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Quote from: Bill S on 02/06/2013 23:06:00It 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.

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

Quote from: lightarrow on 05/06/2013 22:37:17Quote from: Bill S on 02/06/2013 23:06:00It 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.

Quote from: ligharrowOk. 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...

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.

I don't know whether Cantor used the set of all infinite sets in his calculations (do you have a source for this?),

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?

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.

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

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".

Quote from: lightarrow on 06/06/2013 10:01:17Ok. 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

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.

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

You have to go over calculus?

Quote from: JP on 07/06/2013 13:36:43Lightarrow 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...

QuoteWhat 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).

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.

Quote from: JP on 07/06/2013 19:10:45Of 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 1/2^{n}, 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.

Were you talking about Cantor's Absolute 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.

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

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..

... 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

... we are finite, ... so how can we make infinite judgements about something which according to Cantor's insights into mathematical infinities, cannot exist?

fine as long as you acknowledge that necessarily makes it metaphysics, not mathematics.

mathematics exists in a world of thought and doesn't necessarily reflect anything physical.

Cantor thought Absolute Infinity was mathematical.

Interesting that you say " metaphysics, not mathematics", rather than "metaphysics, not physics".

Have we reached a point where physics is so ruled my mathematics that a mathematical "reality" automatically becomes a physical reality?

Quote from: JP mathematics exists in a world of thought and doesn't necessarily reflect anything physical. It seems very easy to lose sight of that fact.

Mathematics is, undoubtedly, the language of nature, but I suspect that is because mathematics is the best language we have found to describe nature, rather than because it actually governs nature.

Quote Cantor thought Absolute Infinity was mathematical. He also established that it could not exist.

A multiplicity can be of such nature, that the assumption of the togetherness/combining of its elements leads to a contradiction, so that it is impossible to conceive the multiplicity as a unity, as a ﬁnished/completed thing. I call such multiplicities absolutely inﬁnite or inconsistent multiplicities. [Letter to Dedekind]

I haven't seen anything to suggest he abandoned it altogether

Have you read: " Barrow. John D. The Infinite Book. Vintage, Random House, London 2005"?

It is not unusual to find references to "infinite speed". How would one define infinite speed? Can it exist?

I'm not sure where "infinite speed" gets referenced

One of the characteristics of infinite speed must be that it would be immeasurable. Consider what this implies: Prior to becoming infinite the tachyon’s speed would be measurable.

... That would lead to the absurd situation in which a tachyon would accelerate from the speed of light to the speed of light.

A tachyon, in this context, is an hypothetical faster-than-light particle with imaginary mass. How would its speed be measurable at all?

As I understand it, a tachyon would never be able to reach the speed of light; its energy-velocity relation would be a mirror of normal particles, its velocity increasing as its energy decreases. It would require infinite energy to decelerate to c (just as a normal particle would require infinite energy to accelerate to c), so it could only exist by moving FTL.

I'm not sure what this means intuitively, but that's what the equations say.

if it exists, has mass...

... the tachyon accelerates away from c, where the photon, and possibly the tachyon, experience no time, and arrives at a point where its experience of time is identical to that at its starting point. Does that make sense?

.....and possibly the tachyon, experience no time, and arrives at a point where its experience of time is identical.....

If you're saying the tachyon arrives at some point having accelerated from c, but no time has elapsed in its frame of reference, I would have to query your definition of acceleration.

"... if you consider a photon to have its own valid frame of reference, its 'journey' in that frame would appear to be instantaneous".

I've mentioned this before, but you have to be very careful about applying the idea of reference frame to a photon (and presumably tachyons).

Quote from: JP on 22/06/2013 00:14:23I've mentioned this before, but you have to be very careful about applying the idea of reference frame to a photon (and presumably tachyons).I agree - perhaps I should have emphasised the 'if' in ".. if you consider a photon to have its own valid frame of reference...". I suppose it's an intuitive attempt to understand photons in familiar terms.

my set theory is a bit limited and rusty.

Possibly the discussion of infinity is wearing a bit thin. No anticipated broadside from Pete as yet.

Possibly the discussion of infinity is wearing a bit thin. No anticipated broadside from Pete as yet. I think this (in part) is where I am at present.

1. Infinity is not just a very big number.2. Eternity is not just a very long time.

3. Something that is finite can never become infinite.

4. Mathematical infinities are theoretical concepts that are unbounded, but not infinite.

5. Cantor’s “absolute infinity” may be infinite, but this cannot be proved nor disproved.

6. Unbounded entities may be subjected to mathematical processes, but attempting to do this to infinity leads to nonsensical answers.

7. There cannot be more than one true (absolute?) infinity, because it must contain everything.

8. Multiplying or dividing infinity makes no practical sense because the result would have to be infinite, and there cannot be more than one "everything".

9. Practically, nothing can be added to infinity, because it is already everything.

10. Nothing can be taken away from infinity, because the remaining quantity would still be infinite, therefore it makes no sense to talk of something being taken away.