Actual Infinity in Reality?

[polymath257]

You really can't mean that. Mathematics (and certainly NOT metaphysics) is the prototypical logical subject.

I clearly said "Mathematical axiomatically-driven abstractions" are not sources of logic. The noun in that phrase is abstractions not mathematics. 

Here is an argument:
1. It is possible that space is flat (in fact, the observed evidence suggests this).
2. Spatially flat manifolds can be infinite in extent. (R^4 is an appropriate example)
3. Hence it is possible that space is infinite in extent.
4. As far as we can see, space is homogeneous (the same at all locations and in all directions)
5. Hence, it is possible that there are an infinite number of stars.

Now, what is you 'metaphysical' issue with either mathematical 'abstractions' or physical possibilities?

That's a mess in addition to not being a valid argument. 5 does not have anything to do with the other 4. At most, 1-4 is trying to argue for infinite space. Stars came out of left field. 

I don't think 3 follows from 1 and 2. In other words, 1 and 2 could be true but that does not give any indication that 3 is true.

OK, let's take things slower.
2. it is possible for spatially flat manifolds to be infinite in extent.
2a. By 1, space is a spatially flat manifold.
3. So it is *possible* for space to be infinite in extent.

4a. Space has at least one star.
4b. Since it is possible for space to be homogeneous, stars are distributed evenly (on the large scale) throughout the universe.
4c. from 3, it is possible that space is infinite in extent.
4d. If space is infinite in extent and stars are distributed evenly in space (on the large scale), then the number of stars is infinite.
5. It is possible that thwe number of stars is infinite.

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How many virtual photons are emitted and reabsorbed across the universe every Plank interval ??

I suppose it's just a large, very large number, but not infinite . . . .

I'm not sure that is a meaningful question, quantum mechanically.

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You really can't mean that. Mathematics (and certainly NOT metaphysics) is the prototypical logical subject.

I clearly said "Mathematical axiomatically-driven abstractions" are not sources of logic. The noun in that phrase is abstractions not mathematics.

ALL of math is abstractions. Even the number 2 is an abstraction. But, for that matter, logic is an abstraction.

[GrandizerII]

I don't think the B Theory of time solves the underlying problem of having a series of cause/effect relationships. It seems to me that even if all points of time are equally real, they are still ordered by a structure we call cause/effect--a tangible series of objects we can use in thought experiments. [NOTE: I say this to start because there are some here who deny even this].

Like I keep saying, cause/effect relationships are only meaningful to talk about in a certain context (from a temporal perspective). When we're discussing the fundamental nature of reality, especially if we're assuming B-theory of time (and/or eternalism), you have to be willing to accept that it may be logically possible that causality is just an illusion. If there is no time flow, then there is not really change or motion happening. And no causality. Which possibly leaves us with simply an eternal 4D (or higher) static structure of which every time moment is a part of. I'm just saying.

Perhaps a variation of Hilbert's Hotel:

We can conceive of a possible world (much like the one you are proposing) with a beginningless series of discrete successive events of equal duration leading up to the present (real or perceived present). 

[ ...en, ... e5, e4, e3, e2, e1, e0]

We can conceive of another possible world with exactly the same events in the same order, but in between each of those events, another event occurs.

[ ...en, En, ... e5, E5, e4, E4, e3, E3, e2, E2, e1,E1, e0]

In this series, an infinite number of additional events have been added to an already infinite series of events. Are there more events? No. Infinity + infinity = infinity. We can also do the subtraction example from Hilbert, and imagine all the events prior to e3 could have been left out of the chain. 

[e3, e2, e1, e0]

In this series, we have subtracted an infinite number of events from an infinite number of events. Infinity - Infinity = 4.  Alternately, every other event could have been left out:

[ ...en, ... e4, e2, e0]

In this series, we have left out an infinite number of events from an infinite number of events. Infinity - Infinity = Infinity. 

This is not just "counter-intuitive". Actual infinities of real objects leads to absurdities (metaphysical impossibilities). Therefore an actual infinite is not logically possible. 

[Example language from a paper from Wes Morrison - http://spot.colorado.edu/~morristo/EndlessFuture.pdf]

I already addressed this earlier, Steve. Go back a few pages to find my post where I show how inf - inf is indeterminate. You are doing subtractions with different instances of infinite sets. There's no contradiction here.

In the first case of subtraction, you took out everything up until e4 (including e4), so of course you will end up with just 4 events.

In the second case, you took out an infinite set from another infinite set in a way where an infinite set remains (by taking out one event for every two events we go through).

In the addition case, you seem to be missing some important assumptions here, so I'm not going to comment on that until I get a clearer picture of what you mean by "discrete" and such.

[polymath257]

3. Hilbert's hotel applies to sets that are countably infinite.  If time is continuous and infinite, it would seem that the set of all possible moments is uncountably infinite.  In that event, Hilbert's hotel simply wouldn't apply.  As long as we're throwing around burden of proof questions, I think you are obligated to either show that time is not continuous, or that even if it is, that the set of all possible moments is a countable infinity.  Otherwise, we can simply dispense with Hilbert's hotel, as it does not cover all the possibilities for a temporally infinite universe that I have raised.  An objection which only applies to some of the possibilities but not all cannot possibly demonstrate that all cases are impossible.

Just a technical point. If we model time using the real numbers (a continuum), then there will be a countable number of *intervals* of any given size (say, 1 second long).
The standard Hilbert Hotel discussion then works just fine with those intervals.

And, in fact, we have

continuum+finite=continuum
continuum+countable infinity=continuum
continuum+continuum=continuum.

Which, again, just shows that continuum-continuum is not well defined as a cardinal number. We can, however, still do set theoretic differences.

[GrandizerII]

Think of it this way, Steve.

What's finity + finity?

What's finity - finity?

Get my drift?

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When you ask if there are 'more', there are, again, two senses for this.

There is the sense of subsets. When you remove elements, the result is a subset of the original. When you add elements, the original is a subset of the result. We usually say that subsets have 'fewer' elements than the supersets.

In all your cases, the subset relation correctly describes the notion of 'more' that you are seeking.

When when talking about cardinality (infinite in your case, is the countably infinite cardinality), subtraction is not well defined. That means that different situations can lead to different answers. There is a similarity with division by 0: 0*3=0, so 0/0=3. But 0*5=0, so 0/0=5. That isn't a contradiction. it is simply that you used division inappropriately. In the case of infinite sets above, you used subtraction inappropriately.

So, yes, this *is* just counter-intuitive: subtraction is not well-defined. That's all.

Here are three different quotes to consider:

In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity [LINKED TO BELOW] is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps. https://en.wikipedia.org/wiki/Actual_infinity

In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1] https://en.wikipedia.org/wiki/Axiom_of_infinity

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects  with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] https://en.wikipedia.org/wiki/Abstraction_(mathematics)

All emphasis added.

Therefore the the abstraction of actual infinity (from the first quote) is based on an axiom that there exists at lease one infinite set. Appropriately, an axiom in mathematics is defined as: a statement or proposition on which an abstractly defined structure is based. In case we are still unclear, the third quote defines an Abstraction. I highlighted the key theme all the way through this. Abstract. 

You have NOT made an argument (in this or the previous thread) where you show how this abstract concept in mathematics applies to the real world. It should be simple to propose some thought experiments or examples for us all to consider. I understand your point that the concept in mathematics exists--now you need to provide some evidence that it applies to real objects. Re-iterating infinite set theory from mathematics will not further this discussion.

No, stop it, Steve. You've been told this multiple times now. If there is no logical contradiction with infinity in the mathematical world, then unless you demonstrate otherwise, it seems logically possible for actual infinity in the physical world to be a thing (just not necessarily in a finite local universe like this).

Give us an example of a real world constraint preventing this from happening (with evidence of course).

[LadyForCamus]

You guys correct me if I’m wrong, but don’t all theoretical models in physics begin as mathematical models?

[Ravenshire]

The only one thing I can deduce from Steve's mental masturbation is that there are no physical infinities 'cause gawd says it's that way. And, if science ever proves a physical infinity, of coirse, it'll be axactly what gawd ordained

Shows you are not paying attention. Or that your perception does not match reality.

Or, that I've paid attention to enough of Steve's posting history.

[GrandizerII]

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Oh, please.

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Infinite number of elements. Infinite being descriptive term for the endless number of elements in the set. Look at the whole set/collection of numbers/elements, and that there is your infinity. It's not somewhere near "the magical ends".


If you were talking about actual infinity, then why are you struggling with the premise? Actual infinity exists, meaning all elements exist in it without bounds/ends. Complete doesn't mean having ends in this context. It means all elements are present.

You can correct me if I am wrong here.   You are saying that there is an infinity where all elements are present, (there can be no more)  and yet the definition of infinity means that it is without inherent limitation there is always more.  You are saying that there is no more and more at the same time, in the same way.  You can make a rule about a set, where the rule creates an open set where anything more is included.  But you do not have all at any point in the process a list of all.  You have a rule which includes all that may be.  Even in your imagination, you cannot have an infinite number of things (barely a fraction really).  At best you have a shorthand, which includes all that may be.

It doesn't matter how many times you assert that actual infinities exist, you haven't really done anything to show that they do, and saying all but not all, over and over; I don't believe helps your case.

Bolded mine. I told you you're confusing actual infinity with potential infinity. In actual infinity, there cannot be anymore. As all the elements are there. All the hotel rooms are occupied. All time moments exist already and always.

You are making an argument of intuition against infinity. That's not the same as making a logical argument against it.

And if all I'm saying in this thread that actual infinity seems logically possible, unless demonstrated otherwise, then if you contest this, it is up to you to "disprove" actual infinity.

EDIT: Considering that there are sets that are infinite in only one direction, I do have to be more careful with generalizing things. But usually, for the real world context, I'm referring to an infinite set in both ways.

[polymath257]

You guys correct me if I’m wrong, but don’t all theoretical models in physics begin as mathematical models?

Well, that has certainly been the tendency since Galileo. Those that don't manage to produce a mathematical model are generally not taken seriously.

[Anomalocaris]

You guys correct me if I’m wrong, but don’t all theoretical models in physics begin as mathematical models?

Well, that has certainly been the tendency since Galileo. Those that don't manage to produce a mathematical model are generally not taken seriously.

Tell that to Michael Faraday.

[polymath257]

Well, that has certainly been the tendency since Galileo. Those that don't manage to produce a mathematical model are generally not taken seriously.

Tell that to Michael Faraday.

And that's a good point. I'm not sure I'd label his work as a 'theory' as much as an 'experimental investigation', but you do have a point.

Maxwell fixed that deficit, though.