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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?
Zeno's paradox has been sufficiently done away with via the use of calculus. In calculus, no such quantization is necessary.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.
Infinity as a concept in the mind is okThere 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.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?
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.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. That's not how inertia works.there is a physical problem, in an infinite universe it have an infinite mass and hence infinite inertia, no motion would be possible.
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.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.
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?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. 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."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.
Ah, I see.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.
Oh?Infinity as a concept in the mind is ok
but
when applied to/in reality, in my understanding/opinion, does not work.
Which definition? There are a lot of definitions of infinity.The definition itself is self contradictory/limiting.
Actually, in most cases, infinite means unbounded, so there would be no such edge.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.
Thanks for mentioning those, I had never run across the Ant On A Rubber Rope before.Some more interesting paradoxes have been proposed since Zeno. Including the "Ant on a rubber rope" and "Hilbert's paradox of the Grand Hotel".
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.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...
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?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.
How does it remotely resemble this?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.
or finite without boundaries, like a torus.Actually, in most cases, infinite means unbounded, so there would be no such edge.
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 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?
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].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.
Precisely. The infinity problem is just as big in the interval [0,1] as it is in the interval [0,∞].If the 0 to 1 range is problematic, we can do the same with the set of natural numbers...
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.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...?
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.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.