[dlorde]

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.

Cool, let's see the proof.

[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.

Cool, let's see the proof.

Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)^n-1 seconds to reach the n-th term, ...
Summing all the times, I reach the end of the series in 2 seconds.

[Bill S]

Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)n-1 seconds to reach the n-th term, ...Summing all the times, I reach the end of the series in 2 seconds.

No; because when you reach the n-th term, or any other, you are still infinitely far from the (non-existent) end.

[lightarrow]

Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)n-1 seconds to reach the n-th term, ...Summing all the times, I reach the end of the series in 2 seconds.

No; because when you reach the n-th term, or any other, you are still infinitely far from the (non-existent) end.

While you worry about it, I have already reached the end in 2 seconds, as I wrote. If you don't believe it, explain how you could still be in some term of the series after 10 seconds...

[Bill S]

While you worry about it, I have already reached the end in 2 seconds, as I wrote. If you don't believe it, explain how you could still be in some term of the series after 10 seconds...

Presumably, you can't; especially if your name happens to be Zeno, but then, you have not explained how you actually reach 2 seconds. 

I am in the fortunate position of not having to explain anything, because I don't accept the infinite series outside mathematics, nor do I believe you can reach infinity.  If you can arrive at the end of it, it is not infinite.

[Bill S]

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.

Surely this is the definition of a countably infinite set, so, at best it is part of the definition of 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?),

Incautious wording on my part.  It would have been better to have said something like: "In my limited knowledge of 2the work of Cantor, I am not aware that he actually used the set of all infinite sets in his calculations. If anyone can provide an example of his so doing I would give it my attention".  That would make more sense than my trying to give examples of where he didn't. 

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?

I used the term "truly" infinite so as not to confuse what I was talking about with "absolutely" infinite.  I did this because it is easy to argue, as you did, that absolute infinity is a mathematical infinity.  Indeed, I was not " talking about the same thing".   

There is also something of a question mark over the concept of absolute infinity. "...  when it is realized in the most complete form". What is the most complete form of infinity?  Does it contain all other infinities?  If not, in what sense is it absolute?  If it does, how can Barrow say:

"Cantor’s most dramatic discovery was that infinities are not only uncountable, they are insuperable.  He discovered that a never-ending ascending hierarchy of infinities must exist.  There is no biggest of all that can contain them all.  There is no Universe of universes that we can write down and capture." 
 

[dlorde]

Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)^n-1 seconds to reach the n-th term, ...
Summing all the times, I reach the end of the series in 2 seconds.

Ah, Zeno would be proud 
Problem is, you're trying to do an infinite number of actions in a finite time, and each action takes a finite time (also a finite amount of energy).
From a physics viewpoint, the quantisation of energy (and probably time) prevents you going beyond a certain point.

[dlorde]

Surely this is the definition of a countably infinite set, so, at best it is part of the definition of a mathematical infinity.

The strict definition is 'A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number'. So it covers all infinite sets.

I used the term "truly" infinite so as not to confuse what I was talking about with "absolutely" infinite.  I did this because it is easy to argue, as you did, that absolute infinity is a mathematical infinity.  Indeed, I was not " talking about the same thing".

You have yet to define, describe, or distinguish between them.  Were you talking about Cantor's Absolute infinity?

[lightarrow]

Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)^n-1 seconds to reach the n-th term, ...
Summing all the times, I reach the end of the series in 2 seconds.

Ah, Zeno would be proud 
Problem is, you're trying to do an infinite number of actions in a finite time, and each action takes a finite time

Infact it was exactly showing that the total time were finite, that Zeno solved the paradox  []
Every action takes a finite time, indeed. Infact (1/2)(n-1) is *always* a finite time. But their sums is finite as well...
You have to go over calculus?

[JP]

I admit I haven't been following this thread that closely, so if I'm repeating someone else's thoughts, I apologize. 

Lightarrow is correct that at any particular step you take a finite amount of time, but the total time taken if you complete the infinite number of steps is also finite.  The reason its counterintuitive is that our brains aren't built to naturally handle the concept of infinity, so it takes training and practice to get comfortable with these ideas.  (If we could deal easily with infinities, calculus wouldn't be such a painful subject!)

The other question being discussed here seems to be whether infinity is physically real or not: can we find examples of infinity in nature or is the mathematics of infinity just a useful approximation to very big or very small things?  We don't know the answer to that.  What we do know is that current models do assume infinitely small things exist since space is continuous.  If I give you a length, you can always chop it in half in our current models and that is the definition of something getting infinitesimally small.  From that, Lightarrow's example/Zeno's paradox follows automatically.  Since we can't currently measure things past some very small scale, we don't know if space and time are really continuous or if there are some smallest building blocks.  In the latter case, infinitesimally small might not exist.

[dlorde]

Lightarrow is correct that at any particular step you take a finite amount of time, but the total time taken if you complete the infinite number of steps is also finite.

But, of course, you can't complete an infinite series...

What we do know is that current models do assume infinitely small things exist since space is continuous.

Not all current models assume that; for example, Loop Quantum Gravity is quite popular, and potentially resolves the problem of singularities (by removing them).

[dlorde]

You have to go over calculus?

??

[JP]

Lightarrow is correct that at any particular step you take a finite amount of time, but the total time taken if you complete the infinite number of steps is also finite.

But, of course, you can't complete an infinite series...

Of course you can!  Lightarrow just did so mathematically.  You can't write down all the terms explicitly, but you can complete it or write the series symbolically as he did.

What we do know is that current models do assume infinitely small things exist since space is continuous.

Not all current models assume that; for example, Loop Quantum Gravity is quite popular, and potentially resolves the problem of singularities (by removing them).

True, but those are speculative hypotheses currently.  The primary models that have been well tested and accepted (the standard model, general relativity) do assume continuous space and time.  That doesn't mean anything more than that if they are granular, the granularity is small enough that we haven't seen it yet on the scales of those models.  But on the other hand there's no evidence for granularity. 

Taking a firm stance the existence or nonexistence of physical infinities is stating an opinion without evidence to back it up.  Mathematical infinities, on the other hand, clearly exist.

[dlorde]

Of course you can!  Lightarrow just did so mathematically.  You can't write down all the terms explicitly, but you can complete it or write the series symbolically as he did.

Summing it, or writing it symbolically isn't what I had in mind. What exactly do you mean by 'complete'? It seems to me that if you could complete it, you could give me the value of the final term - but there isn't one.

[yor_on]

You can take a stretch of line, 13 cm, then split it in 13 even chunks, then split those into 26, then those into 52. Where should it end? Doesn't this depend on what mathematics you use. Probably also on where those mathematics stop producing meaningful answers. But I think you can define a infinity to a finite stretch.
=

It also goes back to if there is a 'granularity' to SpaceTime I think. If it is a 'flow' then you won't find a stop, and it should be possible to zoom in for ever. Or you can define it from what is meaningful, in which case you get two answers. One where it stops being meaningful at some scale, another that ignores what we find meaningful.
==

There is actually a third too, as I think, and that would be if it was a fractal describing a SpaceTime. On the other tentacle that should be real close to a definition of a flow.

[JP]

Of course you can!  Lightarrow just did so mathematically.  You can't write down all the terms explicitly, but you can complete it or write the series symbolically as he did.

Summing it, or writing it symbolically isn't what I had in mind. What exactly do you mean by 'complete'? It seems to me that if you could complete it, you could give me the value of the final term - but there isn't one.

He completed it by writing down an expression for all the terms and then summing them.  The set 1/2ⁿ, when n is a whole number is the expression for all the terms.  Like I said, explicitly writing out all the terms is impossible physically, since we can't write arbitrarily fast.  But since both of these methods are abstract mathematical notation for the series, why should we afford one more importance than the other in a mathematical sense?  In a physical sense, the existence of infinities is a matter of opinion whether continuous things exist or not or whether the universe is infinitely large or not--at least until we get some evidence to back up theories one way or the other.

[dlorde]

He completed it by writing down an expression for all the terms and then summing them.  The set 1/2ⁿ, when n is a whole number is the expression for all the terms.  Like I said, explicitly writing out all the terms is impossible physically, since we can't write arbitrarily fast.  But since both of these methods are abstract mathematical notation for the series, why should we afford one more importance than the other in a mathematical sense?  In a physical sense, the existence of infinities is a matter of opinion whether continuous things exist or not or whether the universe is infinitely large or not--at least until we get some evidence to back up theories one way or the other.

OK, fair enough. Defeated by the semantics of 'complete', I withdraw my statement.

However, I still can't see how an infinite number of actions can be completed in a finite time, since an action requires energy and there is a minimum energy of action (i.e. the quantum of energy transfer). Though my understanding of 'action' in this context (a force doing work on an object) may be astray too.

[JP]

Certainly there's no problem with the mathematics, since mathematics exists in a world of thought and doesn't necessarily reflect anything physical. 

Physically speaking, if space and time are continuous, then lightarrow's example is physical--if you move 2 meters, you move 1 meter + 1/2 meter +1/4 meter + 1/8 meter, etc.  Moving 1/2ⁿ meters takes 1/2ⁿ time.  Since the time and distance get arbitrarily short, there's no problem doing this.  Is space and time have some granularity and a smallest unit, then lightarrow's example will run into problems since there exists some 1/2ⁿ length that is a minimum and the smaller terms in the series don't represent anything physical.

Whether arbitrarily short time and distance can physically exist in nature is another question, and I agree with you that it may not be physical.  Personally, I tend to think it is not physical and that time and distance are granular on some scale, but I have to admit that there isn't any good evidence to support my opinion, although some hypotheses in physics propose it.  Our best well-supported models right now assume continuous space and time, but we haven't checked that they're valid on arbitrarily short length and time scales.

[Bill S]

Just letting everyone know I have not abandoned this thread.  I've been off line for a few days, and may continue to be for a couple more.  However, I'm looking forward to catching up, and joining in again asap.

[Bill S]

Were you talking about Cantor's Absolute infinity?

What is Cantor's Absolute infinity?  Is it the ultimate infinity, beyond which it is not possible to go?  If not, as I already asked, in what sense is it absolute?  If it is, then it does not exist, because Cantor established that there was no overarching infinity.

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.

If " every property of the Absolute Infinite is also held by some smaller object", then there must be a "smaller object" that is absolutely infinite, which, at best, causes problems; and, at worst, makes no sense.   

Personally, I think this is a step beyond the coherent, but I'm no set theorist.

I am no set theorist, either, but, apart possibly from the creation of a mathematical deity, I see no problem with the concept of an infinity that is transcendent.   It must contain all other infinities, because it must contain everything.  It cannot be manipulated by mathematics, because it must contain mathematics. 

I feel sure you will object to that last assertion, but think about it; it may be perceived by us as capable of being manipulated mathematically, but we are finite, or so it appears to us, so how can we make infinite judgements about something which according to Cantor's insights into mathematical infinities, cannot exist?