[yor_on]

A infinity shouldn't have borders. But then we have mathematical infinities in where we can define one as larger than the other.

Two choices.
1. it's correct
2. it's not

It makes one wonder Bill. Assume that it is correct. Then ask yourself of what magnitude out presumably 'infinite universe' is? And how you would build one.

[Halc]

I’ve put together a few questions, and made some tentative moves towards possible answers, or where they might be found.  Still a long way to go, but comments would be appreciated.

I put in some comments, but have not read the whole thread.
 

1. If infinity is not a number, how can you subtract anything from it?

It isn't meaningful to subtract a number from a not-a-number.  'Infinity' is more of an adjective, meaning 'without limit'.  Much confusion arises when its syntactic usage as a noun leads one to treat it as a number.

2. If the Universe is infinite, it contains an infinite number of galaxies.  How does one define an infinite number?

Again, it simply means there is no limit to the number of galaxies.

3. If one is subtracted from an infinite number of objects, is the remainder still an infinite number?  If not, what is it?

Per point 1, it is not meaningful to do addition and subtraction with 'without limit'.

4. Would an infinite number of (identical) objects contain all the examples of that object that could exist?

If they're identical, how is it not one object?  This sort of gets into the law of identity.
Perhaps you mean something like 'just because there are infinite points along a line doesn't mean that there are not other points that do not fall on that line.  So there are examples of points not in that set, but I'd not call any of the points 'identical' since they're all at different places on the line.

5 Is “absolute infinity” (sensu, Cantor) amenable to mathematical manipulation?

6.

The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor.
It can be thought as a number which is bigger than any conceivable or inconceivable quantity, either finite or transfinite.

How could this concept be expressed without referring to infinity as “a number”?

Since it isn't a number, not sure what you're asking.  It's expressed as Ω, but that's not your question. One infinity isn't larger than another since they're not numbers.  Perhaps it means 'highest cardinality', but cardinality isn't an expression of the magnitude of something.  Only numbers have magnitude.

I'd not trust wiki on this.  They call it a number, but any number is finite.  I think a proper mathematician would not word the description this way.

An infinite universe is possible only if the mean density of matter in the universe vanishes.

Why?  Simple explanation, please.
...
I'd really appreciate some guidance with #23.  I'm trying to tie up loose ends.

Since nobody replied to this, I'll just say that the mean density of matter in the universe is typically presumed to be the same as it is in the parts we see (cosmological principle).  yor-on's statement would only apply to a model where there is infinite space, but there is finite matter, presumably all clumped nearby.  I know of no such model that is seriously considered.

[Bill S (OP)]

Halc, I’ve just managed to read your post and compare your comments with mine.  Allowing for differences of interpretation and understanding, there seemed to be a reasonable concordance.  A couple of points need clarification.

Would an infinite number of (identical) objects contain all the examples of that object that could exist?

If they're identical, how is it not one object? 

How do you define “identical”?  My understanding would be that it means “exactly alike”.  How could one object be “exactly alike”?   

  Perhaps you mean something like 'just because there are infinite points along a line doesn't mean that there are not other points that do not fall on that line.

No, that’s not what I meant.  I consider that to be a valid statement; it involves mathematical infinities, of which there are more than enough to make that work. 

I was trying to achieve some clarity regarding the definitions of infinity. 
If infinity is not a number, how can you have an infinite number of anything?
 In maths, you can talk, legitimately, of “infinite points along a line”, this is treating infinity as a number, but I accept that as a valid mathematical device.  However, if we talk of an infinite universe, and we acknowledge that that universe contains objects, we must be saying that there are “infinite” objects. 
The term, “infinite objects” differs from “an infinite number of objects” only semantically.

I’ll rephrase my question.  If the Universe is infinite, and contains (an) infinite (number of) galaxies; could there be other galaxies, that are not included in this? 

[jeffreyH]

I’ll rephrase my question.  If the Universe is infinite, and contains (an) infinite (number of) galaxies; could there be other galaxies, that are not included in this?

If they are not in the set of galaxies in our universe then where are they?

[Halc]

How do you define “identical”?  My understanding would be that it means “exactly alike”.  How could one object be “exactly alike”?

As I said, it depends on your definition of identical.  Two objects in different places have different coordinates, and thus not 'exactly alike', however much you'd not be able to tell them apart if they were scrambled around.  If they have the same coordinates, they're the same object, no?

Would an infinite number of (identical) objects contain all the examples of that object that could exist?

Perhaps you mean something like 'just because there are infinite points along a line doesn't mean that there are not other points that do not fall on that line.

No, that’s not what I meant.  I consider that to be a valid statement; it involves mathematical infinities, of which there are more than enough to make that work.

I thought I gave a counter example.  The set of points on a mathematical line is not a finite set, and yet there are points not on that line, which are examples of a points that can exist that are not in the first (infinite) set.  So the first infinite set does not contain all examples of that object (points) that can exist since I was able to find object not in the set.  'Infinite' does not imply 'all'.

I was trying to achieve some clarity regarding the definitions of infinity. 
If infinity is not a number, how can you have an infinite number of anything?

You can't.  There is not a number that represents the total count of integers for instance.

In maths, you can talk, legitimately, of “infinite points along a line”, this is treating infinity as a number but I accept that as a valid mathematical device.

You're doing your maths wrong if you treat that as a number.  I do not accept it as a valid mathematical device.

However, if we talk of an infinite universe, and we acknowledge that that universe contains objects, we must be saying that there are “infinite” objects.
The term, “infinite objects” differs from “an infinite number of objects” only semantically.

The latter form is invalid.  Saying 'there are infinite objects' just says that there is no number that represents the count of these objects.  I know people us the phrase 'infinite number' all the time in casual conversation. We know what they mean. But in mathematics, that is invalid.
I don't hold a mathematics degree (despite taking 3rd place in a high-school mathematics competition for a region of about 10 million people)

I’ll rephrase my question.  If the Universe is infinite, and contains (an) infinite (number of) galaxies; could there be other galaxies, that are not included in this?

Get rid 'number of' to let it make syntactic sense.

Yes, there could be galaxies not included in 'the universe'.  Maybe there are other universes. I personally don't buy into the existence of unmeasured galaxies, and hence I would say there can only be a finite number of them, but that's just me and my personal definition of 'existence' and 'universe' and such.  All these words have different meaning to different people.

So I talk about integers when I need an example of something infinite, and there are indeed numbers that are not included in that infinite set.  Those numbers are no more or less identical than 5 is from 13.

[alancalverd]

If the Universe is infinite, and contains (an) infinite (number of) galaxies; could there be other galaxies, that are not included in this? 

That depends on your definition of universe. To most people, it means "everything", or, if you like, "every thing", so there can't be any things that are not members of the universe.

If you are just counting galaxies as "things", then an infinite universe consists of a denumberable infinity of galaxies, which as Cantor pointed out, is one of the smaller infinities - possibly the smallest, but now I'm scraping the barrel of my memory!

[evan_au]

If (two objects) have the same coordinates, they're the same object, no?

In the context of this thread (counting massive galaxies), that is correct, as is its converse.

But in the spooky quantum world of microscopic objects, the converse is not necessarily so.
- In the dual-slit experiment, we have two slits at different coordinates in space
- But the wavicle which passes through these slits is (in some sense) at both coordinates, before it is detected

[Halc]

If (two objects) have the same coordinates, they're the same object, no?

In the context of this thread (counting massive galaxies), that is correct, as is its converse.

But in the spooky quantum world of microscopic objects, the converse is not necessarily so.
- In the dual-slit experiment, we have two slits at different coordinates in space
- But the wavicle which passes through these slits is (in some sense) at both coordinates, before it is detected

There are two slits, but I'd not consider them identical since one is a left slit and the other a right slit.
As for the wavicle (is that a word?  I like it), most interpretations say the light is in superposition of passing through both slits, which is not the same as the counterfactual statement of it actually passing through any particular slit.  An interpretation that makes such counterfactual assertions (like pilot wave theory) says the wave goes through both slits (just like water would), but the particle goes through one of them and then rides the wave pattern on the other side.  All interpretations making such statements also need to posit action at a distance (non-locality).

I'm more in the non-spooky camp that says no action at a distance or alteration of the past, and hence I cannot make such counterfactual statements.  But again, that's just my opinion, not a demonstrable thing.

[Bill S (OP)]

That depends on your definition of universe. To most people, it means "everything", or, if you like, "every thing", so there can't be any things that are not members of the universe.

I agree, but if you differentiate between the Universe and the cosmos (sensu Gribbin); there can be things that are not part of the Universe, but must be part of the cosmos.  In some ways, that complicates things, but in others, is introduces simplicity.

[alancalverd]

I haven't pursued Gribbin (never liked his early books) but Wikipedia is succinct

Using the word cosmos rather than the word universe implies viewing the universe as a complex and orderly system or entity; the opposite of chaos.

I think this stinks of philosophical arrogance and mathematical ignorance!

Everything affects everything else to some extent. We can (a) predict the outcome of an idealised 2-body interaction with plenty of confidence, and also state (b) that for any finite sample of the universe, entropy increases with time. The equation of (b) with chaos, or even the suggestion that the universe is or is increasingly chaotic, is simple defeatism.

We provably cannot predict all the interactions of all the particles in the universe, but we have no reason to think that any one interaction is not subject to the same laws (a) as all the others. Chaos is therefore in the eye of the beholder.   

[yor_on]

Well, a ordered shelf with the socks neatly stacked in pairs is of a low entropy if I remember right. With time those socks will get lost, unordered or otherwise disappearing from the shelf, that's a higher entropy. We have one arrow, it always point in a 'one way direction' and with it we will meet a raising entropy. Using entropy you might define it as going towards a place of a 'higher' unordered entropy, aka 'chaos'. But that's only half a truth, we can have places of low entropy inside a system of high entropy and the end result, of a system of high entropy, becomes pretty 'ordered' to me. It's like the 'heat death' of the universe in where everything is equivalent, nothing more happening. So entropy actually hurts my head.

[Bill S (OP)]

I haven't pursued Gribbin (never liked his early books)

My first Pop Sci book was his “In Search of Schrödinger’s Cat”.  I thought it was a bit like the “curate’s egg”. I tried a couple of others and liked them even less.  However, I think his distinction between Universe, universe and cosmos can be quite helpful; which is, probably, more than be said of the Wiki quote.

Making the distinction between the (apparently) finite Universe and an infinite cosmos can, with some unbiased thought, help to avoid at least three common pitfalls, when thinking about how there can be “something”, now.

1. Treating nothing as though it were something.
2. Treating infinity as though it were a number, while strenuously insisting that such is not the case.
3. Resorting to thaumatology when all else fails.

[Bill S (OP)]

 Like the frequently encountered deck of cards analogy, this can easily give a wrong impression.  Just changing the position or orientation of macroscopic objects from an arrangement that we define as orderly, to one that we define as disorderly doesn’t necessarily change its entropic state.  We have to ask ourselves: does moving a sock alter the number of microstates accessible to its constituent particles?  Does being shuffled, or unshuffled, change the microstates of a deck of cards? 

[alancalverd]

It's all a matter of probabilities.

If you define order in terms of adjacency, then pairs of socks are first-degree ordered: any red sock is more likely (indeed certain) to be adjacent to another red sock than any other color. If you now place the pairs in a rainbow, you have a second-degree order because if you start with a red sock you know exactly where to look for a yellow sock, and indeed any other color. The definition of a random set is that knowledge of A gives you no clue as to the nature of its neighbour B or of any other member of the set, unless the set is bounded (published "random number tables" are limited to 100 or 1000 numbers with an even but wholly unpredictable distribution)

So moving a sock doesn't alter the entropy of the set if the move was deliberate and documented, but shaking the box transforms the distribution from a highly improbable spectral order to a mess which is just as probable as any other mess after the same amount of shaking.

Now here's a fun thought, just as I rush out of the door. We can enumerate all the throws of two dice, and easily conclude that the most probable score is 7. What is the most probable score for 3 dice? for n dice? 

[Bill S (OP)]

So moving a sock doesn't alter the entropy of the set if the move was deliberate and documented, but shaking the box transforms the distribution from a highly improbable spectral order to a mess which is just as probable as any other mess after the same amount of shaking.

I can see the analogy involving the distribution of socks, and the fact that ordered movement and shaking can produce very different outcomes, but, surely, even shaking does no more than change the relative positions of macroscopic objects.  Both processes involve an increase in entropy, but that increase is a property of the agent of change, rather than the objects themselves.   

[yor_on]

Nice question Alan. But there is this thing with probabilities, how many throws will you need to get the right statistics?

[alancalverd]

Both processes involve an increase in entropy

No. Intentionally moving one sock to as known position results in a new but entirely predictable order, hence no increase in entropy. It is just as improbable as the original order!

[alancalverd]

Nice question Alan. But there is this thing with probabilities, how many throws will you need to get the right statistics?

We're back to converging sequences! The more throws, the greater your confidence that your measured distribution corresponds to the "true" distribution for an infinite number of throws.

[jeffreyH]

On chaos, if an irrational number has a constant, but unpredictable, sequence of digits, then it cannot be said to be chaotic. If the universe is deterministic, it also cannot be said to be chaotic. So what part does indeterminacy play in whether or not the universe is chaotic?

EDIT: Is indeterminacy in the eye of the beholder only?

[alancalverd]

I tend to reserve indeterminacy for the properties of quantum systems, in particular electrons. It is an inherent property of the system, not the observer. If indeterminacy depended on observation, hydrogen atoms would collapse to neutrons and organic molecules would collapse to a spherical blob when nobody is looking at them, then miraculously leap back into shape when they discover that you care. Guillaume d'Ochambe would be disgusted.

The universe is clearly ordered locally, to the extent that we can predict idealised two-body interactions to any degree of accuracy allowed by essential quantum indeterminacy, but the fact that we can't know about a cause until we see the effect  means that it is macroscopically chaotic: (a) you don't know about the butterfly until the hurricane hits you, and (b) unravelling the chain of events is unlikely to lead you back to the butterfly.