Question about modeling continuous spacetime

[jaketodd]

Since determining how many points there are in a given volume of continuous spacetime would require divisibility by infinity, is set theory's infinite sets the only way to model continuous spacetime?

Thanks,

Jake

 

[jbriggs444]

Since determining how many points there are in a given volume of continuous spacetime would require divisibility by infinity, is set theory's infinite sets the only way to model continuous spacetime?

A model containing a continuum will feature sets with the cardinality of the continuum. That's kind of what the "cardinality of the continuum" means.

In any case, determining the cardinality of the set of points in a model has nothing to do with making a physical measurement and "dividing by infinity". It would be a feature of the model, not a feature of experimental reality.

 

[Dale]

Since determining how many points there are in a given volume of continuous spacetime would require divisibility by infinity, is set theory's infinite sets the only way to model continuous spacetime?

The cardinality of the set of all events in spacetime is Beth-1. I don't know any important theorem in relativity that makes use of that fact.

 

[jaketodd]

A model containing a continuum will feature sets with the cardinality of the continuum. That's kind of what the "cardinality of the continuum" means.

In any case, determining the cardinality of the set of points in a model has nothing to do with making a physical measurement and "dividing by infinity". It would be a feature of the model, not a feature of experimental reality.

I get the "divide by infinity" thing from the fact that continuous space - any volume of continuous space - has an infinite number of points within it - at least that's what Einstein thought - was indeed the inherent nature of space (not just in models). So if one were to ask: How many points are in that chunk of space over there? You would have to take that particular volume of space, and divide it by infinity, which is undefined, which makes continuous space impossible, unless, infinite sets, with differently-sized infinities are used. But even then, you'd need a lot of infinite sets to fully model continuous space! Since there's no smallest volume, there would be no end to how many infinite sets there would be. Dare I say getting near the Absolute Infinite?

The cardinality of the set of all events in spacetime is Beth-1. I don't know any important theorem in relativity that makes use of that fact.

I think that Einstein would be forced to say (did he ever say anything about set theory and its' application to relativity?): "Beth-1 is the size of the universe", like you say. Here's a another question: How does Beth-1 compare to the Absolute Infinite?

I find a discrete treatment of space much more natural... I'm pretty sure relativity is compatible with quanta instead of continuous??

Thanks,

Jake

 

[jaketodd]

Here's another question: What happens to the cardinality of an infinite set (representing an amount of spacetime) when it is warped by gravity? What happens when you stretch or compress an infinity? If there's no answer to that, I just really want to "fall back" on a discrete model of spacetime; Occam's razor perhaps?

Thanks,

Jake

 

[jaketodd]

Dare I say that one thing we have accomplished here so far is that Einstein's continuous spacetime opinion requires set theory?

 

[jbriggs444]

I get the "divide by infinity" thing from the fact that continuous space - any volume of continuous space - has an infinite number of points within it - at least that's what Einstein thought - was indeed the inherent nature of space (not just in models). So if one were to ask: How many points are in that chunk of space over there?

What physical experiment can you propose which has a result that depends on the answer to that question?

You would have to take that particular volume of space, and divide it by infinity, which is undefined, which makes continuous space impossible, unless, infinite sets, with differently-sized infinities are used. But even then, you'd need a lot of infinite sets to fully model continuous space! Since there's no smallest volume, there would be no end to how many infinite sets there would be. Dare I say getting near the Absolute Infinite?

None of this makes any sense. The cardinalities of the set of points in any continuous volume is still Beth-1 regardless of the size of that volume. You do not use "number of points" to characterize the size of infinite sets. You can use other things, such as "cardinality" or "measure".

I think that Einstein would be forced to say (did he ever say anything about set theory and its' application to relativity?): "Beth-1 is the size of the universe", like you say. Here's a another question: How does Beth-1 compare to the Absolute Infinite?

Dale pointed out that the cardinality of the continuum is Beth-1. But that's not an experimental truth. That is a mathematical truth about the model.

As for "Absolute Infinite", that has no relationship with any of this.

 

[Nugatory]

Dare I say that one thing we have accomplished here so far is that Einstein's continuous spacetime opinion requires set theory?

Only in the trivial sense that any discipline that involves manipulating numbers requires set theory. There's nothing specific to relativity here.

 

[jaketodd]

What physical experiment can you propose which has a result that depends on the answer to that question?

That was a rhetorical question.

None of this makes any sense. The cardinalities of the set of points in any continuous volume is still Beth-1 regardless of the size of that volume. You do not use "number of points" to characterize the size of infinite sets. You can use other things, such as "cardinality" or "measure".

Forgive me for not using proper terminology. By "number of points" I do mean "cardinality." If all continuous volumes had the cardinality of Beth-1, then how would anyone be able to discern one volume from another? Certainly there are differently-sized areas all around us.

Dale pointed out that the cardinality of the continuum is Beth-1. But that's not an experimental truth. That is a mathematical truth about the model.

Is there any "experimental truth" for set theory?

Only in the trivial sense that any discipline that involves manipulating numbers requires set theory. There's nothing specific to relativity here.

Is there any empirical evidence for set theory? I was just saying that continuous spacetime seems to me to require set theory - not that either are correct. Many would say discrete spacetime is just as likely, if not more plausible. Quantum mechanics is a good start for the empirically discrete end of things. So no, I don't think any discipline requires set theory, other than purely-conceptual ones - - but prove me wrong.

Cheers,

Jake

 

[jbriggs444]

Forgive me for not using proper terminology. By "number of points" I do mean "cardinality." If all continuous volumes had the cardinality of Beth-1, then how would anyone be able to discern one volume from another? Certainly there are differently-sized areas all around us.

Measure theory addresses that problem. It associates a real-valued "measure" with a given set. For instance, Lebesgue Measure.

https://en.wikipedia.org/wiki/Measure_(mathematics

Do you know what it means for one set to have the same cardinality as another?

Is there any empirical evidence for set theory?

Set theory is part of mathematics. We use proofs and disproofs to assess the truth of mathematical statements. Empirical evidence does not enter in except in the limited sense that an absence of a proof of inconsistency is [arguably weak] evidence for consistency of the formal system in which those proofs are constructed.

 

[Dale]

You would have to take that particular volume of space, and divide it by infinity, which is undefined,

This quantity you are referring to, taken as a limit, would be zero not infinite. I don't think that is what you want to do. The number of elements in a set is called its cardinality. No division is involved, it is basically counting.

Since there's no smallest volume, there would be no end to how many infinite sets there would be. Dare I say getting near the Absolute Infinite?

Please post a reference for this claim

So if one were to ask: How many points are in that chunk of space over there? You would have to take that particular volume of space, and divide it by infinity, which is undefined, which makes continuous space impossible, unless, infinite sets, with differently-sized infinities are used.

The cardinality of the set of events in a small region of spacetime is the same as the cardinality of the set of events in an infinite spacetime: Beth-1. That is also the cardinality of the set of points on a line segment, the whole real line, or RN. They all have the same cardinality.

Here's another question: What happens to the cardinality of an infinite set (representing an amount of spacetime) when it is warped by gravity? What happens when you stretch or compress an infinity?

That doesn't change the cardinality, it is still Beth-1.

 

[Ibix]

Is there any empirical evidence for set theory?

Why would there be? It's pure maths. On the other hand, if what you actually mean is "is there any evidence for the applicability of set theory to the study of spacetime" then yes. Off the top of my head, the GPS system, the Pound-Rebka experiment, some aspects of the Hafele-Keating experiment, Shapiro delay, gravitational lensing, gravitational waves, and probably more, are all experimental confirmation of predictions of general relativity, a theory built on the notion of spacetime as a manifold - which are sets of points.

 

[weirdoguy]

Certainly there are differently-sized areas all around us.

Area or volume of a given set (manifold for example) is something different than it's cardinality. Every square or triangle have the same cardinality, but they may have different areas.

 

[Herman Trivilino]

I get the "divide by infinity" thing from the fact that continuous space - any volume of continuous space - has an infinite number of points within it - at least that's what Einstein thought - was indeed the inherent nature of space (not just in models).

When you model space as a continuum you don't have a finite number of points in any finite volume.

So if one were to ask: How many points are in that chunk of space over there? You would have to take that particular volume of space, and divide it by infinity,

Dividing a volume by a number of points would give you the volume associated with each point. But doing that calculation to answer that question would be circular, because in the model you are using you've already defined points in such a way that they each have a volume of zero.

I find a discrete treatment of space much more natural... I'm pretty sure relativity is compatible with quanta instead of continuous??

Einstein's theories of relativity ignore quantum theory.

When you speak of things that are "not just in models" you are speaking of things that are just not physics.

 

[weirdoguy]

Einstein's theories of relativity ignore quantum theory.

But it's also important to note that we have quantum fleld theory wchich is fully compatible with special realtivity.

Quantum mechanics is a good start for the empirically discrete end of things.

Not eveyrything in QM is dicrete.

 

[PeterDonis]

if one were to ask: How many points are in that chunk of space over there? You would have to take that particular volume of space, and divide it by infinity

No, you wouldn't. You evidently need to spend some time studying (a) set theory, and (b) the theory of manifolds. When you study those and learn the proper way of formulating such questions, you will find that the answer to the question "how many points are in that chunk of space over there" is $C$ (the cardinality of the continuum), regardless of the volume of the chunk of space. This is because the points in any "chunk of space", regardless of its volume, can be put into one-to-one correspondence with the points in all of space, and any two sets which can be put into one-to-one correspondence have the same cardinality.

 

[Ibix]

the answer to the question "how many points are in that chunk of space over there" is $C$ (the cardinality of the continuum), regardless of the volume of the chunk of space.

As an illustration, consider the set of all even numbers. This is clearly half the size of the set of all integers because we're knocking out every other number. Right?

Wrong. Take any integer $n$ and double it. $2n$ is in the even numbers. So I've established a one-to-one relationship between the members of the integers ($n$) and the members of the evens ($2n$). So the sets have the same cardinality.

 

[PeterDonis]

As an illustration, consider the set of all even numbers.

Note that this set is not a continuum, although it still illustrates the key property of infinite sets: that they can be put into one-to-one correspondence with proper subsets of themselves.

 

[RockyMarciano]

The cardinality of the set of all events in spacetime is Beth-1. I don't know any important theorem in relativity that makes use of that fact.

Indeed it is trivially used all the time in that it is the mathematical scenario of the theory and of the transformations in it.
Perhaps more important to the theory is that this cardinality requires the use of the axiom of choice in the context of relativistic QFT and gauge field theory whenever choice is needed.

 

[DrGreg]

Note that this set is not a continuum, although it still illustrates the key property of infinite sets: that they can be put into one-to-one correspondence with proper subsets of themselves.

For example, consider the function defined by $$
y = \frac{1}{x+1} + \frac{1}{x-1}
$$which gives a one-to-one correspondence between the set $\{x : -1 < x < 1\}$ (a line of length 2) and the real line $\{y: -\infty < y < \infty\}$ (a line of infinite length). The one-to-one correspondence (bijection) establishes that both sets have the same cardinality.

 

[jaketodd]

Suffice to say that modeling continuous spacetime does require set theory. I'm satisfied with the answer to my OP. I might get around to reading all these angry replies, I might not.

 

[PeterDonis]

What happens to the cardinality of an infinite set (representing an amount of spacetime) when it is warped by gravity?

Nothing. "Cardinality of the set of points in some region of spacetime" is simply the wrong concept to use if you want to understand how GR works. That is really the gist of the responses you have been getting.

The reason why cardinality is the wrong concept is that, while a spacetime, or a region of one, is indeed modeled as a "set of points" with the cardinality of the continuum, that set has a lot more structure than just its cardinality--in particular, it has a metric, which is what tells you the "size" of a set of points in the sense you are intuitively thinking of it--length, area, volume. So the difference between the set of points describing, say, a cube 1 meter on a side, and the set of points describing all of space is not that the two sets have different cardinalities; it's that they have different volumes given by the metric.

Similarly, if a region of spacetime is warped by gravity, that affects the metric, not the cardinality of the set of points in that region.

 

[Dale]

I might get around to reading all these angry replies, I might not.

I wonder why you assume the unread replies were angry.

 

[pervect]

Mathematics, in the form of point-set topology, is important in describing concepts such as the continuum.

Thinking that the cardinality of the set of points is the only important issue is, however, is incorrect, as other posters have already mentioned. I'll give a specific example of why this is so. The cardinality of the set of points on an infinite line is the cardinality of the set of real numbers $\mathbb{R}$, because there is a one-one correspondence (or if you prefer bijection) between the real numbers and points on an infinite line. The cardinality of the set of points on an infinite plane is the cardinality of the set of pairs of real numbers, $\mathbb{R}^2$, because there is a one-one correspondence between pairs of real numbers and points on a plane.

But it turns out the cardinality of $\mathbb{R}$ equals the cardinality of $\mathbb{R}^2$. Thus the cardinality of the number of points on a line are the same as the cardinality of the number of points on the plane. Loosely speacking, they have "the same number of points".

The resolution of this isn't trivial, and you'll have to read about point set topology to even get an inkling, it's not something you're likely to stumble over by yourself. It will take some serious study. This is especially true if you want a solid proof without any holes in it.

I can outline the end results, but not the formal proofs. While there is a bijection between $\mathbb{R}$ and $\mathbb{R}^2$, so they have "the same number of points", there is NOT a homeomorphism from points on the line (which is a topological space), and points on the plane (which is a topological space). The topological space has added structure, that contains information about what points are "near" other points. The intuitive notion of "nearness" is more-or-less represented by the mathematical notion of "open balls".

Very very loosely speaking, then, what happens is that while we can map every point of the plane onto a single line, we can't do it in such a way that points that are "neighbors" in the plane are also "neighbors" on the line. Such mappings have the property that two points that are close to each other on the plane may be a long ways apart when we map them onto the line.

Using the same loose and imprecise language, mappings that preserve "neighborhood" are called homeomorphisms, and they are different from bijections (which are arbitrary 1:1 mappings that do not necessariliy preserve the "neighborhoods" structure.

You can try to read up on all the terms I mentioned, but you'll probably need a more formal approach to have it all make sense. You can try the math forums for better suggestions of where to read more if you want to study this area seriously.

There are plenty of other non-intuitive (but interesting) results from point-set topology, among the most famous is the Banach-Tarski "paradox". <<link>>.

 

[jaketodd]

I wonder why you assume the unread replies were angry.

I've read plenty.. to make conclusions that I won't specify, my friend.