Strange but True: Infinity Comes in Different Sizes

[shunyadragon]

Is [it] 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)?

I believe that based on the present knowledge nothing exists less than one Plank length nor time. Than again, this may be interpreted that it is likely nothing would be detectable in our physical existence less than one Plank length nor time.

[Boxing Pythagoras]

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.

The detectable part of our universe may be "foamy," but it is far from certain that this is so. It just as easily may be continuous. Until we have a better understanding of the nature of space-time at quantum scales-- and especially the nature of gravity at such scales-- the question of whether space-time is discrete or continuous remains open.

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)?

Sure, it's possible. Again, we do not yet know whether space-time is discrete or continuous. If it is continuous, then intervals smaller than the Planck length and Planck time are perfectly meaningful.

[klaus54]

The detectable part of our universe may be "foamy," but it is far from certain that this is so. It just as easily may be continuous. Until we have a better understanding of the nature of space-time at quantum scales-- and especially the nature of gravity at such scales-- the question of whether space-time is discrete or continuous remains open.

Sure, it's possible. Again, we do not yet know whether space-time is discrete or continuous. If it is continuous, then intervals smaller than the Planck length and Planck time are perfectly meaningful.

If it's continuous, then any reality less than the Planck limits would be unknowable.

Also, again... "Infinity" is an axiom of set theory. That means it can't be proved. But once it's assumed, higher orders of infinity follow from a theorem about the cardinality of power sets. However, even then another axiom is required to show that next order of infinity is equipollent to the power set of any set of the previous order.

Using uncountably infinite sets as domains and ranges of functions of physics is all well and good, but it doesn't follow that a function's value at an arbitrary member of its domain has a physical meaning. The Planck time and length certainly effect "first order" physical limits.

Perhaps physical phenomena exist at time intervals less than 10^(-44) s and 10^(-36) m, but I don't see how they would be detected.

My main point is: Mathematical abstractions don't necessarily map to physical reality. E.g., Cauchy-Riemann manifolds are wonderfully rich mathematically yet have no application to our universe of matter and energy and space-time.

[Boxing Pythagoras]

If it's continuous, then any reality less than the Planck limits would be unknowable.

Not necessarily. Our current models seem to indicate that the Planck scales place hard limits on what we can measure, but it doesn't therefore follow that phenomena at those scales would be wholly unknowable. As an analogue from Classical physics, we'll never be able to measure any phenomena at the event horizon of a Black Hole, but it does not therefore follow that we cannot know anything about phenomena which occur at that event horizon.

Bringing it back to quantum scales, even if the Classical idea of spacetime is "foamy," it's still possible that physical dimensions at the quantum scale are continuous. For example, the one-dimensional strings of M-Theory may be continuous, despite the fact that they result in the Planck scale limitations which we predict on the ability to measure reality.

Also, again... "Infinity" is an axiom of set theory. That means it can't be proved. But once it's assumed, higher orders of infinity follow from a theorem about the cardinality of power sets. However, even then another axiom is required to show that next order of infinity is equipollent to the power set of any set of the previous order.

I completely understand that set theory is axiomatic. All formal logic is axiomatic. That doesn't imply that it is therefore inapplicable to reality.

Using uncountably infinite sets as domains and ranges of functions of physics is all well and good, but it doesn't follow that a function's value at an arbitrary member of its domain has a physical meaning. The Planck time and length certainly effect "first order" physical limits.

I agree. This is, again, why I explicitly stated that I was assuming a continuous space, and that the Real-valued function accurately modeled reality (as opposed to approximating it). Yes, it is entirely possible that space is discrete and that uncountably infinite quantities are wholly inapplicable to the real cosmos. However, it is just as possible that space is continuous and that uncountably infinite quantities are applicable to the real cosmos.

My main point is: Mathematical abstractions don't necessarily map to physical reality. E.g., Cauchy-Riemann manifolds are wonderfully rich mathematically yet have no application to our universe of matter and energy and space-time.

Certainly! I have not claimed, nor would I claim, that mathematical abstractions necessarily map to physical reality. I simply claimed that it is possible that certain mathematical abstractions regarding infinite sets could map to physical reality.

[klaus54]

Not necessarily. Our current models seem to indicate that the Planck scales place hard limits on what we can measure, but it doesn't therefore follow that phenomena at those scales would be wholly unknowable. As an analogue from Classical physics, we'll never be able to measure any phenomena at the event horizon of a Black Hole, but it does not therefore follow that we cannot know anything about phenomena which occur at that event horizon.

Bringing it back to quantum scales, even if the Classical idea of spacetime is "foamy," it's still possible that physical dimensions at the quantum scale are continuous. For example, the one-dimensional strings of M-Theory may be continuous, despite the fact that they result in the Planck scale limitations which we predict on the ability to measure reality.

I completely understand that set theory is axiomatic. All formal logic is axiomatic. That doesn't imply that it is therefore inapplicable to reality.

I agree. This is, again, why I explicitly stated that I was assuming a continuous space, and that the Real-valued function accurately modeled reality (as opposed to approximating it). Yes, it is entirely possible that space is discrete and that uncountably infinite quantities are wholly inapplicable to the real cosmos. However, it is just as possible that space is continuous and that uncountably infinite quantities are applicable to the real cosmos.

Certainly! I have not claimed, nor would I claim, that mathematical abstractions necessarily map to physical reality. I simply claimed that it is possible that certain mathematical abstractions regarding infinite sets could map to physical reality.

Possible, yes.

Formal logic works in analyzing arguments. The set of formal logic constructs is finite (pretty sure) though.

Real and complex numbers (power of continuum cardinal) and higher order cardinality sets (e.g. the set of all real-valued functions of a real variable) are dubiously applied across the spectrum of a physical phenomenon. So your assumption ('glad you said that!) the a real-valued function accurately models reality is dubious -- especially since we can't measure intervals in space and time shorter than the Planck limits.

Yes, almost anything is possible, but empiricism is what separates maths and natural science.

To me knowable and detectable are the same. Also to me detectable and at least potentially measurable are the same.


[I know you agreed with this, but thought I repeat it.] As I leave this thread I reiterate that infinity and increasing orders of infinity are axiomatic concepts in maths. That we use real and complex numbers to describe physical phenomena does not imply that ANY physical phenomenon is defined for all numbers in a topologically dense set.


This is an issue for the philosophies of mathematics and natural science. We already know that many advanced math notions have no connection to reality.


This topic is interesting (and fun!), but I need more data and testability for my scientific taste. My abstract taste has already been satisfied by pure maths.


You can have the last word.

K54

[Jedidiah]

Strange but True: Infinity Comes in Different Sizes

Different sizes? Hmm, so how large is infinity? And how large is the larger infinity?

[klaus54]

Different sizes? Hmm, so how large is infinity? And how large is the larger infinity?

Physically? I have no idea.

"Infinity" is an axiom of set theory (in mathematics everything exists in a set or a "proper class" if there "too many", such as the "Set of all sets").

The Axiom of Infinity looks something like this:

Mathematically, a countably infinite set can be put in a 1 to 1 correspondence with the set {1, 2, 3, ...} and is Aleph_null "big"...

The set of real numbers is bigger than any countable set, it is in fact as big as the power set (set of all subsets) of , which is

And so on, ad infinitum as it were.

Confusing, ain't it?

It's axiomatic stuff, which is my biggest hang-up in applying orders of infinity to the physical world.

[little_monkey]

Ha, mathematicians are no good... just kidding...

In physics, we deal with infinities slightly different. It's like snowflakes - no to snowflakes are alike, so no two infinities are alike. The real interesting thing in QFT is that you take two infinities, subtract them, and you get a finite answer. That in a sense defies common sense. There are various infinities, and those pop up in masses, charges, coupling constant that measure the strength of an interaction. Nevertheless the results we get with this procedure, which goes under the name of renormalization, are extraordinary. You can calculate masses, like the mass of the Higgs boson discovered a few years ago, the anomalous magnetic moment of electron, the Lamb shift in the spectrum of hydrogen, decay rates, the Casimir force, jet productions in high energy colliders, to name a few. The whole concept behind Cantor's idea of different infinities doesn't play any role in physics. It seems like math and physics are on two different planets when it comes to infinities.

[klaus54]

Ha, mathematicians are no good... just kidding...

In physics, we deal with infinities slightly different. It's like snowflakes - no to snowflakes are alike, so no two infinities are alike. The real interesting thing in QFT is that you take two infinities, subtract them, and you get a finite answer. That in a sense defies common sense. There are various infinities, and those pop up in masses, charges, coupling constant that measure the strength of an interaction. Nevertheless the results we get with this procedure, which goes under the name of renormalization, are extraordinary. You can calculate masses, like the mass of the Higgs boson discovered a few years ago, the anomalous magnetic moment of electron, the Lamb shift in the spectrum of hydrogen, decay rates, the Casimir force, jet productions in high energy colliders, to name a few. The whole concept behind Cantor's idea of different infinities doesn't play any role in physics. It seems like math and physics are on two different planets when it comes to infinities.

Excellent!

You clarified to a "T" my point of not conflating the abstract notion of orders of infinity with the physical notion of infinity.

The physics notion of infinity is like the "sideways 8" idea in the calculus - unlimited varieties or duplications.

[klaus54]

Ha, mathematicians are no good... just kidding...
...
The real interesting thing in QFT is that you take two infinities, subtract them, and you get a finite answer. That in a sense defies common sense.
...

Let N={1, 2, 3, ...} (the "natural numbers"),
M={2, 3, 4, ...}
Then Cardinality(N) = Cardinality(M) =

Let "\" be set subtraction, i.e., A \ B = {x : x in A, x !in B}

Then Card(N\M) = Card({1}) = 1

So - = 1 in this case.

Let E = {2, 4, 6, ...}

Then N \ E = {1, 3, 5, ...}

So, Card(N \ E) =

- =

Confused?

[little_monkey]

Let N={1, 2, 3, ...} (the "natural numbers"),
M={2, 3, 4, ...}
Then Cardinality(N) = Cardinality(M) =

Let "\" be set subtraction, i.e., A \ B = {x : x in A, x !in B}

Then Card(N\M) = Card({1}) = 1

So - = 1 in this case.

Let E = {2, 4, 6, ...}

Then N \ E = {1, 3, 5, ...}

So, Card(N \ E) =

- =

Confused?

Interesting but the infinities we encounter in physics are not dealt in that fashion. To give an example, without going into too much details as it would require several pages of demonstration. But one major equation is the Klein-Gordon equation (K-G),

(🜨 + m²) φ(x) = 0

The term m is the mass of the particle. Though that equation looks simple, in most cases, you can't solve it exactly. You need to approximate the solution through Perturbation theory, which is based on the idea of a Taylor series expansion. IOW, φ(x) will be of the kind,

φ(x) = a₀ +a₁x +a₂x₂ + a₃x³ +...

The idea is that the coefficient a₀, a₁, a₂, a₃... get smaller and smaller so that your solution will converge. But in many cases, some of those coefficients will turn out to be infinite - they are really integrals from 0 to ∞, so it will happen that you'll get infinities. So what is done is to redefine m into a bare mass m₀ and a physical mass, m _phys. This adds a counter term to the KG equation, and subtracts the infinity. Initially, many people were dissatisfied with such a procedure, but then it was a lightning rod into what was the physical sense of these counter terms. It turns out that a particle like an electron is surrounded by a cloud of particles, antiparticles, photons, called virtual particles which interacts not only with the bare mass of the electron but also with the vacuum energy, which in some sense is infinite and is subtracted from the equation. The remarkable thing is that the results, which obviously are finite, agree with what is observed to an accuracy of one part in a billion (10⁻⁹) in some cases.

[Boxing Pythagoras]

Let N={1, 2, 3, ...} (the "natural numbers"),
M={2, 3, 4, ...}
Then Cardinality(N) = Cardinality(M) =

Let "\" be set subtraction, i.e., A \ B = {x : x in A, x !in B}

Then Card(N\M) = Card({1}) = 1

So - = 1 in this case.

Let E = {2, 4, 6, ...}

Then N \ E = {1, 3, 5, ...}

So, Card(N \ E) =

- =

Confused?

I don't think this is correct. It does not seem to be the case that Card(N\M)=Card(N)-Card(M), as you imply.

For example, if we were to use finite sets for N and M instead of infinite sets:

Let N={1,2,3,4,5}
Let M={2,3,4,5,6}
Card(N)=Card(M)=5, and therefore Card(N)-Card(M)=0

However, Card(N\M)=Card({1})=1.

[klaus54]

Interesting but the infinities we encounter in physics are not dealt in that fashion. To give an example, without going into too much details as it would require several pages of demonstration. But one major equation is the Klein-Gordon equation (K-G),

( + m²) φ(x) = 0

The term m is the mass of the particle. Though that equation looks simple, in most cases, you can't solve it exactly. You need to approximate the solution through Perturbation theory, which is based on the idea of a Taylor series expansion. IOW, φ(x) will be of the kind,

φ(x) = a₀ +a₁x +a₂x₂ + a₃x³ +...

The idea is that the coefficient a₀, a₁, a₂, a₃... get smaller and smaller so that your solution will converge. But in many cases, some of those coefficients will turn out to be infinite - they are really integrals from 0 to ∞, so it will happen that you'll get infinities. So what is done is to redefine m into a bare mass m₀ and a physical mass, m _phys. This adds a counter term to the KG equation, and subtracts the infinity. Initially, many people were dissatisfied with such a procedure, but then it was a lightning rod into what was the physical sense of these counter terms. It turns out that a particle like an electron is surrounded by a cloud of particles, antiparticles, photons, called virtual particles which interacts not only with the bare mass of the electron but also with the vacuum energy, which in some sense is infinite and is subtracted from the equation. The remarkable thing is that the results, which obviously are finite, agree with what is observed to an accuracy of one part in a billion (10⁻⁹) in some cases.

I was just showing that diddling around with set theory "infinity" - "infinity" can be finite of any cardinality from 0 on up, or infinity.

Your examples are interesting. But the bolded part is confusing. If a_n converges to 0 how can one a's be unbounded?

[klaus54]

I don't think this is correct. It does not seem to be the case that Card(N\M)=Card(N)-Card(M), as you imply.

For example, if we were to use finite sets for N and M instead of infinite sets:

Let N={1,2,3,4,5}
Let M={2,3,4,5,6}
Card(N)=Card(M)=5, and therefore Card(N)-Card(M)=0

However, Card(N\M)=Card({1})=1.

Good point.

I think it "works" for countable sets, and I put "works" in quotes because the example shows that aleph_n - aleph_n is undefined.

OTH, - Card(any finite set) =

But, - = ???

However - =

And so on.

Transfinite cardinal arithmetic is weird.

But, again -- this all ABSTRACT.

And I apologize for the diversion, since this and the previous post had nothing to do with physics -- and I guess that's why I posted it in response to LM's infinity - infinity comment.

I still don't see how the (non-abstract!) notion of infinity (mathematically this is limit, not a cardinal), in particular "infinity" - "infinity" works in physics.

[Boxing Pythagoras]

I still don't see how the (non-abstract!) notion of infinity (mathematically this is limit, not a cardinal), in particular "infinity" - "infinity" works in physics.

We both seem to agree that there is a way, in the real world, in which the abstract notion of infinity might not be applicable to the real world. I think we also agree that the question of whether space-time is fundamentally discrete or continuous remains an open question, but please correct me if I am wrong in this regard.

If we are in agreement on these two, we can construct a hypothetical situation in which the abstract notion of infinity would be applicable to real world physics.