[dlorde]

Popular science books often point out that, in eternity, everything that can happen, must happen.

It's not quite as simple as that; it depends precisely what you mean by 'can happen'. There may well be possible states for a system (e.g. the universe) to be in that cannot be reached by any stepwise change or progression because there is a dependence on priors (see Does Everything Possible Have To Happen?).

[Bill S]

Why does it seem apparently intrinsically indivisible to you? It seems to me that infinity is divisible into any number of parts, including an infinite number of parts. In any division, there will be at least two infinite parts. Consider a road that stretches away from you to infinity in either direction. You can paint a line across it and divide it into two infinite lengths, then paint another line across, making two infinite lengths and one finite length. You can do this an infinite number of times in either direction.

It seems that, in Part 4, I didn't cover the reasoning behind the opening statement of Part 5.   My bad!

" Consider a road that stretches away from you to infinity in either direction."

Mathematically, this may be an acceptable thing to ask, but in reality, you are asking the impossible.

How could you possibly know that the road went to infinity?  There is certainly no way to prove that it does.
OK, you could argue that this is only a thought experiment, but it pertains to something that, almost certainly, cannot exist.  However, let's stick with it for the time.

" You can paint a line across it and divide it into two infinite lengths,"

Literally, "infinite" means "without end".  When you paint your line you mark an end to the first part of your quasi-infinite road.  Beginning and end are dependent on subjective viewpoint, so all you need to do is turn round and your line marks an end to the other half of your road.  It was not my intention to get into etymological discussion, but you rather invite it in your comment about time. 

"..... then paint another line across, making two infinite lengths and one finite length. You can do this an infinite number of times in either direction."

No, you can't.  As you rightly point out; one length is finite, so however many times you repeat the action, you will never reach infinity, in fact, you will always be infinitely far from it.  How could something finite become infinite?

[Bill S]

It's not an 'all embracing now', because you're outside of time in that perspective; 'now' is a subjective experience of observers traversing the time dimension.

One of the difficulties involved in talking about infinity is that our terminology is rooted in linear time.  Suggest a better term for a timeless state and that will be a big step in the right direction. 

[Bill S]

(see Does Everything Possible Have To Happen?).

That's an interesting link, but most of what it says amounts to "everything that can happen, will happen, but not is it can't happen for some reason".

Also, it seems to assume that the same laws (e.g. gravity) that apply in our seemingly finite Universe would automatically in infinity.  Can that be justified?

[dlorde]

" Consider a road that stretches away from you to infinity in either direction."

Mathematically, this may be an acceptable thing to ask, but in reality, you are asking the impossible.

How could you possibly know that the road went to infinity?  There is certainly no way to prove that it does.
OK, you could argue that this is only a thought experiment, but it pertains to something that, almost certainly, cannot exist.

Well of course. All discussion about infinity is either mathematical or thought experiment. 

Literally, "infinite" means "without end".  When you paint your line you mark an end to the first part of your quasi-infinite road.  Beginning and end are dependent on subjective viewpoint, so all you need to do is turn round and your line marks an end to the other half of your road.

That's a semantic straw man. An infinite extent can start wherever you like. Consider the integers, or the real numbers; consider Hilbert's Hotel.

It was not my intention to get into etymological discussion, but you rather invite it in your comment about time. 

I don't follow you - which comment and how is it relevant?

"..... then paint another line across, making two infinite lengths and one finite length. You can do this an infinite number of times in either direction."

No, you can't.  As you rightly point out; one length is finite, so however many times you repeat the action, you will never reach infinity, in fact, you will always be infinitely far from it.  How could something finite become infinite?

I didn't say you will reach infinity, simply that you can repeat the operation an infinite number of times in either direction. Consider the integers as an analogy for the road. You can move in the positive and negative directions, 'marking' every 5th integer to infinity in either direction. You'll get an infinite number of finite sequences of 5 integers. You might also consider that between every pair of integers there is an infinite number of real numbers. The integers are countably infinite, the reals are uncountable; the positive integers make the smallest ordinal infinity, the uncountably infinite real numbers are a bigger ordinal infinity. It's fascinating stuff.

[dlorde]

One of the difficulties involved in talking about infinity is that our terminology is rooted in linear time.  Suggest a better term for a timeless state and that will be a big step in the right direction.

What's wrong with 'timeless state', or '4D block'? I sometimes use 'Parminidean block universe', because it's a reminder that these ideas are ancient, but it's a bit clumsy and can sound pompous. When you stand outside time in this way, you need to look at time as just another dimensional axis; 'now' and 'then' and 'future' and 'past' are points and directions relative to observers on that axis.

[dlorde]

(see Does Everything Possible Have To Happen?).

That's an interesting link, but most of what it says amounts to "everything that can happen, will happen, but not is it can't happen for some reason".

Not really. It's main point is that not all possible states of a system will necessarily occur even given an infinite time. That's why I linked it.

Also, it seems to assume that the same laws (e.g. gravity) that apply in our seemingly finite Universe would automatically in infinity.  Can that be justified?

It just took a particular example that used gravity to illustrate the point; in that hypothetical universe familiar laws applied (surely it would only confuse matters to try and illustrate a point with totally unfamiliar physical laws?).
Is there any reason to suppose that the physical laws familiar to us would not operate in a universe of infinite extent? As far as I know, we still have no definitive evidence that our own universe isn't infinite in extent beyond the observable horizon.

[Bill S]

Well of course. All discussion about infinity is either mathematical or thought experiment.

True, but this misses the salient point that infinite roads, infinite divisions and all other forms of the infinite series exist only in the (presumably finite) minds of those who think about these things.

That's a semantic straw man.

No. something that may be considered to have no end in one direction, but be clearly limited in the other may be said to be unbounded in one direction, but not infinite.  It is possible to argue reasonably and logically that something is unbounded, but to describe any physical thing, in our 4D reality, as infinite, without stipulating that you are talking about a mathematical, or pseudo, infinity is presumptuous and usually inaccurate.

I recall that a few years ago I wrote some notes about the Hilbert hotel.  Unfortunately I can't find them at the moment.  However, my recollection is that it is a clever mathematical illusion. 

Time has caught up with me again, but I'll try to pick up the thread tomorrow.  Sorry that responses are rather bitty, but that's how things are at the moment.

[dlorde]

Well of course. All discussion about infinity is either mathematical or thought experiment.

True, but this misses the salient point that infinite roads, infinite divisions and all other forms of the infinite series exist only in the (presumably finite) minds of those who think about these things.

By pointing out that it's either mathematical or thought experiment, I was emphasising the point that they are abstractions and not (necessarily) real-world considerations. But, whatever.

No. something that may be considered to have no end in one direction, but be clearly limited in the other may be said to be unbounded in one direction, but not infinite.  It is possible to argue reasonably and logically that something is unbounded, but to describe any physical thing, in our 4D reality, as infinite, without stipulating that you are talking about a mathematical, or pseudo, infinity is presumptuous and usually inaccurate.

There is a difference between unboundedness and infinity; infinity has the property of unboundedness in some respect (e.g. along some particular vector), but not all unboundedness is infinite, e.g. the surface of a sphere is unbounded but not infinite. Coincidentally, I'm currently on a short course at the University of Cambridge, on 'Philosophical Paradoxes', and we've just had a session on paradoxes of infinity. In particular, talking about Kant's paradox that there are compelling arguments both that the universe must be infinite in time of existence (the requirement for 'sufficient reason' for starting at some point), and that it cannot be infinite in time (this would make the present the end of an infinite series of events, and an infinite series cannot be completed). Much of the discussion of the second argument involved the choice of starting point for infinite sequences, in time (events), spatial extent, and in numbers (e.g. the positive integers start at 1 (or 0) and extend to infinity in unit increments; there is a lower bound, but no upper bound). We had no problem with starting points for an infinite series, sequences, or extents. Nobody suggested that they necessarily corresponded to any real-world contexts, these were all metaphysical abstractions, thought experiments.

[yor_on]

sounds fun dlorde. And on a totally unrelated question made by Pete. Try to search on < 'jammer.pdf' Max Jammer > for a taste. Not that I would advice anything more than a search naturally, but I did find something from that book in the middle of the search.

[Bill S]

There is a difference between unboundedness and infinity; infinity has the property of unboundedness in some respect (e.g. along some particular vector), but not all unboundedness is infinite, e.g. the surface of a sphere is unbounded but not infinite

I couldn't have put it better myself.

Unfortunately, it is quite common to see the surface of a sphere referred to as infinite.

this would make the present the end of an infinite series of events, and an infinite series cannot be completed

Since directionality is subjective, what sense does it make to talk of being able to start an infinite series? 

Nobody suggested that they necessarily corresponded to any real-world contexts, these were all metaphysical abstractions, thought experiments.

In maths and philosophy reality can be ignored.

"Gradually mathematicians lighted upon a new concept of existence.  Mathematical ‘existence’ meant only logical self-consistency and this neither required nor needed physical existence to complete it.  If a mathematician could write down a set of non-contradictory axioms and rules for deducing true statements from them, then those statements would be said to ‘exist’."    John D Barrow.
 
Enjoy your course.

[dlorde]

Since directionality is subjective, what sense does it make to talk of being able to start an infinite series?

I don't see what direction has to do with it - e.g. the positive integers are an infinite series starting at 0 (or 1); likewise the negative integers are an infinite series starting at 0 (or -1). Why should starting an infinite series be a problem?

Enjoy your course.

Thanks, it's been good so far...

[Bill S]

I don't see what direction has to do with it - e.g. the positive integers are an infinite series starting at 0 (or 1)

Consider your positive integers; move from 0 to 100.  Now turn round and go back the other way.  When you reach 0, you have come to the end of an infinite series.  It should take infinite time to reach the end of an infinite series. 

There are two problems here:

1.  There is no such thing as an infinite series.

2.  There is no such thing as infinite time.

It's not an 'all embracing now', because you're outside of time in that perspective; 'now' is a subjective experience of observers traversing the time dimension.

Does that not support statement 2?

[dlorde]

Consider your positive integers; move from 0 to 100.  Now turn round and go back the other way.  When you reach 0, you have come to the end of an infinite series.  It should take infinite time to reach the end of an infinite series.

As I said, it's the start of an infinite series. Just as 3.1415... is the start of the infinite series of digits that is the decimal representation of pi.

There are two problems here:

1.  There is no such thing as an infinite series.

2.  There is no such thing as infinite time.

I wouldn't mention 1. to a mathematician, they use infinite sequences and series all the time. Archimedes had a method for summing a decreasing infinite series before 212 BC; that task is now done with calculus. Perhaps you're taking the Aristotlean/Intuitionist view that infinities are potential rather than actual? 

Statement 2. is debatable - that's what Kant's paradox is about - he argued that logically, time both must be and could not be infinite. But he used separate arguments to do so.

Do you have any argument to support assertions 1. or 2. ?

It's not an 'all embracing now', because you're outside of time in that perspective; 'now' is a subjective experience of observers traversing the time dimension.

Does that not support statement 2?

I don't see it; how?

[Bill S]

I wouldn't mention 1. to a mathematician

Too late!  as far back as the late 1960s, when I worked in a residential school, I had several discussions with the maths teacher about infinite series.  Eventually he conceded that although the infinite series was a valid mathematical concept, with which I have no problem, the concept was not valid in the real world.

[Bill S]

One of the problems with long posts, especially multiples of long posts, is that people are inclined to skim over them, or even lose patience and not read them.  I am ashamed to confess, I sometimes do that myself when time is very short. 

We may have to be more selective in our points, and work towards added clarity, if we are ever to get anywhere with this.

Perhaps we could start with an opinion from any interested poster on the following:

Do the positive integers 1,2,3...., the negative integers -1,-2,-3..... and the real numbers (eg) between 0 and 1 constitute three infinite series?

[dlorde]

Do the positive integers 1,2,3...., the negative integers -1,-2,-3..... and the real numbers (eg) between 0 and 1 constitute three infinite series?

As I understand it, yes.

However I would much prefer that you outline your arguments to support statements 1 and 2 than go through some Socratic dialogue; perhaps the dialogue could follow the arguments so we know what we're debating.

[Bill S]

Do you have any argument to support assertions 1. or 2. ?

Let's start with an extract from "Part 5". If you can't find what you want we can progress from there:

"
Consider Cantor's infinities: the whole numbers constitute an infinite series, so do the even numbers and the odd numbers.  Thus, Cantor demonstrated that, not only were there numerous infinities, but they were not all the same size.  It is evident that the infinity containing the even, or odd, numbers must be half the size of the infinity containing the whole numbers.  Could it be that question is answered, that we can divide infinity and that any parts into which we divide it will be infinite?  There seem to be at least two reasons why this cannot be the case.  The first is that even Cantor does not seem to have performed mathematical calculations with the infinite set of all infinities; this appears to be the only one of his infinities that is not actually a mathematical infinity.  The other is that, practically there is the complication that anything that is truly infinite must contain everything; there cannot be two infinities, because each would have to contain the other. 

Applying the Reflection Principle to the infinite set of all infinities would lead to the following contradiction:  The reflection principle holds that within a universal set, containing all sets, it must be possible to find a set that contains any property found in the universal set.  The obvious contradiction is that the universal set contains all other sets (that is one of its properties), but this property cannot be found in any of the other sets.

Wikipedia says:  " In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[2] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite."

Cantor defined a countable infinity to be one that can be put into one-to-one correspondence with the list of natural numbers, whereas an uncountable infinity cannot.  Useful as these concepts may be to the mathematician, none is an "absolute" infinity, and cannot therefore be considered as more than "unbounded"."

[dlorde]

... It is evident that the infinity containing the even, or odd, numbers must be half the size of the infinity containing the whole numbers.

You may think it's evident, but what is intuitive isn't necessarily correct. The definition of an infinite set is that any proper subset has the same size as the whole set. The elements of the subset can be mapped one-to-one with the members of the whole set.

Could it be that question is answered, that we can divide infinity and that any parts into which we divide it will be infinite?

I already covered this.

... even Cantor does not seem to have performed mathematical calculations with the infinite set of all infinities; this appears to be the only one of his infinities that is not actually a mathematical infinity.

I don't know whether Cantor used the set of all infinite sets in his calculations (do you have a source for this?), but there are an infinite number of infinite sets, so it must be a mathematical infinity. What is your argument that it is not?

...anything that is truly infinite must contain everything; there cannot be two infinities, because each would have to contain the other.

What precisely does the truly in 'truly infinite' mean? It is generally accepted that there are multiple infinite sets; e.g. the real numbers are infinite, the whole numbers are infinite, neither set contains the other. If you introduce your own concept of 'truly infinite' that way, you're not talking about the same thing; and I don't see how it has any coherent meaning - can you explain? 

Applying the Reflection Principle to the infinite set of all infinities would lead to the following contradiction:  The reflection principle holds that within a universal set, containing all sets, it must be possible to find a set that contains any property found in the universal set.  The obvious contradiction is that the universal set contains all other sets (that is one of its properties), but this property cannot be found in any of the other sets.

That simplistic version of the Reflection principle is clearly self-contradictory for all classes of universal sets, and so is useless in that form. A description of a non-contradictory formulation is given here: Motivation for reflection principles.

Cantor defined a countable infinity to be one that can be put into one-to-one correspondence with the list of natural numbers, whereas an uncountable infinity cannot.  Useful as these concepts may be to the mathematician, none is an "absolute" infinity, and cannot therefore be considered as more than "unbounded".

Another custom infinity... so what does the qualifier 'absolute' mean in this context?

Infinity is a particular kind of unboundedness, it doesn't require 'more' than that.

Cantor himself defined an Absolute Infinite as that which transcended the transfinites (all other infinities). He said:

"The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type".
 
For him it was a kind of mathematical deity, possessing a reflection principle that every property of the Absolute Infinite is also held by some smaller object. Personally, I think this is a step beyond the coherent, but I'm no set theorist.

Was that what you had in mind? if not, what? and is your 'absolute' infinity different from what you call 'truly' infinite? if not, why use two names for it?

[lightarrow]

It should take infinite time to reach the end of an infinite series. 

I can do it in a finite time, and I can even prove it.