Actual Infinity in Reality?

[Succubus]

I am not asking if there is a concept in mathematics that deals with infinity or if there exists sets with an infinite number of members (although you might use the concept in a larger argument). I am not asking if we can theoretically divide something an infinite amount of times (although you might use the concept in a larger argument). I am not asking about a potential infinite. 

I am asking about an actual infinite of something concrete (not abstract). Can it logically exist? Why or why not? 

No mention of God either. Philosophy subforum--let's stick with pure metaphysics.

Bolstered by your success in other threads in instigating thousand word replies to your bullshit, you've expanded the strategy, ie, this thread. And a very successful strategy it will be.

Page one hundred ten of this thread will prove my point.

I am not asking about a potential infinite.

Jabba, is that you?

[RoadRunner79]

You don’t see a problem in having all the elements present, and it being infinite.  If the are all there, then they are limited and thus finite.   You still have a contradiction.

They are all there, but there is an infinite number of them. That's what makes it an infinity.

I'm not arguing that it's intuitive, btw. Just saying that there is no logical problem with it.

I don’t think that addresses the contradiction I mentioned. Also where you not just recently telling Steve that infinity wasn’t a quantity. Have you changed your mind?

[GrandizerII]

They are all there, but there is an infinite number of them. That's what makes it an infinity.

I'm not arguing that it's intuitive, btw. Just saying that there is no logical problem with it.

I don’t think that addresses the contradiction I mentioned. Also where you not just recently telling Steve that infinity wasn’t a quantity.  Have you changed your mind?

Infinity is not a number in the sense that it's not an element of the set of real numbers, it's that which encompasses the number of elements in it. It's the set itself when the set contains infinite elements. Pay attention to the grammar of words before you charge me with contradicting myself.

And you didn't state any contradiction. All you did was show that you continue to confuse actual infinity with potential infinity.

[Angrboda]

Yes, of course, it is *logically* possible. There is no internal contradiction to the concept of an actual infinity.

Now, I don't know if you consider time to be abstract or concrete, but there is certainly no internal contradiction to the concept of an infinite time.

There is also no contradiction in the possibility of an infinite number of stars in our universe if the universe is infinite in spatial extent.

My naive intuitions balk more at the idea of a first moment than at the idea that there will always be another moment before any moment you choose.  To insist that there was once a brand spanking new, first ever moment before which there was no before .. certainly requires a lot of support.

The idea is very reminiscent of the same question about space. If there is an end to space, what lies on the other side of that boundary? If there is always an 'other side' to any boundary in space, then surely it must be infinite? Some stomp their feet and declare that there is nothing on the other side, so space is finite. Others point out that this doesn't make sense; how can there not be 'something' on the other side, even if only empty space? It seems to me that our questions about whether time and space are infinite or not rest upon questions which, by their very assumptions, admit of no answer one way or the other. It seems to me an obvious sign that the question is in some sense malformed. That it makes no sense to claim either based upon what appears to be nothing more than questions that seem to simply reflect our ignorance about the matter as a whole. Garbage in, garbage out. If the assumptions, intuitions, and models behind these questions are flawed, surely our responses to them are going to be similarly flawed.

[GrandizerII]

About the seeming contradictions in Hilbert's Hotel, here's to put things in clearer perspective:


inf(1,2,3,4,5,6,7,8,9,10,...) - inf(1,2,3,4,5,6,7,8,9,10...) = 0
In this case, inf(positive integers) - inf(positive integers) = 0
or
inf - inf = 0

inf(1,2,3,4,5,6,7,8,9,10,...) - inf(2,4,6,8,10,...) = inf(1,3,5,7,9,...)
In this case, inf(positive integers) - inf(positive even integers) = inf(positive odd integers)
or
inf - inf = inf

inf(1,2,3,4,5,6,7,8,9,10,...) - inf(4,5,6,7,8,9,10,...) = 3
In this case, inf(positive integers) - inf(positive integers except for 1, 2, and 3) = 3
or
inf - inf = 3


So no contradictions, just different infs we're dealing with.

Also, case 1 proves there is no contradiction (because same collection - same collection is indeed 0).

Either way, without context, that is why inf - inf is indeterminate, much like 0/0 is indeterminate.

[Whateverist]

My naive intuitions balk more at the idea of a first moment than at the idea that there will always be another moment before any moment you choose.  To insist that there was once a brand spanking new, first ever moment before which there was no before .. certainly requires a lot of support.

The idea is very reminiscent of the same question about space.  If there is an end to space, what lies on the other side of that boundary?  If there is always an 'other side' to any boundary in space, then surely it must be infinite?  Some stomp their feet and declare that there is nothing on the other side, so space is finite.  Others point out that this doesn't make sense; how can there not be 'something' on the other side, even if only empty space?  It seems to me that our questions about whether time and space are infinite or not rest upon questions which, by their very assumptions, admit of no answer one way or the other.  It seems to me an obvious sign that the question is in some sense malformed.  That it makes no sense to claim either based upon what appears to be nothing more than questions that seem to simply reflect our ignorance about the matter as a whole.  Garbage in, garbage out.  If the assumptions, intuitions, and models behind these questions are flawed, surely our responses to them are going to be similarly flawed.

Well said.  Thinking time goes back limitlessly certainly isn't anything I'm greatly invested in.  The main thing is that for any reasonable question involving the past there would always be a time before that.  Perhaps there is something we are missing given our frame of reference which prevents us conceptualizing time in a manner which renders the eternal/true beginning question moot.  Perhaps.  But I'm quite content to concede that the matter is indeterminable and therefore useless as at the premise in any argument.

[Kernel Sohcahtoa]

About the seeming contradictions in Hilbert's Hotel, here's to put things in clearer perspective:


(1) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(1,2,3,4,5,6,7,8,9,10...) = 0
In this case, inf(positive integers) - inf(positive integers) = 0
or
inf - inf = 0

(2) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(2,4,6,8,10,...) = inf(1,3,5,7,9,...)
In this case, inf(positive integers) - inf(positive even integers) = inf(positive odd integers)
or
inf - inf = inf

(3) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(4,5,6,7,8,9,10,...) = 3
In this case, inf(positive integers) - inf(positive integers except for 1, 2, and 3) = 3
or
inf - inf = 3


So no contradictions, just different infs we're dealing with.

Also, case 1 proves there is no contradiction (because same collection - same collection is indeed 0).

Either way, without context, that is why inf - inf is indeterminate, much like 0/0 is indeterminate.

Regarding the examples posted above, another way to see it is to define the positive integers as the universal set.  Thus, when we take the complement of each of the examples (the set of positive integers, the set of positive even integers, and the set containing 4,5,6,7,8,9,10,...) we obtain the empty set, the set containing the positive odd integers, and the set containing 1,2,and 3 respectively.

P.S. For anyone interested

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Let A be a set with a universal set U. We denote the complement of a set as A^c where A^c=U–A , which is the set containing all elements x with the defining property that x is not in A.

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[GrandizerII]

About the seeming contradictions in Hilbert's Hotel, here's to put things in clearer perspective:


(1) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(1,2,3,4,5,6,7,8,9,10...) = 0
In this case, inf(positive integers) - inf(positive integers) = 0
or
inf - inf = 0

(2) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(2,4,6,8,10,...) = inf(1,3,5,7,9,...)
In this case, inf(positive integers) - inf(positive even integers) = inf(positive odd integers)
or
inf - inf = inf

(3) inf(1,2,3,4,5,6,7,8,9,10,...) - inf(4,5,6,7,8,9,10,...) = 3
In this case, inf(positive integers) - inf(positive integers except for 1, 2, and 3) = 3
or
inf - inf = 3


So no contradictions, just different infs we're dealing with.

Also, case 1 proves there is no contradiction (because same collection - same collection is indeed 0).

Either way, without context, that is why inf - inf is indeterminate, much like 0/0 is indeterminate.

Regarding the examples posted above, another way to see it is to define the positive integers as the universal set.  Thus, when we take the complement of each of the examples (the set of positive integers, the set of positive even integers, and the set containing 4,5,6,7,8,9,10,...) we obtain the empty set, the set containing the positive odd integers, and the set containing 1,2,and 3 respectively.

P.S. For anyone interested

---

Let A be a set with a universal set U.   We denote the complement of a set as A^c where A^c=U–A , which is the set containing all elements x with the defining property that x is not in A.

---

Thanks. Your approach is definitely a better alternative. The important thing to take away from this is that Hilbert's Hotel is about set operations not merely number operations. Both infinity and "finity" (is this a word?) are not numbers, after all.

[polymath257]

We have *two* ways to talk about the relative sizes of two sets:

1. Is set A a subset of set B? In other words, is everything in A also in B? We write A⊆B..

2. Can set A be paired in a one-to-one way with a subset of B? In this case, we write card(A)<=card(B).

Now, the first (subset) always implies the second (size is not larger).

For *finite sets, if A is a subset of B and A is not equal to B, then card(A)<card(B), i.e, the 'size' is strictly smaller. The common phrase is 'the whole is greater than any part'.

But for *infinite* sets, it is possible for A to be a 'proper subset' of B and yet have card(A)=card(B), so they are the 'same size'. In fact, this is often used as the *definition* of what it means to be an infinite set (due to Dedekind).

This fact that there are two different ways to compare the size of sets is the basis of a lot of the paradoxes of infinite sets. There is no *contradiction* in the Hilbert Hotel. But it shows that set differences and sizes differences are different things. That is why *for sizes*, the expression infinity-infinity is given no meaning.

NOBODY has come up with an actual contradiction in the notion of an infinite set. Not that people haven't tried. But there is well over 100 years of mathematics based on the properties of infinite sets and *no* contradiction has ever been found.

As for 'concrete things in real life', we simply do not know if space or time or the number of stars are infinite. It is possible that all of these are finite. It is possible that al are infinite. It is possible that time is finite and space is infinite or vice versa. They are all *logical* possibilities.

BUT WE DO NOT KNOW IF THEY ARE REALITY.

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The term actual infinity is contradictory.

The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential, but not to be mistaken for physically existing.  [ wiki ]

Infinite of course refers to limitless or endless.   It cannot be ended, or completed.  It cannot be contained or actualized in it's entirety or limited.  Therefore when you put the two together, it is a contradiction.  

Infinity turns out to be the opposite of what people say it is. It is not 'that which has nothing beyond itself' that is infinite, but 'that which always has something beyond itself'. (Aristotle)

Infinity means that there is no limit, that there is always something more.   You cannot bind it and say "here is infinity"  at any given point.  You will have a potential infinity, which may be increased.

So, infinite, in your personal language, is not the same as simply not being finite, i.e, countable with some positive integer.

What, precisely, do you mean by 'completed'? Is there some process going on? Why do you assume that? What do you mean by 'actualized'?

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Might I suggest that you update your definition of the term 'infinity' to the more modern one for discussion sake?

[RoadRunner79]

The term actual infinity is contradictory.


Infinite of course refers to limitless or endless.   It cannot be ended, or completed.  It cannot be contained or actualized in it's entirety or limited.  Therefore when you put the two together, it is a contradiction.  


Infinity means that there is no limit, that there is always something more.   You cannot bind it and say "here is infinity"  at any given point.  You will have a potential infinity, which may be increased.

So, infinite, in your personal language, is not the same as simply not being finite, i.e, countable with some positive integer.

1.) What, precisely, do you mean by 'completed'?  2.) Is there some process going on? 3.) Why do you assume that? 4.) What do you mean by 'actualized'?

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5.) Might I suggest that you update your definition of the term 'infinity' to the more modern one for discussion sake?

If you mean that it is simply uncountable as in really large and impossible to quantify then no.  And I don't believe that this view would make sense in regards to the discussion on actual vs potential infinite.

1.)  I mean that it is a completed or whole thing, that you can say here is infinity. 
2.)  By definition an endless one, or one without limit.
3.)  See definition.
4.)  To be made actual or real
5.)  If you have difficulty, then propose how you would like to discuss it.  As long as the ideas are the same, it doesn't matter to me.  Just because something is new or novel however, doesn't mean that it is better.