The problem with infinity.

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.

 

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 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.

 

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.

 

No. That's not how inertia works.

 

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.

 

Ah, I see.

 

Oh?

Which definition? There are a lot of definitions of infinity.

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.

 

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

 

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 [itex]\aleph[/itex] 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.

 

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...?

 

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.