Exploring the Science of Infinity: Is It Possible?

[Nugatory]

I think the confusion is between (a) the absence of a limit, or (b) having a infinite value for a property.

Much of the problem here is in the informal English-language term "having an infinite value", which is somewhere between too vague to reason accurately with and just plain meaningless. Infinity is not a value (unless you happen to be an IEEE 854 nerd - irrelevant here) so anyone who is using that phrase is necessarily saying something else... and this is one of the times when the mathematicians' insistence on rigor produces better results than the physicists' lawless coercion of the math.

 

[Buzz Bloom]

It's a reasonable conjecture that some other physics is involved at sufficiently small values of rr to keep everything finite, but that is certainly not observationally confirmed and there is no compelling candidate theory.

Hi Nugatory:

Have two questions regarding the quote.
1. Would it be correct to say that it is impossible to observe any phenomena inside the event horizon?
2. Is it theoretically possible for there to be a compelling theory? That is, what could make a theory about what goes on inside the event horizon compelling?

Regards,
Buzz

 

[jbriggs444]

1. Would it be correct to say that it is impossible to observe any phenomena inside the event horizon?

If you dive in, you can observe. You just can't report back.

 

[Buzz Bloom]

Much of the problem here is in the informal English-language term "having an infinite value", which is somewhere between too vague to reason accurately with and just plain meaningless. Infinity is not a value (unless you happen to be an IEEE 854 nerd - irrelevant here) so anyone who is using that phrase is necessarily saying something else... and this is one of the times when the mathematicians' insistence on rigor produces better results than the physicists' lawless coercion of the math.

Hi Nugatory:

In what way is the following assertion "too vague to reason accurately with and just plain meaningless"?
Assuming that (a) the GR models of the universe are correct, and (b) that future astronomical data results in a mathematically correct calculation that the average curvature of the universe is positive with a confidence of 99.99%, it would then be also be known with a confidence of 99.99% that the volume of the universe is a value that is an infinite number of cubic meters.

Regards,
Buzz

 

[Buzz Bloom]

If you dive in, you can observe. You just can't report back.

Hi jbriggs:

Thank you for the insight. I will change my question to take your observation into account.

Would it be correct to say that it is impossible to observe from the outside any phenomena inside the event horizon, and also from the inside any phenomena happening closer to the center of the black hole than you are?

Regards,
Buzz

 

[jbriggs444]

Hi Nugatory:

In what way is the following assertion "too vague to reason accurately with and just plain meaningless"?
Assuming that (a) the GR models of the universe are correct, and (b) that future astronomical data results in a mathematically correct calculation that the average curvature of the universe is positive with a confidence of 99.99%, it would then be also be known with a confidence of 99.99% that the volume of the universe is a value that is an infinite number of cubic meters.

Regards,
Buzz

Putting a mathematical hat on, I'd be more comfortable saying that 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].

That avoids the need to treat "infinity" as a value.

 

[sysprog]

and this is one of the times when the mathematicians' insistence on rigor produces better results than the physicists' lawless coercion of the math.

[rant]
I think that mathematicians should accept some of the responsibility for non-rigor; for example, treating the infinitesimal as non-zero when it's among the infinity of infinitesimals being summed, and then treating it as exactly zero when it's being disregarded at the asymptote, is formally inconsistent, and leads to absurdities that offend both rigorous and non-rigorous understanding, such as saying that having zero probability is not the same thing as being impossible, but impossible things are among those which have zero probability, and insisting on using zero to designate both of those kinds of zero without acknowledging that there must be more than one meaning in mathematics for zero, that zero in fact means one of the numbers on the continuum from the negative infinitesimal to the positive infinitesimal inclusively, with what is ordinarily meant by zero being the midpoint of that interval, and what is meant by zero in the statement 'the probability of choosing a rational number at random from within any interval within the reals is zero', is actually some number greater than the midpoint of the interval from the negative infinitesimal to the positive infinitesimal, whereas the impossible has a probability which is not even infinitesimally greater than zero, and so has what may be called a 'strictly not greater than zero' probability.
[/rant]

 

[Buzz Bloom]

[We would have to nail down with some precision what sort of subsets we had in mind and what volume measure we'd use].

Hi jbriggs:

I may be mistaken, but it seems to me to be impossible to define any such subset without saying either (a) "the subset is infinite in some appropriately defined dimension, or (b) the length value of some dimension is greater than any finite number.

Would it be satisfactory to you saying, "The volume of a GR universe with positive curvature is greater than any finite volume"? This avoids "infinite" by using instead a definition for infinite.

Regards,
Buzz

 

[ZapperZ]

It's common knowledge that there are infinities in the mathematics that describe physical phenomena. We don't know whether anything physical is actually infinite. Your post questioned why there appeared to be a fixation on extension of space as to whether it is finite or infinite, and I presented an example of similarly unknown matters on the smallness scale as distinguished from the largeness scale; just as we don't know whether the universe is infinitely large, we don't know whether distances or durations can be infinitely or infinitesimally small.

Then this is an issue that is not falsifiable, and thus, a topic in philosophy. Have fun going around in circles!

Zz.

 

[Nugatory]

2. Is it theoretically possible for there to be a compelling theory? That is, what could make a theory about what goes on inside the event horizon compelling?

"Compelling" is a matter of opinion and aesthetic preference, but I'd take as an operational definition that a theory is compelling if only a crackpot would argue, in the absence of a well-reasoned alternative, that we shouldn't be applying it.

I personally would not hesitate to trust classical E&M (applied within a sufficiently small region in a free-fall frame) on either side of the event horizon... for the same reason that we were willing to trust it to work on the far side of the moon before the era of space-travel. True, it hasn't yet been tested in that domain, but only a total crackpot would have suggested that it wasn't applicable and in that sense the theory was compelling.

We don't have a theory that as convincingly extends to extreme densities and pressures... but there's no reason why such a thing couldn't be discovered tomorrow.

 

[sysprog]

Then this is an issue that is not falsifiable, and thus, a topic in philosophy. Have fun going around in circles!

Zz.

A theory that a particular something is infinite may well be falsifiable. We know that some things once theorized to be infinite are in fact finite, for example, the speed of light. Similarly, we might in future learn of a minimum distance, a minimum time interval, and a consequent maximum frequency of light. I think we may not ever be able to prove anything physical to be infinite; however, some things may long resist every effort to prove them finite.

 

[jbriggs444]

I may be mistaken, but it seems to me to be impossible to define any such subset without saying either (a) "the subset is infinite in some appropriately defined dimension, or (b) the length value of some dimension is greater than any finite number.

I think you have mistaken the notion I was describing. None of the subsets involved need be infinite.

I am suggesting something very much like a limit. We play a game. You specify a volume (one cubic meter, one cubic light year, one hundred megaparsecs cubed, a cube with each side the factorial of Graham's number meters in length or whatever). Then I say "yes, there is a subset of the universe with at least that volume".

"For every volume V > 0, there is a subset with a volume at least that great".

That avoids not just the need to speak of infinity but also the need to speak of a numerical volume of the universe.

Edit: The above is just about definitions. I make no claim whether the universe really is or is not infinite in this sense.

 

[Matt Benesi]

I would appreciate seeing an example of such physical infinities. I am guessing there may be another misunderstanding that I may be able to explain.

I find myself wondering how many directions an averaged magnetic field is oriented in as I spin a magnet (relative to something else).

 

[sysprog]

You specify a volume (one cubic meter, one cubic light year, one hundred megaparsecs cubed, a cube with each side the factorial of Graham's number meters in length or whatever). Then I say "yes, there is a subset of the universe with at least that volume".

"For every volume V > 0, there is a subset [of the $\mathbb R^3$ spatial universe] with a volume at least that great".

(bracketed referent added)

Clearly, if and only if the universe is infinite (not volume-bounded), you and the other player (the immortal versions) can forever keep playing this game. But if it's finite, you'll at some point be blocked from completing your next turn, unless the last volume specified by the other player and acceded to by you were to be the actual finite limit, at which point the game would be concluded, unless the size of the universe were to increase.

That avoids not just the need to speak of infinity but also the need to speak of a numerical volume of the universe.

I don't see how how it accomplishes either of these. The game progressively increases the lower volumetric bound of the hypothetical $\mathbb R^3$ spatial universe, but it doesn't eliminate the consideration of whether there is physically an upper bound to the volumetric size. In ordinary vernacular, if there is, the universe is finite, and if not, it isn't.

Edit: The above is just about definitions. I make no claim whether the universe really is or is not infinite in this sense.

Me neither.

 

[jbriggs444]

The game progressively increases

There is no progression. The question is which player has a winning strategy.

 

[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]

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]

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]

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:

10 goto 10

.
What's wrong with posing the question 'does the universe have infinite volume?'?

 

[jbriggs444]

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:

10 goto 10

.

A course in real analysis might be of use.

 

[Klystron]

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'

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.

Please explain why.

 

[Mark44]

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.

 

[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.

 

[Mark44]

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'.

Yeah, I don't see anything wrong with asking this question.

 

[Buzz Bloom]

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"?

https://en.wikipedia.org/wiki/Parallel_postulate​

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

 

[jbriggs444]

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]

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]

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:

if Aguess > Uvolume then Awins;
else Bwins;

A pseudocode version of the game with more than one iteration:

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]

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.