[dlorde]

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..

Yes. That's why I said I thought it was a step beyond the coherent. It is possible he was poorly paraphrased and meant each property of the Absolute Infinite is also held by some smaller object, but even this seems incoherent unless you explicitly exclude the property of 'absoluteness' (and probably some others).

... 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

It's not my cup of tea, but - fine as long as you acknowledge that necessarily makes it metaphysics, not mathematics. Cantor thought Absolute Infinity was mathematical.

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

I don't know what you mean by 'infinite judgements'.

[Bill S]

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

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? 

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.

Cantor thought Absolute Infinity was mathematical.

He also established that it could not exist.

This is why I have been trying to stress the difference between mathematical infinities and physical infinity. 

[dlorde]

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

Why interesting? It seems to me that infinity is a mathematical concept that can be used in physics. There may or may not be infinities in the real world - I don't see how we could know.

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

Depends what you mean by a 'mathematical "reality"'; but assuming your question isn't tautologous (if something is 'real', that usually means physically real), I'd say no, not at all - that's why I said 'mathematics' rather than 'physics'.

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

It seems very easy to lose sight of that fact.

If you say so.

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.

I wouldn't argue with that, although it seems slightly loaded - i.e. if the universe does fundamentally operate in a mathematical way (or if maths does describe it's behaviour precisely), is it governed by maths or does it define maths? It seems a question of semantics. After all, where did the fundamentals of mathematics originate, if not observation of, and interaction with, the universe.

Eugene Wigner had quite strong views on this: The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Cantor thought Absolute Infinity was mathematical.

He also established that it could not exist.

Do you have a source for that? my understanding as that he initially called the set-theoretic universe (including transfinites) 'the Absolutely Infinite', and thought that it could be acknowledged but not known, or even approximated. Later, he revised this, calling the set-theoretic universe and other proper classes 'inconsistent multiplicities' or 'absolutely infinite':

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. Not that it really matters...

[Bill S]

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

Look closely at the wording of the quote, it would be possible to say almost anything without abandoning it altogether,  

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

[dlorde]

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

No.

[Bill S]

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

[dlorde]

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

Speed is distance over time, and increases as time decreases. So speed tends to infinity as time tends to zero; although mathematically dividing by zero is undefined, not infinity, one could define infinite speed as traversing a distance in no time, i.e. instantaneously.

If by 'can it exist?', you're asking if something in the real world can traverse a distance instantaneously, there are relativistic considerations. No physical object can accelerate to or past the speed of light, but if you consider a photon to have its own valid frame of reference, its 'journey' in that frame would appear to be instantaneous. The only other apparently instantaneous effect over distance I can think of is the decoherence of quantum entanglement.

[JP]

Average speed is distance traveled over time taken to travel that distance.  Instantaneous speed is distance over time in the limit as time goes to zero.  As always, dealing with limits requires care.  As time decreases to zero, distance traveled also decreases to zero.  Their ratio as they get tiny determines instantaneous speed.  An "infinite" speed would correspond to an object whose distance traveled did not decrease to zero as the time interval over which it was traveling did decrease to zero.  As an example, if an object moved at least 1 meter no matter how tiny a time interval you measured, that object would have infinite speed.  This would violate relativity  and doesn't seem particularly physical, so I doubt it exists in reality. 

I'm not sure where "infinite speed" gets referenced, but I can't recall seeing any references to it. 

[Bill S]

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

Here's one:   “So if a tachyon were created in some violent event in space, it would radiate energy away furiously…..and go faster and faster, until it had zero energy ……and was travelling at infinite speed”.

      Gribbin John.  Companion to the Cosmos.  Phoenix (Orion Books Ltd.), London.  1996.

[Bill S]

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.  It seems hardly credible that it would suddenly reach a point where its speed would no longer be measurable.  What could happen to bring about this change?   

We might argue that we already have the answer to that:  " An "infinite" speed would correspond to an object whose distance traveled did not decrease to zero as the time interval over which it was traveling did decrease to zero."  [JP].  Would that not imply that it must be everywhere at once?  Could we not argue that if tachyons exist, either there is only one, that is everywhere; or, every tachyon in existence is "here" all the time?

[Bill S]

   What if we consider infinite acceleration?  In that scenario the tachyon would just continue to accelerate for ever.  Certainly this would make more sense, but then, it would never actually reach infinite speed, it would simply continue striving for it, and a more accurate term might be unlimited/ unbounded acceleration, rather than infinite acceleration, because, although we could imaging the acceleration going on for ever, it could never reach a point where we could say: "now it is infinite", unless we accept JP's definition (above), in which case, there are some scientists who would equate this with the speed of light.  That would lead to the absurd situation in which a tachyon would accelerate from the speed of light to the speed of light. 

[dlorde]

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.

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

[dlorde]

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

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.

[Bill S]

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

I'll rethink! 

Tachyons are hypothetical particles that, hypothetically, travel faster than light.  Any measurement of their hypothetical speed would, hypothetically, have to be made by some hypothetical measuring device that, hypothetically, inhabited the hypothetical realm of which tachyons are hypothetical denizens.

Does that put us on the same page? 

Back to the serious stuff a little later, I hope.

[JP]

Our classical concept of speed probably doesn't work that well for tachyons, even if they somehow did exist.  All Bradyons (particles with real, non-zero mass) travel slower than light.  One explanation for this is that they can achieve zero momentum but still have energy (from E=mc²).  An object with energy moves through time and an object with momentum moves through space.  So a Bradyon always moves through time but can stop moving through space, obtaining a zero velocity.

A tachyon is in some ways the opposite.  It can obtain zero energy, but it always has non-zero momentum.  At zero energy, it satisfies (mc=p where p is momentum).  This means it can somehow stop moving through time and move only through space.  I'm not sure what this means intuitively, but that's what the equations say.

[Bill S]

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.

we're definitely on the same page here.  However, you seem to have missed the point, perhaps because I was not clear enough.

The point I was aiming for was that if we accept JP's definition of infinite speed, which seems quite reasonable to me, a tachyon (at full speed?!) would not experience time.  Some people, including some scientists, believe that this is the case with the photon, which travels at c.  Since we cannot prove, either that the photon does not experience time, or that the tachyon, if it exists, has mass, it must be acceptable to theorise that 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?
 

[Bill S]

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

Intuitively, it means very little to me, but, sadly, the equations would probably mean even less.

Congratulations on yet another explanation that does mean something!

I've been having some thoughts about tachyons and their relationship to the Universe.  If they still make sense to me when I get them together, and if I can work out how to include diagrams in posts, I might have a go at a new theory.

[dlorde]

if it exists, has mass...

Imaginary 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?

It's a bit opaque (see bolding - surely either it 'experiences' time or it doesn't). 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. Acceleration apart, it looks like a way of saying for a zero-energy tachyon what I proposed earlier for the photon: "... if you consider a photon to have its own valid frame of reference, its 'journey' in that frame would appear to be instantaneous".

[Bill S]

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

What is identical to experiencing no time?  It must be experiencing no time.

That's why I asked: "Does that make sense?"

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.

Rightly so.  I was simply pointing out the two definitions in question.

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

Convenient as it would be, at times, to consider a photon to have its own valid frame of reference, such would seem not to be the majority view in scientific circles. 

"Therefore, just as bradyons are forbidden to break the light-speed barrier, so too are tachyons forbidden from slowing down to below c, because infinite energy is required to reach the barrier from either above or below."   (From your link)

Presumably, they would be able to travel at c if they were massless; but then, would they be able to travel at any other speed? 

Of course, we must not forget that this is all hypothetical, but I think I'm beginning to like the possibility of a link between c and infinite speed.  That could be good for some crackpottery! 

[JP]

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).  The reason we can discuss what something experiences is that we can compare it's reference frame (a frame in which it is at rest) to another reference frame (in which it's in motion).  Length and times change as measured from the ref. frame of the moving observer.  We cannot do this for a photon since there exists no reference frame in which its at rest--we can't find such a frame to compare to other observers.  It doesn't exist in special relativity.  Perhaps a post-SR theory will describe it.

The usual claim that photons are timeless (which I've occasionally seen in pop-sci books) comes from mis-applying the equations of special relativity which assume that the photon is at rest in some reference frame.

I'm not sure how to think about a tachyon's experience, since it can't ever exist in the reference frame of a bradyon (which we are).  However, there should be frames in which it's at rest, so you might be able to compare it to other tachyons and figure out what it experiences.  I'll have to think about it more.