# I Is Infinity Possible?

#### sysprog

There is no progression. The question is which player has a winning strategy.
The progression I envisioned would be that each volume magnitude proposed by player 1 and acceded to as a lower bound by player 2 would be greater than that on the previous turn.

As to the winning strategy question, the second player always gets to make use of the first player's efforts, and gets to use the 'greater than or equal to' property to ensure that he is never wrong unless the first player has already erred.

Citing the 'high confidence' conjecture from your post ...
... we would have high confidence that there are subsets which have more than any specified finite volume. [We would have to nail down with some precision what sort of subsets we had in mind and what volume measure we'd use]
... you appear to be postulating a universe whose size is at least countably infinite, wherein there would be no winning strategy for such a game, but in the case of a finite universe, the advantage would be to the second player, who would always come out with at least a draw.

#### jbriggs444

Homework Helper
The progression I envisioned would be that each volume magnitude proposed by player 1 and acceded to as a lower bound by player 2 would be greater than that on the previous turn.
There are no turns. You get one go.

#### sysprog

There are no turns. You get one go.
That's an arbitrary and unwarranted constraint.

#### jbriggs444

Homework Helper
That's an arbitrary and unwarranted constraint.
Multiple turns is unnecessary.

#### sysprog

Multiple turns is unnecessary.
In that case, my earlier answer (from post #51) seems sufficient to me:
... you appear to be postulating a universe whose size is at least countably infinite, wherein there would be no winning strategy for such a game, but in the case of a finite universe, the advantage would be to the second player, who would always come out with at least a draw.
Looking at the case in which player 1 specifies 1 stere ($1m^3$) as the volume, player 2 wins, for any reasonable implementation of your earlier proviso:
[We would have to nail down with some precision what sort of subsets we had in mind and what volume measure we'd use]
Assuming that you aren't just being facetiously or smugly pedantic about expositing the idea that if the universe is volumetrically finite, it is apt to be much larger than any finite number we could specify, I don't see much of a point to your game.

#### jbriggs444

Homework Helper
I don't see much of a point to your game.
The point is to produce a definition of "infinite" which does not use the term "infinity"

#### sysprog

The point is to produce a definition of "infinite" which does not use the term "infinity"
I don't see how or why that's helpful. There are many such definitions. For example, to specify infinity iterations, one could say: the number of iterations that will occur before the following program halts:
Code:
10 goto 10
.
What's wrong with posing the question 'does the universe have infinite volume?'?

#### jbriggs444

Homework Helper
I don't see how or why that's helpful. There are many such definitions. For example, to specify infinity iterations, one could say: the number of iterations that will occur before the following program halts:
Code:
10 goto 10
.
A course in real analysis might be of use.

#### Klystron

Gold Member
It would be remiss to not mention here the name Georg Cantor who showed there are infinitely many kinds of infinity. He has pre-worried about some of this for you. Wrap your head around that........here's a start:
https://www.britannica.com/science/transfinite-number
His primary works are relatively approachable without too much pain.
It bears reiteration that Einstein and fellow scientists had the benefit of German mathematician Georg Cantor's development of set theory. Understanding sets, particularly construction and metrics, answers the basic question asked in this thread.

#### sysprog

The point is to produce a definition of "infinite" which does not use the term "infinity"
I don't see how or why that's helpful. There are many such definitions. For example, to specify infinity iterations, one could say 'the number of iterations that will occur before the following program halts'
Code:
10 goto 10
.
What's wrong with posing the question 'does the universe have infinite volume?'?
A course in real analysis might be of use.

#### Mark44

Mentor
The point is to produce a definition of "infinite" which does not use the term "infinity"
I don't see how or why that's helpful.
A definition of a term should not use that same term, or wording that is only slightly different. Your definition involving a Basic example seems fine to me to, conveying as it does the idea of endless repetition.

#### sysprog

It bears reiteration that Einstein and fellow scientists had the benefit of German mathematician Georg Cantor's development of set theory. Understanding sets, particularly construction and metrics, answers the basic question asked in this thread.
I don't see how it does that. We don't know, for example, whether the physical analog of the $\mathbb R^3$ space does or does not conform to the axiom of completeness that the mathematical $\mathbb R^3$ space conforms to. If it does, then physical space is continuous, and there is no physical minimum distance greater than the infinitesimal. We also don't know whether the physical universe has the same size as $\mathbb R$. If it does, then it's infinite. But even if it's not continuous, it could still be volumetrically infinite, or not.

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#### sysprog

A definition of a term should not use that same term, or wording that is only slightly different. Your definition involving a Basic example seems fine to me to, conveying as it does the idea of endless repetition.
No-one in this thread started out to give a tautological definition, and I don't think that's what @jbriggs444 was driving at, but he's being at best cryptic when he suggests a course in real analysis in response to being asked what's wrong with posing the question 'is the universe volumetrically finite or not'. [Edit: when making that suggestion, he quoted my post only up to the end of the 1-line program, and didn't quote the question that I posed after that, so maybe he was just trying to suggest that my definition by specification of infinite process was too naive for purposes of this discussion.] Also, I understand that an endless process is not the only way to conceive of the infinite, and I was using one only as an example. I think the question whether the physical universe is finitely or infinitely large, if it can be resolved at all, cannot in either case legitimately be dismissed as a matter of definition.

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#### Mark44

Mentor
but he's being at best cryptic when he suggests a course in real analysis in response to being asked what's wrong with posing the question 'is the unverse volumetrically finite or not'.
Yeah, I don't see anything wrong with asking this question.

#### Buzz Bloom

Gold Member
The point is to produce a definition of "infinite" which does not use the term "infinity"
Hi jbriggs:

I may well be misunderstanding your concept, but it seems to me that you are replacing the word "infinity" or a definition of "infinity" with a process that takes an infinite number of steps in order to demonstrate that an infinite volume is in fact infinite. I am OK with this from the point of view that this approach may be more aesthetic to you than defining the concept of a physical infinite volume in terms of a definition of an infinite value for a physical attribute. However, my own personal aesthetic is the opposite. How do you feel about the parallel postulate using the concept of "indefinitely" which also avoids the use of "infinity"?

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Wikipedia presents a list of 15 equivalent postulates. Here is #9.
There exists a pair of straight lines that are at constant distance from each other.​

This concept is true for a space with zero curvature, and that implies an unbounded (infinite) volume. For a finite universe, this #9 would not be true.

Regards,
Buzz

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#### jbriggs444

Homework Helper
replacing the word "infinity" or a definition of "infinity" with a process
There is no sequential process. You win or lose in one round. Either player A can win (by presenting a volume that won't fit) or player B can win (by showing that player A's proposed volume will fit).

#### Buzz Bloom

Gold Member
There is no sequential process. You win or lose in one round. Either player A can win (by presenting a volume that won't fit) or player B can win (by showing that player A's proposed volume will fit).
Hi jbriggs:

I apologize if I have misunderstood your descriptions of the contest. Please explain how it can be determined in one round that a universe has an unbounded volume?

Regards,
Buzz

#### jbriggs444

Homework Helper
I apologize if I have misunderstood your descriptions of the contest. Please explain how it can be determined in one round that a universe has an unbounded volume?
It is a definition, not a procedure.

#### sysprog

There is no sequential process. You win or lose in one round. Either player A can win (by presenting a volume that won't fit) or player B can win (by showing that player A's proposed volume will fit).
In that case, if player B wins, all that has been shown is that the universe is at least as large as the volume specified by player A. It hasn't thereby been shown that it's larger than any volume that could have been specified by player A. Repeated iterations with larger volumes specified could establish progressively larger minimum volumes as long as B keeps winning. Unless player A wins, the game cannot tell us whether the universe is finite or infinite.

Simulating the game in pseudocode:
Code:
if Aguess > Uvolume then Awins;
else Bwins;
A pseudocode version of the game with more than one iteration:
Code:
do while not(done);
Aguess = AGuess + 1;
if AGuess > Uvolume then done = 1;
end
If the program halts, the universe is finite, and if it doesn't, it isn't, but we already know that the physical universe is volumetrically at least much larger than we can factually test for, so neither procedure can really tell us whether the universe is finite or infinite; only that it's at least as large as A's latest guess.

#### jbriggs444

Homework Helper
In that case, if player B wins, all that has been shown is that the universe is at least as large as the volume specified by player A.
Again, you fail to understand. The question is who has the winning strategy. If there is a winning strategy, one round is all it takes.

#### sysprog

Again, you fail to understand. The question is who has the winning strategy. If there is a winning strategy, one round is all it takes.
If player A has the winning strategy, then it must be because he knows how to specify a number that is larger than than that needed to exceed the volumetric size of the Universe, while if player B has the winning strategy, it must be because he knows that the universe has a size at least as large as any size that could be denumerated by A.

Wherefore, A could have a winning strategy only if the universe is finite, and B could have a winning strategy only if B knows its size to be at least as large as anything A could specify.

I don't see how this is equivalent to the question whether the universe is finite or infinite; if you do, please elaborate, instead of merely telling me that I don't understand.

#### jbriggs444

Homework Helper
If player A has the winning strategy, then it must be because he knows how to specify a number that is larger than than that needed to exceed the volumetric size of the Universe, while if player B has the winning strategy, it must be because he knows that the universe has a size at least as large as any size that could be denumerated by A.

Wherefore, A could have a winning strategy only if the universe is finite, and B could have a winning strategy only if B knows its size to be at least as large as anything A could specify.

I don't see how this is equivalent to the question whether the universe is finite or infinite; if you do, please elaborate, instead of merely telling me that I don't understand.
For about the third or fourth time, this is a definition of what it would mean for the universe to be infinite, not an operational test to decide the question.

#### sysprog

For about the third or fourth time, this is a definition of what it would mean for the universe to be infinite, not an operational test to decide the question.
I disagree with your contention that it is such a definition. I think that none of your responses adequately addresses the issue I've raised regarding what I perceive to be its deficiency in that regard. In particular, it appears to me that by the parameters you've specified, it's possible for the universe to be finite, but ineffably large. I think it's possible that there are finite numbers larger than any that we could specify other than procedurally.

Haha, I think the OP has been totally confused now. He hasn't replied at all.

#### cmb

Speculating what is outside our measurable universe is contrary to the rules of the forum.

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