Strange but True: Infinity Comes in Different Sizes

[oxmixmudd]

I was not annoyed, your hypothetical situation remains unlikely, and weak as klaus54 noted.




The above is a little unclear and needs to explained. First, the universe does not 'somehow support nor yield to them.' Math concepts are derived in the abstract, and than they are used as part of a tool box of math in support hypothesis and theories.

IYO, Do the ideas of mathematics exists separate from the universe itself? Or are they derived from it? or Both?

Jim

[klaus54]

IYO, Do the ideas of mathematics exists separate from the universe itself? Or are they derived from it? or Both?

Jim

Both. Math is a great tool to use to model physical behavior. But higher maths is far more extensive than that. Much of higher math has zero application to physical reality, yet it's 100% consistent due to its axiomatic nature. E.g., I know of guy who wrote his Ph.D. dissertation on compact strictly-pseudoconvex Cauchy-Riemann manifolds. The math was entirely consistent, involving lots of tensor calculus, but no one but a complete nutter would believe that C-R manifolds have any connection to the physical world.

Back to SciFi-ish speculations:

If a you have an exact Doppelgaenger with your exact history in another universe, wouldn't that universe need to have the same physical constants as the one you're in? In particular it would discreteness via the Planck length/quantum foam limitations? In particular every bounded universe would be finite.

Calculus uses the limit concept but it's an abstraction applied to reality. In reality there are no topologically "dense" spaces -- or dense in "what" is a better question. BTW, limits of sequences work just as well using the rationals of the approach due the denseness of the rationals in the reals.

[Boxing Pythagoras]

Calculus uses the limit concept but it's an abstraction applied to reality. In reality there are no topologically "dense" spaces -- or dense in "what" is a better question. BTW, limits of sequences work just as well using the rationals of the approach due the denseness of the rationals in the reals.

I don't have a dog in the whole "multiverse" fight, but as someone with an interest in the philosophy and mathematics of infinities, I'll drop my two cents, here.

If space-time is continuous, rather than discrete, it would seem perfectly clear that it could be described by the Real numbers. If such is the case, then it follows pretty naturally that the properties of the Reals, including transfinite cardinality, are similarly applicable to the actual world.

As for the multiverse, it depends upon the particular multiverse model under discussion. If we were to look at, say, the Many Worlds interpretation of QM, then it is fairly simple to imagine a scenario in which uncountably many infinite parallel universes coexist. If a Wave function accurately describes a particular quantum event, and that Wave function is over a field of Real numbers, then on Many Worlds there would be an uncountably infinite number of universes being described by that Wave function.

[klaus54]

I have no problem with using uncountably infinite sets as domains and ranges of functions applied to the the real universe(s).

My point was that our knowledge in this universe at least is limited by the Planck time and length. Gravitation and electromagnetism smear into quantum foam at that scale. The Planck beezwax combined with the limitation of data storage makes the bounded part of our universe finite -- not even countably infinite.

That's all I was trying to get across.

[klaus54]

I don't have a dog in the whole "multiverse" fight, but as someone with an interest in the philosophy and mathematics of infinities, I'll drop my two cents, here.

If space-time is continuous, rather than discrete, it would seem perfectly clear that it could be described by the Real numbers. If such is the case, then it follows pretty naturally that the properties of the Reals, including transfinite cardinality, are similarly applicable to the actual world.

As for the multiverse, it depends upon the particular multiverse model under discussion. If we were to look at, say, the Many Worlds interpretation of QM, then it is fairly simple to imagine a scenario in which uncountably many infinite parallel universes coexist. If a Wave function accurately describes a particular quantum event, and that Wave function is over a field of Real numbers, then on Many Worlds there would be an uncountably infinite number of universes being described by that Wave function.

BTW, if there were an unlimited number of universes that (cardinal) number would of mathematical necessity be "only" countably infinite.

I'd love to see a sci-fi scenario that would refute this and be true to the mathematical notion of transfinite numbers.

N.B., this is a different notion than using the uncountably infinite sets of reals and complex numbers and their higher dimensional Cartesian products as domains and ranges of functions describing our universe.

[Boxing Pythagoras]

BTW, if there were an unlimited number of universes that (cardinal) number would of mathematical necessity be "only" countably infinite.

I'd love to see a sci-fi scenario that would refute this and be true to the mathematical notion of transfinite numbers.

N.B., this is a different notion than using the uncountably infinite sets of reals and complex numbers and their higher dimensional Cartesian products as domains and ranges of functions describing our universe.

I'm confused, here, because I actually gave an example of a scenario which is consistent with transfinite mathematics and which would result in an uncountably infinite number of actual universes.

Again, on the Many Worlds interpretation of QM, the Wave function describes a probability distribution. Every value in the range of that function is representative of a collection of actual, existing universes, on Many Worlds. Since the range of the function is an interval of Real numbers, it therefore follows that there exists an uncountably infinite quantity of extant universes, on Many Worlds.

I don't see any reason to think that an unlimited number of universes would be countably infinite by "mathematical necessity."

[seer]

I'm confused, here, because I actually gave an example of a scenario which is consistent with transfinite mathematics and which would result in an uncountably infinite number of actual universes.

Again, on the Many Worlds interpretation of QM, the Wave function describes a probability distribution. Every value in the range of that function is representative of a collection of actual, existing universes, on Many Worlds. Since the range of the function is an interval of Real numbers, it therefore follows that there exists an uncountably infinite quantity of extant universes, on Many Worlds.

I don't see any reason to think that an unlimited number of universes would be countably infinite by "mathematical necessity."

In case you missed it Boxing:

http://www.theologyweb.com/campus/sh...uestion/page38

[klaus54]

I'm confused, here, because I actually gave an example of a scenario which is consistent with transfinite mathematics and which would result in an uncountably infinite number of actual universes.

Again, on the Many Worlds interpretation of QM, the Wave function describes a probability distribution. Every value in the range of that function is representative of a collection of actual, existing universes, on Many Worlds. Since the range of the function is an interval of Real numbers, it therefore follows that there exists an uncountably infinite quantity of extant universes, on Many Worlds.

I don't see any reason to think that an unlimited number of universes would be countably infinite by "mathematical necessity."

Are these universes separated from each other? If so, is the concept of distance valid in the "inter-universe" space? If both of these are true, then how "in the world" could the set of universes be uncountably infinite?

Also, in what units are the domain and range of the wave function?

It would seem that if time and distance are involved then your scenario would involve domain interval values smaller than the quantum foam.

[klaus54]

Are these universes separated from each other? If so, is the concept of distance valid in the "inter-universe" space? If both of these are true, then how "in the world" could the set of universes be uncountably infinite?

Also, in what units are the domain and range of the wave function?

It would seem that if time and distance are involved then your scenario would involve domain interval values smaller than the quantum foam.

I guess I'm behind the times of the weird world of quantum physics. My mind still has a hard time NOT making a distinction between Multiverse and Many Worlds. The old-fashioned notion of the Multiverse as discrete "bubble universes" in a unending "bubble wrap" still pervades my thinking. In this scenario, the set of bubbles would be mapped 1-1 to {1, 2, 3, ...}

Sorry but I can't conceive of multiple universes overlapping in the same inflationary space-time.

Source: http://www.preposterousuniverse.com/blog/2011/05/26/are-many-worlds-and-the-multiverse-the-same-idea/

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© Copyright Original Source

[oxmixmudd]

Are these universes separated from each other? If so, is the concept of distance valid in the "inter-universe" space? If both of these are true, then how "in the world" could the set of universes be uncountably infinite?

Also, in what units are the domain and range of the wave function?

It would seem that if time and distance are involved then your scenario would involve domain interval values smaller than the quantum foam.

An interesting question Klaus. I don't know the answer, but one way of expressing Aleph 1 is 'all possible infinite subsets of Aleph 0' or the 'power set' of Aleph 0.

So I guess one requirement for 'uncountably many' universes would be a mechanism that produces infinitely many at 'a time', and all possible variations of them. Doesn't sound too plausible. But, as you pointed out, there is no problem with describing aspects of the universe in terms of the reals, which means that at least theoretically there could be aspects of each universe that could be represented from a set of transfinite numbers. I still go back to the circle. Except for the fact that we believe the universe to be quantized (which means any real circle will a some level look like we are setting up for some calculus proof of the area under a curve), the ratio of diameter to circumference IS a transfinite number in terms of its unquantized actual value. So it appears realistic to conclude that the properties of a universe, at least some of them, could be considered 'sourced' from the Reals.

As an extension then, if universes can have, in fact, real properties that can be sourced from (or map back to) any possible Real number, then the number of potential universes could be considered Aleph1, even though number of actual extant universes might never exceed Aleph0.



Jim

[shunyadragon]

I guess I'm behind the times of the weird world of quantum physics. My mind still has a hard time NOT making a distinction between Multiverse and Many Worlds. The old-fashioned notion of the Multiverse as discrete "bubble universes" in a unending "bubble wrap" still pervades my thinking. In this scenario, the set of bubbles would be mapped 1-1 to {1, 2, 3, ...}

Sorry but I can't conceive of multiple universes overlapping in the same inflationary space-time.

Source: http://www.preposterousuniverse.com/blog/2011/05/26/are-many-worlds-and-the-multiverse-the-same-idea/

...

© Copyright Original Source

I likewise cannot easily conceive of overlapping space times or universes. I believe that some distance is most likely but not necessary because of the influence of gravity involved would cause a distortion of the universes. There is no apparent influence of of the gravity of our universe from any other universes.

Yes there could be a difference between the emergent properties of possible universes, except it is present not known the range of possible differences between universes.

[klaus54]

An interesting question Klaus. I don't know the answer, but one way of expressing Aleph 1 is 'all possible infinite subsets of Aleph 0' or the 'power set' of Aleph 0.

So I guess one requirement for 'uncountably many' universes would be a mechanism that produces infinitely many at 'a time', and all possible variations of them. Doesn't sound too plausible. But, as you pointed out, there is no problem with describing aspects of the universe in terms of the reals, which means that at least theoretically there could be aspects of each universe that could be represented from a set of transfinite numbers. I still go back to the circle. Except for the fact that we believe the universe to be quantized (which means any real circle will a some level look like we are setting up for some calculus proof of the area under a curve), the ratio of diameter to circumference IS a transfinite number in terms of its unquantized actual value. So it appears realistic to conclude that the properties of a universe, at least some of them, could be considered 'sourced' from the Reals.

As an extension then, if universes can have, in fact, real properties that can be sourced from (or map back to) any possible Real number, then the number of potential universes could be considered Aleph1, even though number of actual extant universes might never exceed Aleph0.

Jim

Good post.

A point of clarification of the abstract: The power set of aleph_null has cardinality 2^aleph_null (AKA the Power of the Continuum =c), and to prove 2^aleph_null = aleph_1 requires the Axiom of Choice. With or without AoC, 2^aleph_null >= aleph_1. Similarly for 2^c >= aleph_2, so forth.

This highlights one of the my issues with applying the abstraction of transfinite cardinals (or ordinals!) to the "real" Cosmos.

The whole schmeer relies on axioms, the axiom of infinity being first and foremost. And "infinity" doesn't exist in our experience and ability to detect.

[klaus54]

I likewise cannot easily conceive of overlapping space times or universes. I believe that some distance is most likely but not necessary because of the influence of gravity involved would cause a distortion of the universes. There is no apparent influence of of the gravity of our universe from any other universes.

Yes there could be a difference between the emergent properties of possible universes, except it is present not known the range of possible differences between universes.

Yes indeed. What does "overlapping" or "superimposed" even mean in a (macro) physical sense?

If there is a space-time separation among universes, then mathematically that "Universal Set" would be at most countably infinite. If, as Jim pointed out, that perhaps an "infinite number" of universes exist in each discrete bubble, then the cardinality of the set of all universes within the "Universal Set" would equal the maximal cardinality within all of the bubbles, as in the strangeness of transfinite arithmetic, aleph_0 X aleph_n = aleph_n, n>=0.

Anyhoo, doesn't the possibility of a (power of the) continuum of universes within a bubble depend on the Many World Interpretation which is kinda sorta sketchy at best?

[Boxing Pythagoras]

Are these universes separated from each other? If so, is the concept of distance valid in the "inter-universe" space? If both of these are true, then how "in the world" could the set of universes be uncountably infinite?

They are separated in Hilbert space. It is possible to discuss a concept analogous to "distance" or displacement in Hilbert space. If both are true, then the set of universes would be uncountably infinite provided the Hilbert space which housed them was continuous.

Also, in what units are the domain and range of the wave function?

The dimension of the Wave function is , where N is the number of particles in the system and d is the dimensionality of space.

It would seem that if time and distance are involved then your scenario would involve domain interval values smaller than the quantum foam.

As I noted, I was presuming that space-time is continuous. If space-time is not continuous, but rather discrete and "foamy," then this example would not be valid.

I guess I'm behind the times of the weird world of quantum physics. My mind still has a hard time NOT making a distinction between Multiverse and Many Worlds. The old-fashioned notion of the Multiverse as discrete "bubble universes" in a unending "bubble wrap" still pervades my thinking. In this scenario, the set of bubbles would be mapped 1-1 to {1, 2, 3, ...}

Completely agree, here. This sort of multiverse would seem to be countably infinite, it seems to me.

Sorry but I can't conceive of multiple universes overlapping in the same inflationary space-time.

I would say that Many Worlds doesn't really propose such a thing. Rather, on Many Worlds (and, again, providing that we are operating upon a continuous space), it would seem that the possible values in the range of the Wave function represent wholly distinct space-times. Imagine a Real number line in which every single point on the line represented a distinct space-time. That's fairly analogous to what I am saying.

[klaus54]

They are separated in Hilbert space. It is possible to discuss a concept analogous to "distance" or displacement in Hilbert space. If both are true, then the set of universes would be uncountably infinite provided the Hilbert space which housed them was continuous.

The dimension of the Wave function is , where N is the number of particles in the system and d is the dimensionality of space.

As I noted, I was presuming that space-time is continuous. If space-time is not continuous, but rather discrete and "foamy," then this example would not be valid.

Completely agree, here. This sort of multiverse would seem to be countably infinite, it seems to me.

I would say that Many Worlds doesn't really propose such a thing. Rather, on Many Worlds (and, again, providing that we are operating upon a continuous space), it would seem that the possible values in the range of the Wave function represent wholly distinct space-times. Imagine a Real number line in which every single point on the line represented a distinct space-time. That's fairly analogous to what I am saying.

Well, the detectable part of our universe is "foamy" due to the breakdown of gravitation and EM at the Planck level, thus finite.

The wave function's domain is space and time (I think), both of which are "foamy" in our universe.

Is possible for "anything" to exist at lengths less than the Planck Length or time intervals less than the Planck Time (roughly 10^(-35) m and 10^(-43) s, respectively)?