can something exist literally 'infinitely' far away?

Originally posted by twhitehead
...
While looking at that page, I came across this page:
https://en.wikipedia.org/wiki/Monstrous_moonshine
which honestly goes way over my head.

Oh my god. I have just looked at it and think lets not go there.
I bet it is more above my head than yours.
I find it extremely horridly hard enough dealing with the basic algebra of modular form
(see https://en.wikipedia.org/wiki/Modular_form ) , which I just touched on with my university maths courses and found extremely difficult, which that "Monstrous moonshine" involves.

Originally posted by humy
And yes, you were right in implying that that bit of what I said there didn't make sense.

The reason your statements about infinity eventually lead to a contradiction is they do actually assume ∞ is a real number. Mathematicians, when dealing with the same problems you were looking at avoid using ∞ as a real number by going to set theory, and instead of asking whether the set of integers has the same 'number' of elements as the set of even numbers, instead ask whether or not the two sets have the same cardinality. The key being that cardinal numbers include numbers that are not real numbers at all and shouldn't be confused with them.

https://en.wikipedia.org/wiki/Cardinality
https://en.wikipedia.org/wiki/Cardinal_number

Originally posted by twhitehead
The reason your statements about infinity eventually lead to a contradiction is they do actually assume ∞ is a real number. Mathematicians, when dealing with the same problems you were looking at avoid using ∞ as a real number by going to set theory, and instead of asking whether the set of integers has the same 'number' of elements as the set of even num ...[text shortened]... hem.

https://en.wikipedia.org/wiki/Cardinality
https://en.wikipedia.org/wiki/Cardinal_number

Oh I forgot all about cardinality.
I fear I could get horribly bogged down with my current research into probability and epistemology if I cannot find a way of avoiding infinite sets and the dreaded infinity.

Originally posted by humy
Oh I forgot all about cardinality.
I fear I could get horribly bogged down with my current research into probability and epistemology if I cannot find a way of avoiding infinite sets and the dreaded infinity.

The secret to dealing with infinity, is to constantly remind yourself that despite all our intuitions to the contrary, it isn't a real number.
A similar tactic must be used for quantum mechanics. As long as you keep firmly in mind that a photon is not a particle or a wave, but rather can have particle like or wave like behaviour, then you avoid many of the pitfalls. The problem is our intuition just keeps on dragging us back into thinking it is one or the other.

Originally posted by twhitehead
What do you mean by 'mathematical philosophy'? I see it as mathematical fact.

[b]Basically, we have no way of knowing, we can in principle experimentally rule out theories that allow open topologies, however, current evidence is that the universe is expanding which implies infinite spatial extent.
How does an expanding universe imply infinite spat ...[text shortened]... ually seem to assume it is finite, so it seems unlikely that the implication is obvious to them.[/b]

Mathematics provides the additional language for physics theories. Some mathematicians called ultra-finitists claim that there are no numbers larger than - for example - the number of electrons in the universe - numbers larger than that have no meaning and so do not really exist. Other mathematicians will happily accept that a number can be infinite. Most have a position in between. The difference is one of mathematical philosophy. So questions such as whether two objects can be infinitely far from one another given some theory depends on mathematical philosophy rather than anything intrinsic to the theory.

Just a point, my procedure for picking two objects and measuring the distance between them may assume an Axiom of Choice (I don't know, this is one for the mathematicians here). In which case whether the theory disallows this hangs on something that is undecidable, and is therefore a matter of philosophy rather than internal consistency.

Regarding the second point: There are three possible cosmologies and they imply different global topologies. The curvature can be positive (like a sphere), negative, or flat. This is easiest to picture using some two dimensional examples - in the positive curvature case a sphere curves back on itself and so is both closed, finite and borderless. The flat case is a plane, either it has an edge or it extends for ever. It doesn't curve back on itself. The third case is difficult to imagine, essentially you have a surface where every point looks like a saddle. If it is symmetric (in the same way that a sphere has the same curvature at each point) then it cannot curve back and join up again, like the plane it either has to have an edge or be of infinite extent (see hyperboloid on Wikipedia for a picture). Cosmologies with accelerating expansion look like a four dimensional version of the third of these.

A universe of infinite extent clashes with the notion that the universe started off smaller than a proton. One possible way to resolve this is that the accelerating expansion is local. Although it affects the entire observable universe there are regions beyond that of intense positive curvature which cancel out the negative curvature here and allow the entire universe to curve back on itself and be finite. A shape which illustrates this is a torus. On the innermost surface the curvature is negative, if the curvature were like that everywhere on the torus it could not curve back to join up with itself, however on the outermost part (if the torus in question was a tire it would be the point in contact with the ground) the curvature is positive and cancels out the negative curvature and allows the shape to curve back on itself and join up again (if you integrate the curvature over area you get the Euler Characteristic and for a Torus that is zero so that cancelation is exact in this case).

Originally posted by humy
No, I am not.

As I have said twice already, it depends on what you mean by ∞. There are certainly definitions under which the expressions may make sense.

Can you give me an example of a definition of ∞ which would render my previous argument in that post invalid?


[quote] This is similar to the fact that

1+2+3+4+... = -1/12 ...[text shortened]... _series link implies it cannot converge because the individual terms don't tend to zero.

...”

All series have a principle value (I can't remember if I got the jargon right). For series where the typical term is 1/n^s where the sum is over all integers and s is a complex number then the series converges for all s such that the real part of s is strictly greater than 1. Riemann defined a function called the Zeta function that is equal to the series where it converges and then analytically extended it to the entire complex plane. The resultant function has a pole at s = 1, and where its zeros are is one of the Millenium prizes. It provides a consistent way of assigning a number to a series:

zeta(s) = 1 + 1/(2^s) + 1/(3^s) + ... + 1/(n^s) + ...

and if s is -1 we have:

zeta(-1) = 1 + 2 + 3 + 4 + ...

Using some contour integration Riemann found a reflection formula that relates zeta(s) to zeta(1 - s), and some other well understood functions. We can therefore assign a canonical value to the divergent sum 1 + 2 + 3 + ... and it's -1/12. The only sum of this form which cannot have a finite answer assigned to it is the harmonic series as the zeta function has a pole there.

Whether you regard this analytic continuation trick as a valid move or not is philosophical, the answers are consistent and useful (which is what tends to win it).

Originally posted by DeepThought
Some mathematicians called ultra-finitists claim that there are no numbers larger than - for example - the number of electrons in the universe - numbers larger than that have no meaning and so do not really exist.

They sound like physicists / philosophers not mathematicians.

Other mathematicians will happily accept that a number can be infinite.
But not a Real number.

Most have a position in between. The difference is one of mathematical philosophy.
Nevertheless, the question in the OP, is not about whether infinite numbers can exist, it is asking whether or not an infinite dimension has a subset of infinite extent. And I say no, that is not possible.

So questions such as whether two objects can be infinitely far from one another given some theory depends on mathematical philosophy rather than anything intrinsic to the theory.
No, it doesn't. If the theory includes finite dimensions, then clearly objects cannot be infinitely far away, and if the theory includes infinite dimensions then it depends on what the theory has to say about the structure of dimensions.

A universe of infinite extent clashes with the notion that the universe started off smaller than a proton.
Which is why such a notion is not founded on evidence - and you will find that smart cosmologists never say that, but instead say 'the observable universe' as that caters for both finite and infinite universes.

Originally posted by twhitehead
They sound like physicists / philosophers not mathematicians.

[b]Other mathematicians will happily accept that a number can be infinite.
But not a Real number.

Most have a position in between. The difference is one of mathematical philosophy.
Nevertheless, the question in the OP, is not about whether infinite numbers can exist, it is ...[text shortened]... but instead say 'the observable universe' as that caters for both finite and infinite universes.[/b]

Look up L.E.J. Brouwer who amongst other things invented the fixed point theorem.

You've missed what I was getting at. When we make claims about the world it is either a claim generated by the theory or an experimental result. To make where I'm making the distinction I'm making more clear, consider the discovery of the Higgs boson at LHC. What they actually saw were behaviours in their detectors. Their claim is along the lines of: "Our detectors exhibited a behaviour consistent with two photons with a combined energy of 125GeV a statistically significant number of times. We therefore conclude that a particle with the right properties to be the Higgs boson exists and has a mass of 125GeV.". The empirical part of that statement is: "Our detectors exhibited a behaviour.", the whole of the rest of the statement is theoretical. That what they saw in their detectors corresponds to two photons is a theoretical claim (a very well justified theoretical claim, but nevertheless theoretical). That the theory being verified is being compared with a null claim means that other theories which have scalars about the right mass and decay path, but not part of the Higgs mechanism are not ruled out.

Any answer to the question in the OP cannot be straightforwardly empirical. According to what we think our best theory is says, it's impossible for us to see everything that is a finite distance away, never mind verify the existence of things infinitely far away. So we have to look at the theories to see if they allow it. The theories have empirical justification, but that doesn't stop them being theories. Most of the time physicists implicitly assume that the quantities they are using are correctly described by the real numbers. Consider distances, we assume that the correct way to denote distances is to use the reals. For almost all applications it doesn't matter that it might not correctly represent distances at a fundamental level. So the question of whether the reals are the right set of numbers to use in fundamental physics is an open one.

That General Relativity uses the real numbers doesn't mean that it could not use the surreals, it doesn't make much difference at the level GR makes its predictions. This might matter for Quantum Gravity theories, but the discussion isn't about them. So, if we can use some number system that allows infinite distances in General Relativity then the answer to the OP would be "Yes, the theory allows objects infinitely far apart.". I don't think empirical evidence helps particularly, all we can say is that this region of the universe (the observable to us part) appears to be expanding at an accelerating rate. That doesn't really rule out any of the options. If it's possible to give a definitive answer it hangs on what numbers correctly describe things like distance and we can't be sure of the answer to that, but that hinges on whether things like surreal numbers are allowed in mathematics. If mathematics does not allow such structures it's hard to see them as an allowable object in a physics theory.

I didn't read all the posts where you and humy were discussing set theory. What I did notice is that you seem to have missed out the notion of measure. This is an additional property you impose on a set. On Wikipedia the place to start is probably the page on Lebesgue Integrals. Roughly, measure is a non-negative number assigned to a set, the interval [0, 1] has measure 1. There are a collection of axioms (basically to make it additive in the right way). Let m(S1) be the measure of the set of points S1 and S2 also a set with measure m(S2) and, S1 U S2 is the union and S1*S2 is the intersection of S1 and S2 then:

m(S1 U S2) + m(S1 * S2) = m(S1) + m(S2).

So, you are looking for a subset of the reals with infinite measure. Let's make a cut on the real line and exclude the point at the origin. This divides the set of reals into two subsets of infinite measure. Now, because we are using the reals then there is no point on the real line where we can make a cut to generate three subsets where one of them does not have finite measure (the bit between the origin and where we made our second cut). But, that the reals do not have this property does not mean that some space could not be constructed so that subsets with infinite measure can be found which are adequately continuous (see Banach-Tarski paradox for an example of what I'm trying to rule out with the words "adequately continuous".), think of the distance along a fractal curve for a simple example. Questions like this tend to require one to worry about things like the Axiom of Choice and how you've gone about constructing your sets, and I think you may have made implicit assumptions about the sets you are looking at which mean that your conclusions aren't as general as you think they are.

Originally posted by DeepThought
You've missed what I was getting at.

I am sure I have, but I am also sure that I have done so because you are stating it incorrectly.

Any answer to the question in the OP cannot be straightforwardly empirical.
Obviously not, given that the OP starts with a hypothetical. But the answer can be straightforwardly logical. No observations are needed to answer the question, only maths. It would appear, however, that you have misread the question.

According to what we think our best theory is says, it's impossible for us to see everything that is a finite distance away, never mind verify the existence of things infinitely far away.
I just look to logic, and it says 'no', loud and clear.

Most of the time physicists implicitly assume that the quantities they are using are correctly described by the real numbers. Consider distances, we assume that the correct way to denote distances is to use the reals. For almost all applications it doesn't matter that it might not correctly represent distances at a fundamental level. So the question of whether the reals are the right set of numbers to use in fundamental physics is an open one.
Now you are on to something. But this is quite different from what you said in previous posts. I fully realise that quantum mechanics probably uses complex numbers not reals. Nevertheless, given that we currently define distance in reals, to say something is infinitely far away is incoherent and thus impossible. If distances are not actually measurable in real numbers, then there may be objects whose distance cannot be measured. But they are not infinitely far away.

So, if we can use some number system that allows infinite distances in General Relativity ...
Except we don't.

If it's possible to give a definitive answer it hangs on what numbers correctly describe things like distance and we can't be sure of the answer to that, but that hinges on whether things like surreal numbers are allowed in mathematics.
No, I don't think it does. I think it hinges on the structure of space time.

But, that the reals do not have this property does not mean that some space could not be constructed so that subsets with infinite measure can be found which are adequately continuous (see Banach-Tarski paradox for an example of what I'm trying to rule out with the words "adequately continuous".), think of the distance along a fractal curve for a simple example. Questions like this tend to require one to worry about things like the Axiom of Choice and how you've gone about constructing your sets, and I think you may have made implicit assumptions about the sets you are looking at which mean that your conclusions aren't as general as you think they are.
Given that the only set we have been looking at is the Reals, I don't think we have made that error. I do accept that physical distance may not be measurable using the Reals. And I still strongly disagree with some of your earlier claims which don't really reflect what you have said in this post.

I have concluded that;

1, at the very least in valid and completely reasonably constructed systems of pure mathematics, there exists such thing as infinities (unfortunately. It would MASSIVELY simplify my research if I could just simply dismiss all infinities as total nonsense, period ).

2, expressions like " ∞ = ∞ " are neither true or false but gibberish partly because they lead to all sorts of paradoxes and partly because there are different orders of infinity (which can be thought of as different cardinalities of infinite sets ) and the "∞" symbol doesn't imply which order of infinity it is referring to (it needs to be the cardinality of an infinite set for that) thus the LHS of " ∞ = ∞ " may or may not be referencing the same order of infinity to the RHS of " ∞ = ∞ " thus " ∞ = ∞ " is too ambiguous to imply whether it is true or false.

However;

3, if A and B are infinite sets then it does make sense to write about their cardinality with expressions like |A| = |B| or |A|<|B| etc thus implying something about the 'equality' or 'inequality' of different infinities. But here we must be very careful in how we define 'equality' or 'inequality'.

Originally posted by humy
I have concluded that;

I agree with your conclusions.

Originally posted by twhitehead
I am sure I have, but I am also sure that I have done so because you are stating it incorrectly.

[b]Any answer to the question in the OP cannot be straightforwardly empirical.
Obviously not, given that the OP starts with a hypothetical. But the answer can be straightforwardly logical. No observations are needed to answer the question, only maths. ...[text shortened]... ree with some of your earlier claims which don't really reflect what you have said in this post.[/b]

No, I don't think it does. I think it hinges on the structure of space time.

Then you are confusing the thing with the theory about the thing. Since we only know the structure of space-time through our theories what is sayable about it depends on what is allowed in the theories. Selecting the reals is just making the simplest choice, but it's a choice and without any empirical input an arbitrary one.

Things like complex numbers aren't really a different structure, any theory with complex numbers in can be rewritten in terms of reals with a small sacrifice of elegance. This isn't available if we are trying to get between the surreals and the reals, the infinitesimals don't map to anything.

I looked at my previous post. Apart from the section in my first post where I pointed out that anti-deSitter spaces are open which was not connected with this point, I have not altered my argument between posts. My argument is that first, any answer to this is purely theoretical, the theory might have empirical backing but it's not close to parts of the theory that have been tested, and second it hangs on the number system used in the theory, whether two points can be separated by an infinite measure set.

Originally posted by humy
actually, I take that back. I mean " an infinity number " by 'number' there. I am aware that conventional maths terminology says 'infinity' is not a 'number' but don't know how to say it without either implying infinity is a real number or implying there exists infinity, which is a concept I am unsure of.
And yes, you were right in implying that that bit of what I said there didn't make sense.

I've just noticed this, a common way of expressing this is to talk about the extended reals the normal reals with extra points at +/- infinity. The problem is that the real numbers are so called because at one time they were believed to be "the numbers nature uses", so we get this equivocation problem when we try to talk about what number system is "real". I'd suggest the word "realized" (as in realized in nature) or "actual", to express what you're trying to.

Originally posted by DeepThought
I looked at my previous post. Apart from the section in my first post where I pointed out that anti-deSitter spaces are open which was not connected with this point, I have not altered my argument between posts.

The things I object to are things like this:

Some mathematicians called ultra-finitists claim that there are no numbers larger than - for example - the number of electrons in the universe - numbers larger than that have no meaning and so do not really exist.

Numbers don't 'exist'. And anyone who thinks a number has no meaning because it is larger than the number of electrons in the universe is just nuts. In addition, the OP starts off with the assumption that the universe is infinite.

I do agree that if space cannot be measured in reals then little can be said at this time to answer the OP. However, if space can be measured in reals, then the answer to the OP is clearly 'no'.

Originally posted by twhitehead

Numbers don't 'exist'. [/b]

I could be completely wrong here, but I suspect you may have subtly misunderstood his exact intended meaning.
Personally, if I said a number 'exists', depending on which sense I mean 'exists', I personally would mean either merely something along the lines of 'it makes sense' in pure mathematical sense or 'it makes sense' to talk about, say, n number of things physically existing in the external world. But what I wouldn't intend to imply by saying a number 'exists' is that the number itself can be or is in some sense literally 'there' floating about in the external world independent of both all physical variables in the external world and potential/actual concepts of the mind (not implying you would think that I would mean such a thing).