a bias against 'abstract' set theory. I would suggest that is misplaced. By showing the abstract ideas have no internal contradictions, we see the range of *logical* possibilities and find that range is much, much broader than those suggesting actual infinites are problematic.