# The problem with infinity.

There are physicists who insist that the universe is finite and has a distinct geometry. So what'd be the problem if the universe were infinite?

Drakkith
Staff Emeritus
Those physicists need to look into modern cosmology then. It is not known whether the universe is finite or infinite, and it doesn't matter, according to current theory, which one is correct. If the universe is finite then it must be much larger than the observable universe. If it is infinite, well, then it is infinite. As for the geometry of the universe, it has been observed to be very very close to being flat, but this does not rule out non-flat geometry that is too subtle to detect currently.

Well due to Zeno's paradox of motion depicted by achilles and the tortoise, infinity does not exist. Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller.
Infinity is a number that cannot be divided, cannot be measured, and cannot be contained. This infinite universe obviously does not exist due to the fact that all pieces of space are made of this ultimately small unit.
Every quantity can be described by this unit, thus making the concept of infinity null.

Matterwave
Gold Member
Well due to Zeno's paradox of motion depicted by achilles and the tortoise, infinity does not exist. Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller.
Infinity is a number that cannot be divided, cannot be measured, and cannot be contained. This infinite universe obviously does not exist due to the fact that all pieces of space are made of this ultimately small unit.
Every quantity can be described by this unit, thus making the concept of infinity null.
Zeno's paradox has been sufficiently done away with via the use of calculus. In calculus, no such quantization is necessary.

There has been no successful quantization of space-time so far.

There are physicists who insist that the universe is finite and has a distinct geometry. So what'd be the problem if the universe were infinite?
Infinity as a concept in the mind is ok
but
when applied to/in reality, in my understanding/opinion, does not work.

The definition itself is self contradictory/limiting.

however its possible that if you were to reach the "edge" of say, time-space or any other variable/dimension, you could extend it further but it would still remain finite.

I feel that QM might actually be solving the problem of infinity.......at least for some concepts such as the infinite loop of a cause having a cause.

There are physicists who insist that the universe is finite and has a distinct geometry. So what'd be the problem if the universe were infinite?
there is a physical problem, in an infinite universe it have an infinite mass and hence inﬁnite inertia, no motion would be possible.

think rather in a finite universe without boundaries.

Well due to Zeno's paradox of motion depicted by achilles and the tortoise, infinity does not exist. Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller.
Infinity is a number that cannot be divided, cannot be measured, and cannot be contained. This infinite universe obviously does not exist due to the fact that all pieces of space are made of this ultimately small unit.
Every quantity can be described by this unit, thus making the concept of infinity null.
No. What Zeno never mentions as his intervals are halved it the time taken for each step. Hence, with the development of calculus it became easy to prove that even if the absolute number of steps are infinite it is easily accomplished in a finite amount of time.

Some more interesting paradoxes have been proposed since Zeno. Including the "Ant on a rubber rope" and "Hilbert's paradox of the Grand Hotel". Vitali sets are cool and easier to understand than Banach–Tarski paradox. Also the Hausdorff paradox is interesting. Basically trying to avoid infinities is every bit as paradoxical as infinities themselves.

The reason calculus is traditionally defined in terms of limits was to avoid infinities, because people inappropriately assumed too much, like invalid Zeno stuff. It was thus better to avoid it than to throw around Zeno stuff a priori. Once that was done, and proofs about the properties of infinities could be proven anyway, the need to avoid infinities lost its relevance. Hence non-standard calculus was born. If you want to make sense of it you'll need to drop the Zeno stuff and stick with more rigorous justifications, such as provided by non-standard calculus.

This being a physics forum it's also interesting to point out some consequences. If everything must be quantized to avoid infinities then General relativity must be quantized. If everything must be quantized then this is tantamount to the claim that Quantum computers are not fully scalable, i.e., have fundamental limits. Of course you can offload that onto other Universes, but then you are stuck with an infinite number of finite Universes.

That's some fairly serious empirically accessible claims, because it still only avoids infinities by making Quantum computers unscalable.

there is a physical problem, in an infinite universe it have an infinite mass and hence inﬁnite inertia, no motion would be possible.
No. That's not how inertia works.

Drakkith
Staff Emeritus
there is a physical problem, in an infinite universe it have an infinite mass and hence inﬁnite inertia, no motion would be possible.

think rather in a finite universe without boundaries.
Perhaps if you think of the entire universe as a system moving relative to another system. Per the usual definition of the universe, that it is everything that exists, this is not possible.

This being a physics forum it's also interesting to point out some consequences. If everything must be quantized to avoid infinities then General relativity must be quantized.
I don't follow you. How did you go from the fact that Calculus solves Zeno paradox (wich it does, at least partially) to the claim that everything must be quantized?
I don't see the connection between Calculus and quantization in physics or in general. Precisely Calculus allows us to deal with infinities in a continuous form.

I don't follow you. How did you go from the fact that Calculus solves Zeno paradox (wich it does, at least partially) to the claim that everything must be quantized?
I don't see the connection between Calculus and quantization in physics or in general. Precisely Calculus allows us to deal with infinities in a continuous form.
I don't. I was reacting to what Pianoasis stated the claim was made that: "Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller."

Even so, my words did not constitute a "claim" as it was prequalified with an "if". Certainly I did go a step beyond Pianoasis's words when I associated their notion of units of space which cannot be subdivided with quantization, though Quanta are not things in the usual sense. But it was a trivial extension which laid the groundwork for what I was rejecting, rather than claiming.

I don't. I was reacting to what Pianoasis stated the claim was made that: "Zeno proved that space is made up of an ultimately small piece that can not be divided any smaller."

Even so, my words did not constitute a "claim" as it was prequalified with an "if". Certainly I did go a step beyond Pianoasis's words when I associated their notion of units of space which cannot be subdivided with quantization, though Quanta are not things in the usual sense. But it was a trivial extension which laid the groundwork for what I was rejecting, rather than claiming.
Ah, I see.

Infinity as a concept in the mind is ok
but
when applied to/in reality, in my understanding/opinion, does not work.
Oh?

The definition itself is self contradictory/limiting.
Which definition? There are a lot of definitions of infinity.

however its possible that if you were to reach the "edge" of say, time-space or any other variable/dimension, you could extend it further but it would still remain finite.
Actually, in most cases, infinite means unbounded, so there would be no such edge.

If the universe were infinte, we'd likely not know it. The earth would most likely remain in orbit about the sun, and our galaxy would be indistinguishable from the one we observe....
in fact all observations would be the same.

Flatness or near flatness that is usually assumed is relevant to the observable universe. Our inflationary cosmology says nothing of the global geometry of the universe....yet.

Some more interesting paradoxes have been proposed since Zeno. Including the "Ant on a rubber rope" and "Hilbert's paradox of the Grand Hotel".
Thanks for mentioning those, I had never run across the Ant On A Rubber Rope before.
Might Hilbert's Hotel have a flaw in the premise that relates to problems with infinity? If an infinite number of rooms are each occupied by a guest, where does the new guest come from? Some think that an infinite collection must necessarily contain all instances...

Actually, infinite means unbounded, so there would be no such edge

Thanks for mentioning those, I had never run across the Ant On A Rubber Rope before.
Might Hilbert's Hotel have a flaw in the premise that relates to problems with infinity? If an infinite number of rooms are each occupied by a guest, where does the new guest come from? Some think that an infinite collection must necessarily contain all instances...
It was Georg Cantor that demonstrated that not only was actual infinities perfectly logical but that it necessarily entailed orders of infinity, called aleph numbers $\aleph$ or cardinality (a powerset). Cardinality is basically the size of an infinite set. This was controversial in Cantor's day since the only infinity acceptable prior to that was unbounded sets, or potential infinities.

You can look up his work, including his diagonal argument, for a more quantitative treatment. I'll just give a more intuitive description. If you have a finite interval, i.e., distance between two points, then logically you can divide it into an infinite set of infinitesimal points. To answer the above question one way, it cannot be said that the infinite set of points between 0 and 1 contain all possible numbers. There is not only more than one infinite set, there are an infinite set of infinite sets. There is no flaw in Hilbert's Hotel.

There is also countably infinite sets and uncountable sets. There is also open sets and closed sets, which differ only in whether they contain the infinitesimal boundary points or not. There are also dense sets, which play a role in many quantum foundation arguments, including EPR and the scalability of quantum computing.

The one thing you cannot do is make a priori generalized statements, like infinity must contain the entirety of the whole Universe to be infinite. Any finite subset of the Universe also contains an infinite set of infinitesimal points. Neither can you, for the same reason, make the claim that two infinite sets must be the same size. You must restrict your statements about infinity to those statements that can be mathematically demonstrated to be consistent, and avoid the intuitively implied and inconsistent properties Zeno's arguments depended on.

To answer the above question one way, it cannot be said that the infinite set of points between 0 and 1 contain all possible numbers. There is not only more than one infinite set, there are an infinite set of infinite sets. There is no flaw in Hilbert's Hotel.
I'm not thinking that the infinite set of points between 0 and 1 contains all possible numbers, only that it contains all possible numbers between 0 and 1. It seems to me by definition, the set of points between 0 and 1 must include every point between 0 and 1. Are you suggesting otherwise?

I'm thinking that any arbitrary number I specify between 0 and 1 must already be included in the set of points between 0 and 1; so I don't see how any possible number between 0 and 1 is not already a member of the set of points between 0 and 1.

The "new guest" coming to Hilbert's Hotel's is like a point between 0 and 1 that is not a member of the set of points between 0 and 1... I see this as a flaw in the premise.

If the 0 to 1 range is problematic, we can do the same with the set of natural numbers... I'm thinking that the set of natural numbers must include any and all arbitrary natural numbers that I may specify... this seems clear by definition.
If each occupied room is mapped to a natural number, an infinite number of rooms means all the natural numbers are mapped, as are their corresponding guests... the "new guest" would need to represent an unmapped natural number, but there are none, by definition.

Maybe I'm missing something...?

The "new guest" coming to Hilbert's Hotel's is like a point between 0 and 1 that is not a member of the set of points between 0 and 1... I see this as a flaw in the premise.
How does it remotely resemble this?

EDIT: Upon closer inspection, the paradox is simply mapping natural numbers to new natural numbers. The set ##\mathbb{Z}\cup\left[1,\infty\right)## has the same cardinality as the set ##\mathbb{Z}\cup\left[2,\infty\right)##. This should solve the paradox quite easily.

Actually, in most cases, infinite means unbounded, so there would be no such edge.
or finite without boundaries, like a torus.

I'm not thinking that the infinite set of points between 0 and 1 contains all possible numbers, only that it contains all possible numbers between 0 and 1. It seems to me by definition, the set of points between 0 and 1 must include every point between 0 and 1. Are you suggesting otherwise?
In the original post I addressed it was implied that maybe if there was an infinite number of hotel rooms, then these rooms being occupied implied infinite guest such that there could be no new guest. If there are an infinite set of point between the two points, [0,1], and each of these correspond to a hotel room occupied by a point, then the original suggestion to work around the hotel hotel paradox implies that this infinity of points, [0,1], contains all points that might occupy the infinity of hotel rooms. Hence I made the suggestion in order to provide proof by contradiction that the hotel paradox was not flawed.

I'm thinking that any arbitrary number I specify between 0 and 1 must already be included in the set of points between 0 and 1; so I don't see how any possible number between 0 and 1 is not already a member of the set of points between 0 and 1.

The "new guest" coming to Hilbert's Hotel's is like a point between 0 and 1 that is not a member of the set of points between 0 and 1... I see this as a flaw in the premise.
If the points between 0 and 1 are an infinite set of occupied hotel rooms, and yet "new guest" are still available from members that are not member of the set of points between 0 and 1, why is this a special case? The original suggestion was that an infinite number of guest implied no more guest exist, but here you add a special case to say there are more guest available from sets other that [0,1].

If the 0 to 1 range is problematic, we can do the same with the set of natural numbers...
Precisely. The infinity problem is just as big in the interval [0,1] as it is in the interval [0,∞].

I'm thinking that the set of natural numbers must include any and all arbitrary natural numbers that I may specify... this seems clear by definition.
If each occupied room is mapped to a natural number, an infinite number of rooms means all the natural numbers are mapped, as are their corresponding guests... the "new guest" would need to represent an unmapped natural number, but there are none, by definition. Maybe I'm missing something...?
Only problem is that I can pull new guest from the infinite set of real number which you didn't included here. Note that the numbers are merely name tags on the guest, and it make no difference which ones you label with which numbers. I can relabel an infinite number of guest labeled with even numbers with odd numbers, and visa versa, and the count remains the same. I can also relabel all natural numbers as real numbers simply by multiplying their name tags with an irrational number and assigning them that number. It changes nothing about the total number of guest.

Cantor showed that all these infinities existed, but we should not lose sight of the fact that they are mathematical infinities. Mathematically, what does it mean to say that something exists? If a mathematician can write down a set of non-contradictory axioms, and set down rules for deducing mathematically true statements from them, then those statements can be said to ‘exist’. This existence requires only logical self-consistency. Physical existence is completely unnecessary.

If there can be a profound difference between physical and mathematical “existence” then it seems reasonable to identify a similar difference between physical and mathematical “truth”. Cantor’s infinities were all mathematical infinities, as are the rooms and guests in Hilbert’s Hotel. They may bear no relation to any possible physical infinity, which would include an infinite universe.

The actual, ancient, fear of infinity was not removed; it was just that Cantor provided the world with a “label” that could be attached to infinity, which reads: “this is a mathematical infinity – it doesn’t bite”.

my_wan,
Your responses support my thinking that the Hilbert Hotel premise is flawed.

A "complete" mapping to the natural numbers would be to include all of them by taking the natural numbers in order... 1,2,3... any other mapping scheme like 2,4,6... is an obvious mechanism to skip some numbers, yet there would be objection to a scheme that skipped points between 0 and 1.
Claiming that the new guest could be from the set of real numbers when the set of guests is represented by the natural numbers misses the whole point. If the points between 0 and 1 represent the rooms, it is the guests that are mapped to the natural numbers. The paradox is based on the assumption that all guests, including any possible new guests, are all of the same class of thing - natural numbers, and violating that by suggesting a new guest could be a real number is like solving the question of the origin of the new guest by finding a chair and checking that chair into the hotel as a new guest... no, the new guest has to be a person like all the other guests.

Claiming that the new guest could be from the set of real numbers when the set of guests is represented by the natural numbers misses the whole point.
It doesn't miss the point any more than saying that the set of all hotel customers must consist of all possible hotel customers, and that is the only way you can claim there is nobody remaining to request a room in the hotel.

If the real numbers represent the set of all present hotel customers, and the natural numbers represent the set of all people that might request a room, then there are an infinity of people that may request a room even after an infinite number of people have already filled the hotel. To assume otherwise is effectively an attempt to impose a boundary condition on an unbounded variable.