[dlorde]

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).

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.

[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).

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.

What I'm trying to point out is that there is a scientific problem with doing so.  If we're going to talk about the properties of this hypothetical frame in scientific terms, we need to have a scientific theory that covers it, i.e. the theory needs to describe the properties of that frame and be testable somehow.  Special relativity explicitly does not cover that reference frame and there is no way we know of to test the properties of such a frame (if it were to exist).  The best we can say scientifically is that we don't know if such a frame exists and don't know its properties if it were to exist.

[Bill S]

What did cantor do?

Cantor revolutionised the mathematical concept of infinity.  He lifted infinity from the realms of philosophy and theology and handed it to mathematicians and scientists.  He identified infinities that were "countable" and "uncountable"; that is, infinities that could be put in one-to-one correspondence with the list of natural numbers 1,2,3,4,5,6… , and infinities that could not.  So, for example, the even numbers are countably infinite, so are all the odd numbers.
   
Cantor defined all countable infinities as being the same size.  This seems to introduce something of a paradox, because the list of natural numbers is infinite, and the list of odd numbers is infinite, intuition would suggest that one of these must be half the size of the other.  We know that intuition is not always our best guide, so we should put aside intuition and look at how we can claim that these two infinities are the same size. the obvious answer is that we can go on putting one into one-to-one correspondence with the other for ever and we will not run out of either; but is that what infinity is really about?  We can do this in theory, but never in practice.   These infinities, undoubtedly, have their uses in the more esoteric realms of set theory and other branches of mathematics, but are of very dubious value when applied to the real world.  One of the major problems with trying to integrate general relativity and quantum theory is that the equations of one, when applied to the other tend to lead to infinities, so the equations become nonsense.

Initially, the mathematics community was not over enthusiastic about Cantor's work, but after a time, mathematicians said "cool", or "hoc frigidulum est" or whatever the expression of the time might have been.  Since then, Cantor has been widely quoted as having established that infinity was mathematically manipulatable. 

What did Cantor really do to infinities?  He discovered that there were ways of making infinities manageable by mathematicians.  However, even he accepts that this is only a partial victory.   

Barrow says: “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."

Cantor called this "Absolute infinity", he likened it to "God", but at the same time established that this absolute infinity did not exist.  Here is another paradox.  If we argue that "There is no Universe of universes that we can write down and capture." There is no greatest infinity.  Then, surely, we must argue that our so-called infinite series are not infinite, because they lead to no infinity "that we can write down and capture".  They are unbounded, because we can neither see nor imagine an end, but we cannot say that they are infinite in any sense other than mathematical.

[JP]

I suppose picking a "largest infinity"is a bit like picking a "largest number."  No matter what number or infinity you tell me, I can give you a bigger one.  You can get around this by defining a process rather than a number, such as "my number is X, which is defined as X=1/Y as Y tends to zero."  I can no longer pick a single number that is bigger than your X.  I wonder if you could find a "biggest infinity" in a loose sense through a limiting process like this.  For example, start with the set of all real numbers, and replace each number with the set of real numbers--then repeat without end.  I don't know if there are bigger infinite sets--my set theory is a bit limited and rusty. 

[Bill S]

my set theory is a bit limited and rusty. 

You're lucky, I don't have enough set theory to go rusty.

I suspect you are right about applying the same technique to infinities as to numbers.  Of course, you would not identify a "biggest infinity", any more than you would identify a biggest number. 

If cantor is right, there is no "biggest infinity", there are mathematical infinities, which are actually unbounded mathematical concepts.  Then there is "absolute infinity" which cannot be reached by any finite process, so, as far as mathematics is concerned, does not exist.

[yor_on]

A frame of reference practically, is you measuring locally (clock and ruler) relative some other. If you you use 'c' for the one measuring, then nothing makes sense anymore. The arrow of time must 'disappear', as the Lorentz contraction should be infinite, etc. But we 'see' photons, and they follow causality, macroscopically defined. It's all about coordinate systems, and their limits. 'c' is one.

[Bill S]

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.   

[Pmb]

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

Huh? I don't get it. Fill me in on the joke.

I have strong feelings about a lof of this stuff. Some of it's based in the article

Quantum Theory Needs No 'Interpretation, Christopher A. Fuchs and Asher Peres, Physics Today, March 2000

For anyone who'd like to read it PM your e-mail address to me and I'll e-mail it to you.

[JP]

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.

I can give you a broadside on this one. 

The problem with many of your points is that you're not being precise.  There are different ways to get at the concept of infinity.  Most common in physics is to mean something very big, which is represented by allowing numbers to increase without bound.  Very small is another option, in which things decrease without bound.  It's unknown if these concepts match nature: is the universe infinite in size, or can space be broken down into infinitesimally small pieces?  Regardless, the theories based upon them are accurate enough that we can get away with these uses of infinity.

1.  Infinity is not just a very big number.

2.  Eternity is not just a very long time.

Yes, I agree.  They are more along the lines of concepts of things or times increasing without bound.

3.  Something that is finite can never become infinite.

This is a blanket claim and I'm not sure it's provable.  I suspect that in physics, this is probably the case--at least for most things, but making a blanket statement like this is metaphysics, not physics.

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

I'm not sure what you mean by "mathematical" infinities.  There are certainly infinities that are in a sense bounded but infinite: for example the set of real numbers between 0 and 1 is bounded below by 0 and above by 1, but has an infinite number of elements.

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

I don't know enough math to derive a proof either way, but if absolute infinity contains all other infinite sets, then it has to at least be as big as any single one of them.  Since all those sets are infinite, absolute infinity must be infinite, if it exists.

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

As stated, this claim is false.  It is a mathematical process to add elements to a set.  I can add all real numbers to the set of all rational numbers and end up with a valid set, for example.  What you can't do is to pretend that "infinity" can stand in as a real number, since it's a concept, not a number.  1+infinity doesn't make sense since the "+" operation isn't defined for the concept infinity.

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

Don't know on this one.

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".

Kind of.  It's more general to go back to the idea that infinity is a concept, not a number, so you can't expect to apply operations which are defined for numbers to it. 

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

This is false.  You can add elements to an infinite set.  This is precisely what Cantor realized when he established different types of infinity.

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.   

Again, false.  You can take an element out of an infinite set and are left with an infinite set without that element.

Edit: Realized I had quoted your list twice.  :p

[Bill S]

Thanks JP.

Bit pushed tonight, but will have some comments asap.

[Bill S]

Fill me in on the joke.

No joke.  Just thought you would have damning things to say.