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January 21, 2020

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TOPIC: Is the universe infinite or just really really big? A proposed way to distinguish the two possibilities [refresh]

TOPIC: Is the universe infinite or just really really big? A proposed way to distinguish the two possibilities [refresh]

As promised, this is a follow-up to one of the summary posts from the FQXi conference in Vieques. If you have read those, you may recall that Anthony Aguirre asked the intriguing question: is there any way for us to tell if the universe is infinite or simply really, really big? (George Musser has also blogged about this.) In this blog post I suggest one possible way in which this might be accomplished. I emphasize that much of the content of this post is entirely speculative but it does offer suggestions on how to more rigorously determine the validity of the conjecture. It relies, however, on an assumption that runs entirely counter to my own conclusion in my most recent FQXi essay: that bits (in some fashion) constitute the basic building blocks of the universe.

**The Setup**

Consider the following string of binary digits

…011010010101101…

where, for the moment, we assume that the full string is infinite but that we only have knowledge of the sub-string shown above. Now consider the following maps:

00 —> a

11 —> a

01 —> b

10 —> b

Notice that each digit in the original string ends up getting counted twice.

Thus the original string can be mapped to

…XbabbbabbbbabbY…

where the X and the Y are unknown given that we do not know what precedes the 0 on the left and what follows the 1 on the right. Formally, this second string is doubly-infinite meaning its cardinality is double the cardinality of the first string since every digit in the first string is counted twice (yes, there are actually*levels* of infinity!).

Now suppose that we have complete knowledge of the doubly-infinite string of a’s and b’s with the exception of one value. It should be relatively clear that, if we know the distribution of 0’s and 1’s in the original string (e.g. 60/40 or 30/70), then the unknown value in the second string should be immediately known unless the original distribution is 50/50 in which case we have no idea (try it!).

So now take any two locations, X and Y, on what we call the “image” string (the strings of a’s and b’s) with the caveat that there are at least two letters between them. Suppose that we have complete knowledge of the doubly-infinite string*other* than these two locations. From the argument in the preceding paragraph, it should be clear that the instant we know one of these two values (say X), we immediately know the other (Y). A formal proof of this appears in Section 5 of this article by Steve Shea, which is open access.

**Do we really need an infinite amount of knowledge?**

The way I have described the problem (courtesy of Steve Shea) would seem to imply that we need to know absolutely every value in the doubly-infinite string, i.e. we need an infinite amount of knowledge, if we are to correctly predict the value of Y given the value of X (which also assumes that the original string of 0’s and 1’s is not a 50/50 split). But this gets to the heart of Anthony’s problem: do we really need an infinite amount of knowledge or just a*whole lot* of knowledge? One could imagine that we could approach a very high probability of correctly predicting Y given our knowledge of X as our knowledge of the doubly-infinite string gets large (though I would like to emphasize that I am not aware of a formal proof of this as yet—Steve’s article has just been published, though he’s been working on it for several years).

Nevertheless, approaching such a high-degree of accuracy with a prediction would also seem to require that the strings really*are* infinite. One can imagine that if they are *not* infinite, there could be some sort of "edge" effect caused by the fact that the second string would be ill-defined at the ends and that such an effect could somehow propagate through the string (again, I emphasize that this is speculation at this point and that there is no formal proof of this). For example, we could interpret the mapping (and thus, to some extent, the original string) as being nothing more than a kind of production rule somewhat akin to those in formal language theories: it’s just a rule for generating the string of a’s and b’s (note that in such theories the "start" symbol need not be at the extreme left or right of a string which means it could still be infinite—if you are unfamiliar with the concept of a "start" symbol, a good read is Douglas Hofstadter’s Pulitzer Prize-winning opus Gödel, Escher, Bach). Given that (and assuming that the ratio of 0’s to 1’s in the original string is *not* unity, i.e. they are not a 50/50 split since, otherwise, there really is no production rule), it should be clear that the a’s and b’s are interdependent which means I could rewrite the production rule simply in terms of the a’s and b’s themselves. Inherent in the original rule, however, there is no way to deal with endpoints. Specifically, the endpoints of the string of a’s and b’s would be ill-defined based on the production rule which would make any neighboring values in the string undefined and so on such that the entire string is ill-defined (e.g. it would be as if there was no "start" symbol). Thus the string must be infinite, at least in this formal system.

I would then*conjecture* that while the strings themselves *must* be infinite, we only need a *finite* (though arguably large) amount of knowledge in order to predict Y given X with a high degree of accuracy. Of course this all hinges on whether or not the original string is completely random. If the original string of 0’s and 1’s is exactly 50/50, the value of Y cannot be predicted with any greater accuracy than 50% (which essentially means it cannot be predicted at all). Likewise, the requirement that the string of a’s and b’s be infinite no longer holds, i.e. it’s really a relic of the production rule.

**A universe of qubits**

What does any of this have to do with the universe? Let’s just suppose that John Wheeler was correct and that, at its core, the universe is built up from bits—binary digits—or, rather, qubits. Since we don’t actually see the world as 0’s and 1’s (or answers to yes/no questions) it is clear that there is some form of mapping that goes on at some deep level from these binary questions to something slightly less fundamental. Let’s take, as a first approximation, the map introduced at the beginning and suppose that the string of a’s and b’s represents the results of measurements of simple two-level quantum systems. In other words, the a’s and b’s represent our*knowledge* of the universe at its most fundamental level (notice that it may or may not suggest a deeper reality of 0’s and 1’s or even something else entirely). We might then imagine that the universe (or, rather, our knowledge of it) can be reduced to a sub-string of a very, very long (possibly infinite) string of outcomes of measurements on qubits. This is not so radical an idea, by the way (see Seth Lloyd’s article on this topic: arXiv:1312.4455).

One glitch in this argument, of course, is that we don’t know the order of the measurement outcomes, i.e. the string of a’s and b’s. We have small strings of correlated outcomes, but we don’t know for certain if all of these small strings are part of one larger string, i.e. if we know, for example,

…abbababa…

…babbabab…

…aaababaa…

is it necessarily true that

…abbababa…babbabab…aaababaa…

or that

…babbabab…abbababa…aaababaa…?

If all of the sub-strings indeed are part of one, larger string it would seem to suggest that all possible qubit measurements in the universe may be correlated in some way since, as long as there exists a production rule, we can connect even the most far-flung elements of the full string of a’s and b’s. This, also, is not such a radical idea if we take the correlations as being equivalent to quantum entanglement (see Buniy and Hsu’s article on this: arXiv:1205.1584v2).

Note that there is one question here that I have not addressed and that is whether knowledge of X necessarily implies that Y is the*opposite* of X. If the a’s and b’s represent measurement outcomes for entangled qubits then one would seem to expect that the outcomes of X and Y must be opposite one another. There is no requirement in the mathematics that this be the case. In addition the mathematics imply a string of somewhat looser correlations (recall they must be at least two letters apart) in between the entangled pair. It might be possible to address the latter through something like quantum teleportation, but I won’t address that here. Instead I will simply assume that the production rule is such that the sub-strings that are accessible to us *appear* to behave in such a way that X and Y always behave as if they are entangled.

**Is the universe infinite or just really, really large?**

Let’s take Buniy and Hsu’s idea as correct in which case all qubit measurements are somehow part of a single, large string. We can access different sub-strings of this larger string via entanglement measurements. Note, however, that X and Y do not necessarily need to be in the same sub-string. So, for instance, an entanglement experiment with qubits might represent the following pair of sub-strings

…abbabaXabbaba…ababaabYbabaaba…

In other words, we are ignorant of some of the intermediary processes that connect them. Nevertheless, we do know that a determination of X immediately tells us what Y must be since this is measurable in a laboratory. This suggests that the intermediary processes, while perhaps not readily apparent to us, nevertheless exist as long as there is a production rule and as long as the full string is doubly-infinite. In fact, if Buniy and Hsu are correct (and note that their argument is based on cosmological models and thus not dependent on anything I suggest here), the mere presence of entanglement suggests that the string of all qubit measurements in the universe is infinite which would itself suggest that the universe is infinite,*but only if the underlying production rule is not completely random*. Recall that none of this works if the 0’s and 1’s in the original string are a 50/50 split. If they are, there is no correlation between X and Y.

I caution against rushing to judgement about these ideas. Clearly we see correlations and entanglement in the universe on some level. Does this automatically imply, then, that the universe is infinite*and* not entirely random at its base level? It clearly does not since we don’t know if Buniy and Hsu are correct. In other words, we have no idea of our measurements of entanglement are merely parts of one large, interconnected string of such measurements. What it does seem to tell us is that if they are correct (and it is a big "if"), then it is highly likely that the universe is both infinite and non-random. (It also would seem to suggest that Max Tegmark might be right after all about the role mathematics plays in the universe.)

Now Anthony was interested in this from the standpoint of the multiverse and one could easily modify these ideas to take that into account. I won’t do it here. I can also imagine someone taking Steve’s results*without* reference to Buniy and Hsu and coming up with a way to measure whether the universe is infinite or merely very large.

There are a lot of assumptions and conjectures in this post but there are also a lot of concrete starting points for further exploration. Can we definitively prove, mathematically, that these strings*must* be infinite for this effect to be possible? I offered a heuristic argument above as to why, but a more formal proof would be welcome. Can we really re-interpret Steve’s original mapping in a way that makes it self-referential to the string of a’s and b’s, i.e. can we find a production rule for the string of a’s and b’s such that it would exactly match what we would have obtained using the original mapping? Conversely, if we can’t, can we definitively rule out the possibility that one exists? Are Buniy and Hsu correct in suggesting that all particles in the universe are ultimately entangled and can we actually reduce everything to a set of qubit measurements? What might the results presented here say about the multiverse? Is there some other way to work with these results that might say something useful about the size of the universe?

In my mind, these are all ideas worth pursuing and so I suppose you could interpret this blog post as a challenge: let’s get some answers to these questions!

this post has been edited by the author since its original submission

report post as inappropriate

Consider the following string of binary digits

…011010010101101…

where, for the moment, we assume that the full string is infinite but that we only have knowledge of the sub-string shown above. Now consider the following maps:

11 —> a

01 —> b

10 —> b

Notice that each digit in the original string ends up getting counted twice.

Thus the original string can be mapped to

…XbabbbabbbbabbY…

where the X and the Y are unknown given that we do not know what precedes the 0 on the left and what follows the 1 on the right. Formally, this second string is doubly-infinite meaning its cardinality is double the cardinality of the first string since every digit in the first string is counted twice (yes, there are actually

Now suppose that we have complete knowledge of the doubly-infinite string of a’s and b’s with the exception of one value. It should be relatively clear that, if we know the distribution of 0’s and 1’s in the original string (e.g. 60/40 or 30/70), then the unknown value in the second string should be immediately known unless the original distribution is 50/50 in which case we have no idea (try it!).

So now take any two locations, X and Y, on what we call the “image” string (the strings of a’s and b’s) with the caveat that there are at least two letters between them. Suppose that we have complete knowledge of the doubly-infinite string

The way I have described the problem (courtesy of Steve Shea) would seem to imply that we need to know absolutely every value in the doubly-infinite string, i.e. we need an infinite amount of knowledge, if we are to correctly predict the value of Y given the value of X (which also assumes that the original string of 0’s and 1’s is not a 50/50 split). But this gets to the heart of Anthony’s problem: do we really need an infinite amount of knowledge or just a

Nevertheless, approaching such a high-degree of accuracy with a prediction would also seem to require that the strings really

I would then

What does any of this have to do with the universe? Let’s just suppose that John Wheeler was correct and that, at its core, the universe is built up from bits—binary digits—or, rather, qubits. Since we don’t actually see the world as 0’s and 1’s (or answers to yes/no questions) it is clear that there is some form of mapping that goes on at some deep level from these binary questions to something slightly less fundamental. Let’s take, as a first approximation, the map introduced at the beginning and suppose that the string of a’s and b’s represents the results of measurements of simple two-level quantum systems. In other words, the a’s and b’s represent our

One glitch in this argument, of course, is that we don’t know the order of the measurement outcomes, i.e. the string of a’s and b’s. We have small strings of correlated outcomes, but we don’t know for certain if all of these small strings are part of one larger string, i.e. if we know, for example,

…abbababa…

…babbabab…

…aaababaa…

is it necessarily true that

…abbababa…babbabab…aaababaa…

or that

…babbabab…abbababa…aaababaa…?

If all of the sub-strings indeed are part of one, larger string it would seem to suggest that all possible qubit measurements in the universe may be correlated in some way since, as long as there exists a production rule, we can connect even the most far-flung elements of the full string of a’s and b’s. This, also, is not such a radical idea if we take the correlations as being equivalent to quantum entanglement (see Buniy and Hsu’s article on this: arXiv:1205.1584v2).

Note that there is one question here that I have not addressed and that is whether knowledge of X necessarily implies that Y is the

Let’s take Buniy and Hsu’s idea as correct in which case all qubit measurements are somehow part of a single, large string. We can access different sub-strings of this larger string via entanglement measurements. Note, however, that X and Y do not necessarily need to be in the same sub-string. So, for instance, an entanglement experiment with qubits might represent the following pair of sub-strings

…abbabaXabbaba…ababaabYbabaaba…

In other words, we are ignorant of some of the intermediary processes that connect them. Nevertheless, we do know that a determination of X immediately tells us what Y must be since this is measurable in a laboratory. This suggests that the intermediary processes, while perhaps not readily apparent to us, nevertheless exist as long as there is a production rule and as long as the full string is doubly-infinite. In fact, if Buniy and Hsu are correct (and note that their argument is based on cosmological models and thus not dependent on anything I suggest here), the mere presence of entanglement suggests that the string of all qubit measurements in the universe is infinite which would itself suggest that the universe is infinite,

I caution against rushing to judgement about these ideas. Clearly we see correlations and entanglement in the universe on some level. Does this automatically imply, then, that the universe is infinite

Now Anthony was interested in this from the standpoint of the multiverse and one could easily modify these ideas to take that into account. I won’t do it here. I can also imagine someone taking Steve’s results

There are a lot of assumptions and conjectures in this post but there are also a lot of concrete starting points for further exploration. Can we definitively prove, mathematically, that these strings

In my mind, these are all ideas worth pursuing and so I suppose you could interpret this blog post as a challenge: let’s get some answers to these questions!

this post has been edited by the author since its original submission

report post as inappropriate

"It should be relatively clear that, if we know the distribution of 0's and 1's in the original string (e.g. 60/40 or 30/70), then the unknown value in the second string should be immediately known..."

This only works if one knows that the odds are EXACTLY 60/40 etc, and not 59.9996/40.0004. In other words, one still needs to have an infinite amount of information (infinite significant digits) about the odds.

"One could imagine that we could approach a very high probability of correctly predicting..."

One can imagine a great many things that end up being untrue.

"Clearly we see correlations and entanglement in the universe on some level. Does this automatically imply, then, that the universe is infinite and not entirely random at its base level?"

No. It does not imply anything at all. You can't pull ten pounds of information out of a one-pound container of information (such as the equations of mathematical physics).

Rob McEachern

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This only works if one knows that the odds are EXACTLY 60/40 etc, and not 59.9996/40.0004. In other words, one still needs to have an infinite amount of information (infinite significant digits) about the odds.

"One could imagine that we could approach a very high probability of correctly predicting..."

One can imagine a great many things that end up being untrue.

"Clearly we see correlations and entanglement in the universe on some level. Does this automatically imply, then, that the universe is infinite and not entirely random at its base level?"

No. It does not imply anything at all. You can't pull ten pounds of information out of a one-pound container of information (such as the equations of mathematical physics).

Rob McEachern

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"One can imagine a great many things that end up being untrue."

Did you miss the parts where I very clearly noted that these are*conjectures* that are meant to stimulate rigorous investigations? If you think any of them are misguided, feel free to prove them wrong (rigorously). Personally, I have no allegiance to the ideas themselves so I think both a positive or a negative result would be just fine with me as long as it is rigorous.

"No. It does not imply anything at all. You can't pull ten pounds of information out of a one-pound container of information (such as the equations of mathematical physics)."

You must not be a fan of Max's ideas...

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Did you miss the parts where I very clearly noted that these are

"No. It does not imply anything at all. You can't pull ten pounds of information out of a one-pound container of information (such as the equations of mathematical physics)."

You must not be a fan of Max's ideas...

report post as inappropriate

Ian,

You are correct, I am not a fan of Max's ideas - Which should come as no surprise, given that the title of my 2012 FQXI essay was "Misinterpreting Reality - Confusing Mathematics for Physics"

Rob McEachern

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You are correct, I am not a fan of Max's ideas - Which should come as no surprise, given that the title of my 2012 FQXI essay was "Misinterpreting Reality - Confusing Mathematics for Physics"

Rob McEachern

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Rob,

"You can't pull ten pounds of information out of a one-pound container of information (such as the equations of mathematical physics)."

10 to 1 compressibility isn't much of a challenge. In fact, all the information in the universe fits in a 1-dimensional string of binary digits as Ian implies.

The questions are whether the string has to be infinitely long and therefore algorithmically incompressible; or of finite length and therefore algorithmically compressible (in principle); or of finite length and sufficiently compressible to predict future events from past conditions, which is what Ian's article is about.

From previous exchanges, I know that you favor incompressibility, in which case the universe is its own algorithm.

I favor the primacy of the continuum. In this case, the infinite scalability of information-containing bits is time dependent; primordial information encoded and compressed into a topological point at infinity is available to 3-dimension decoding at every scale of 4 dimension spacetime. This requires a simply connected topological structure.

Tom

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"You can't pull ten pounds of information out of a one-pound container of information (such as the equations of mathematical physics)."

10 to 1 compressibility isn't much of a challenge. In fact, all the information in the universe fits in a 1-dimensional string of binary digits as Ian implies.

The questions are whether the string has to be infinitely long and therefore algorithmically incompressible; or of finite length and therefore algorithmically compressible (in principle); or of finite length and sufficiently compressible to predict future events from past conditions, which is what Ian's article is about.

From previous exchanges, I know that you favor incompressibility, in which case the universe is its own algorithm.

I favor the primacy of the continuum. In this case, the infinite scalability of information-containing bits is time dependent; primordial information encoded and compressed into a topological point at infinity is available to 3-dimension decoding at every scale of 4 dimension spacetime. This requires a simply connected topological structure.

Tom

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Iaan,

This is a bit out of left field, but it seems to me the situation is that information is conceptually static, but reality is necessarily dynamic. So when information is derived, it requires effectively striking one thing with something else. One, the object being measured and the other, the object being used to measure it. Such as measuring the spin of a particle requires interfering it with some other entity.

The situation then is that 'information' is essentially these contact points between the device and the entity. Or even just a photon striking our retina. If this contact wasn't being made, then there would be no information. Any physical entities would just be floating in their own void.

So from this perspective, it would seem Wheeler's concept of it from bit is meaningless. You need those contact points to have information. The 'bit' is an action, the measurement is an act of contact!

Try as I might, I just don't see how they arise otherwise. Without the action(verb), the bit(nouns), are not entangled, not in contact with each other.

The only possible reason for even thinking of nature as discrete seems to arise from the distinct contact points of these relations.

This then goes to the nature of time. Those physical entities are not strung out along a time dimension because the energy of which they consist is proceeding from one contact to the next and is as much a part of the entity as any measure of events occasioned by the process.

Duration is not a dimension external to the point of the present, but is what is presently occurring between the occurrence of the events. The trough of the waves occurring between the peaks. The problem is this trough is not being recorded, only frequency and amplitude of the peaks. The trough is the wave to the particle of the peaks.

Regards,

John M

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This is a bit out of left field, but it seems to me the situation is that information is conceptually static, but reality is necessarily dynamic. So when information is derived, it requires effectively striking one thing with something else. One, the object being measured and the other, the object being used to measure it. Such as measuring the spin of a particle requires interfering it with some other entity.

The situation then is that 'information' is essentially these contact points between the device and the entity. Or even just a photon striking our retina. If this contact wasn't being made, then there would be no information. Any physical entities would just be floating in their own void.

So from this perspective, it would seem Wheeler's concept of it from bit is meaningless. You need those contact points to have information. The 'bit' is an action, the measurement is an act of contact!

Try as I might, I just don't see how they arise otherwise. Without the action(verb), the bit(nouns), are not entangled, not in contact with each other.

The only possible reason for even thinking of nature as discrete seems to arise from the distinct contact points of these relations.

This then goes to the nature of time. Those physical entities are not strung out along a time dimension because the energy of which they consist is proceeding from one contact to the next and is as much a part of the entity as any measure of events occasioned by the process.

Duration is not a dimension external to the point of the present, but is what is presently occurring between the occurrence of the events. The trough of the waves occurring between the peaks. The problem is this trough is not being recorded, only frequency and amplitude of the peaks. The trough is the wave to the particle of the peaks.

Regards,

John M

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"doubly-infinite meaning its cardinality is double the cardinality of the first string since every digit in the first string is counted twice (yes, there are actually levels of infinity!)."

Aleph_2 squared = aleph_4? What nonsense. I prefer the sane reasoning by Archimedes, Aristotle, Spinoza, Galileo: Infinity is a fictitious quality:

2 times oo = oo.

Eckard

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Aleph_2 squared = aleph_4? What nonsense. I prefer the sane reasoning by Archimedes, Aristotle, Spinoza, Galileo: Infinity is a fictitious quality:

2 times oo = oo.

Eckard

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Ian Durhem,

Being not a mathematician, I nonetheless understood that so called doubly-infinite series which are infinite to both sides, e.g. ...-2, -1, 0, 1, 2, ... and also the rational numbers can be bijected to the natural numbers and have therefore the cardinality aleph_0 of anything that is "countably" infinite. The next larger cardinality is aleph_1 which means the continuum of real numbers.

Did you not accept that?

Reasoning myself I agree with e.g. Spinoza who understood infinity as an unreachable by counting and also inexhaustible quality.

If you are claiming that an infinite Cantorian cardinality can be doubled by counting twice then tell me please which mathematician shares this opinion.

Eckard

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Being not a mathematician, I nonetheless understood that so called doubly-infinite series which are infinite to both sides, e.g. ...-2, -1, 0, 1, 2, ... and also the rational numbers can be bijected to the natural numbers and have therefore the cardinality aleph_0 of anything that is "countably" infinite. The next larger cardinality is aleph_1 which means the continuum of real numbers.

Did you not accept that?

Reasoning myself I agree with e.g. Spinoza who understood infinity as an unreachable by counting and also inexhaustible quality.

If you are claiming that an infinite Cantorian cardinality can be doubled by counting twice then tell me please which mathematician shares this opinion.

Eckard

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That's not what Ian is saying, Eckard. He is talking about unknown values of X and Y connected by an unknown string of digits of which there are at least 2. We don't know the value of X because we don't know what values preceded it, nor the value of Y because we don't know what values follow it. That is the "double infinty," on the separable interval (-oo, +oo).

Let X = 0, Y = 1:

"Now suppose that we have complete knowledge of the doubly-infinite string of a's and b's with the exception of one value. It should be relatively clear that, if we know the distribution of 0's and 1's in the original string (e.g. 60/40 or 30/70), then the unknown value in the second string should be immediately known unless the original distribution is 50/50 in which case we have no idea (try it!)."

He is not comparing infinities; rather, he is calculating finite values bounded by infinity on both sides. A random binary string of weighted distribution (that is, not normal) -- say of a 100-digit string, 60 digits are 0 and 40 are 1 -- it doesn't matter how the digits fall; if we know the weight of the distribution, we know that any string of the same weight will always fall toward one infinity or another by the same degree; i.e. we know the exact values of X and Y in the second string.

If the distribution is normal (50% 0s and 50% 1s) the Xs and Ys fall randomly, -oo or +oo. This is a numerical version of Lamport's Buridan's Ass argument that I have cited before:

"*Buridan's Principle.* A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time."

A normal distribution is identical to a continuous range of values.

Tom

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Let X = 0, Y = 1:

"Now suppose that we have complete knowledge of the doubly-infinite string of a's and b's with the exception of one value. It should be relatively clear that, if we know the distribution of 0's and 1's in the original string (e.g. 60/40 or 30/70), then the unknown value in the second string should be immediately known unless the original distribution is 50/50 in which case we have no idea (try it!)."

He is not comparing infinities; rather, he is calculating finite values bounded by infinity on both sides. A random binary string of weighted distribution (that is, not normal) -- say of a 100-digit string, 60 digits are 0 and 40 are 1 -- it doesn't matter how the digits fall; if we know the weight of the distribution, we know that any string of the same weight will always fall toward one infinity or another by the same degree; i.e. we know the exact values of X and Y in the second string.

If the distribution is normal (50% 0s and 50% 1s) the Xs and Ys fall randomly, -oo or +oo. This is a numerical version of Lamport's Buridan's Ass argument that I have cited before:

"

A normal distribution is identical to a continuous range of values.

Tom

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Tom,

*In a continuous range of values*, is it possible to race from milestone 1 to 5, without passing through milestone 4? Or for those having weight problems, is it mathematically feasible to come down from 100kg to 70kg without passing through 80kg?

Akinbo

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Akinbo

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"Is there any way for us to tell if the universe is infinite or simply really, really big?"

The question that should logically have preceded this unless it is assumed is:

If there was a beginning and the universe had not always existed infinitely in time and size, at the beginning, was the universe infinitely small or zero?

If this question can be reasonably answered beyond speculation then the one posed by Anthony Aguirre can be better formulated in that

- Can something increase mathematically or physically from zero or infinitely small to infinity without passing any size in between?

A yes-or-no answer to this will appear to answer the question posed by Anthony Aguirre.

Akinbo

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The question that should logically have preceded this unless it is assumed is:

If there was a beginning and the universe had not always existed infinitely in time and size, at the beginning, was the universe infinitely small or zero?

If this question can be reasonably answered beyond speculation then the one posed by Anthony Aguirre can be better formulated in that

- Can something increase mathematically or physically from zero or infinitely small to infinity without passing any size in between?

A yes-or-no answer to this will appear to answer the question posed by Anthony Aguirre.

Akinbo

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Rob, I am replying here to support and counter your statements respectively.

You said, "...the distinction between math and physics, that I discussed". And I agree. You said, "no amount of mathematical cogitation can answer the question they pose". I disagree.

Can the size of something (the universe) increase from zero (or 10^{-35}m) to infinity without passing through the sizes in between?

Even the big bang theory gives a time-line with temperatures and sizes of each epoch in the early evolution of the universe.

If we pursue this line of thinking that nothing can increase from a smaller size to a really, really big size as Anthony Aguirre conjectures, there are further enlightening facts that must come to light by that useful thinking method of reductio ad absurdum.

For instance, put Zeno, Ian, Anthony and Max, and maybe Tom, in a room and you will see that when they come out smiling after a tough dialectic session, we may be able to answer the question whether space is discrete or continuous. How? If a universe's size/radius has to pass through the real number line, it cannot even expand from 10^{-35}m in the first place, talk less of the 10^{25}m we assume its size to be today! Moral: Length is ultimately discrete and describable by the natural number system, if not the universe cannot even expand!

And take this from Penrose, p.113, The Emperor's New Mind,*"The system of real numbers has the property, for example, that between any two of them, no matter how close, there lies a third.... We should at least be a little suspicious that there might eventually be a difficulty of fundamental principle for distances on the tiniest scale.... This confidence (in the real number system) is perhaps misplaced."* In other words, at the quantum gravity scale, further division of length may no longer be feasible physically and math and physics become distinguishable.

Akinbo

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You said, "...the distinction between math and physics, that I discussed". And I agree. You said, "no amount of mathematical cogitation can answer the question they pose". I disagree.

Can the size of something (the universe) increase from zero (or 10

Even the big bang theory gives a time-line with temperatures and sizes of each epoch in the early evolution of the universe.

If we pursue this line of thinking that nothing can increase from a smaller size to a really, really big size as Anthony Aguirre conjectures, there are further enlightening facts that must come to light by that useful thinking method of reductio ad absurdum.

For instance, put Zeno, Ian, Anthony and Max, and maybe Tom, in a room and you will see that when they come out smiling after a tough dialectic session, we may be able to answer the question whether space is discrete or continuous. How? If a universe's size/radius has to pass through the real number line, it cannot even expand from 10

And take this from Penrose, p.113, The Emperor's New Mind,

Akinbo

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In my theory our universe is finite but it is contained in an infinite structured and elastic ether called the E-Matrix. Interacting Objects moving in the E-Matrix give rise to all the processes and all the interactions. This model of our universe includes the possibility of the existence of multiverses. A paper of the origin of our universe is proposed in the following link:

http://www.modelmechanics.org/2011universe.pdf

Ken Seto

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http://www.modelmechanics.org/2011universe.pdf

Ken Seto

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I don’t understand if the finite Universe is a discrete Universe or a limited Universe.

I am thinking that a limited Universe have an observable frontier, where there is a leap in the expansion: a change in the spectral emission of the galaxies, an observable abrupt jump in the expansion velocity some equitemporal rings (but this can be too simple: I don’t know if it possible an observation of old frontiers, near the Big Bang).

I am thinking that the discrete Universe, with discrete times, or a discrete spaces (and so a finite number of states), must have an indication in the first lights: if there is a granularity, then in billions of years, for some primeval sources, an initial granularity in the emission must give different path in the signals (across granularity space); for example different distant sources can have a unique statistical feature (amplification of the granularity signal because of the long path).

But all these thoughts can be (certainly) purely speculative.

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I am thinking that a limited Universe have an observable frontier, where there is a leap in the expansion: a change in the spectral emission of the galaxies, an observable abrupt jump in the expansion velocity some equitemporal rings (but this can be too simple: I don’t know if it possible an observation of old frontiers, near the Big Bang).

I am thinking that the discrete Universe, with discrete times, or a discrete spaces (and so a finite number of states), must have an indication in the first lights: if there is a granularity, then in billions of years, for some primeval sources, an initial granularity in the emission must give different path in the signals (across granularity space); for example different distant sources can have a unique statistical feature (amplification of the granularity signal because of the long path).

But all these thoughts can be (certainly) purely speculative.

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Ian,

In relation to Steve's lemma 5, he and/or you might be interested in my lemma on p. 8 of this work in progress.

Erratum: I know 119 isn't a prime. Stupid mistake that doesn't change my argument, but I haven't taken the time to change it. The structure works for all odd integers, but gets its distinctiveness -- as Steve's lemma allows by existence -- from the minimum walk standard, which is positive definite only for pairs of distinct primes regardless of separation.

Best,

Tom

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In relation to Steve's lemma 5, he and/or you might be interested in my lemma on p. 8 of this work in progress.

Erratum: I know 119 isn't a prime. Stupid mistake that doesn't change my argument, but I haven't taken the time to change it. The structure works for all odd integers, but gets its distinctiveness -- as Steve's lemma allows by existence -- from the minimum walk standard, which is positive definite only for pairs of distinct primes regardless of separation.

Best,

Tom

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Is the universe infinite, or just really, really equal to -1/12

Rob McEachern

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Rob McEachern

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Could be. This is a old result due to Ramanujan.

Here is an excerpt from a posting on the sci.physics newsgroup (02/98) by Dan Piponi:

"In (bosonic) string theory via the operator formalism you find an infinite looking zero point energy just like in QED except that you get a sum that looks like:

1+2+3+4+...

Now the naive thing to do is the same: subtract off this zero point energy. However later on you get into complications. In fact (if I remember correctly) you must replace this infinity with -1/12 (of all things!) to keep things consistent.

Now it turns out there is a nice mathematical kludge that allows you to see 1+2+3+4+... as equalling -1/12. What you do is rewrite it as

1+2-n +3-n +...

This is the Riemann Zeta function. This converges for large n but can be analytically continued to n = -1, even though the series doesn't converge there. Zeta(-1) is -1/12. So in some bizarre sense 1+2+3+4+... really is -1/12.

But even more amazingly is that you can get the -1/12 by a completely different route - using the path integral formalism rather than the operator formalism. This -1/12 is tied up in a deep way with the geometry of string theory so it's a lot more than simply a trick to keep the numbers finite.

However I don't know if the equivalent operation in QED is tied up with the same kind of interesting geometry."

Another conjectured geometric result, if one allows a continuous but closed trajectory self limiting to 12 determinants of a positive definite matrix is in my research into Sophie Germain primes, illustrated in figues 2 &3 here.

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Here is an excerpt from a posting on the sci.physics newsgroup (02/98) by Dan Piponi:

"In (bosonic) string theory via the operator formalism you find an infinite looking zero point energy just like in QED except that you get a sum that looks like:

1+2+3+4+...

Now the naive thing to do is the same: subtract off this zero point energy. However later on you get into complications. In fact (if I remember correctly) you must replace this infinity with -1/12 (of all things!) to keep things consistent.

Now it turns out there is a nice mathematical kludge that allows you to see 1+2+3+4+... as equalling -1/12. What you do is rewrite it as

1+2-n +3-n +...

This is the Riemann Zeta function. This converges for large n but can be analytically continued to n = -1, even though the series doesn't converge there. Zeta(-1) is -1/12. So in some bizarre sense 1+2+3+4+... really is -1/12.

But even more amazingly is that you can get the -1/12 by a completely different route - using the path integral formalism rather than the operator formalism. This -1/12 is tied up in a deep way with the geometry of string theory so it's a lot more than simply a trick to keep the numbers finite.

However I don't know if the equivalent operation in QED is tied up with the same kind of interesting geometry."

Another conjectured geometric result, if one allows a continuous but closed trajectory self limiting to 12 determinants of a positive definite matrix is in my research into Sophie Germain primes, illustrated in figues 2 &3 here.

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1 + 2 + 3 + 4 + 5 + ... = -1/12

Another reason why mathematics should not be confused with physics. And I note the use of "nice mathematical kludge", "bizarre sense", "simply a trick", "hocus-pocus", etc all of which are pointers that any bizarre, counter-intuitive, predictions in physics, none of which have ever being observed in reality have their origin in mathematics, not in physics itself.

And Tom browsed through your paper. A demonstration of your deep affection for mathematics! I asked you somewhere whether you run from milestone 1 to 5 without passing 4 or lose weight from 100kg to 70kg without passing 80kg, but you didnt answer. But in your paper I see orderly numbering so I can guess your answer. Therefore, if I my guess s correct, the universe cannot increase from 1m size to 10^{25}m without passing through 10^{10}m size. If you follow this you find the universe can only be really, really big but not infinite.

Akinbo

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Another reason why mathematics should not be confused with physics. And I note the use of "nice mathematical kludge", "bizarre sense", "simply a trick", "hocus-pocus", etc all of which are pointers that any bizarre, counter-intuitive, predictions in physics, none of which have ever being observed in reality have their origin in mathematics, not in physics itself.

And Tom browsed through your paper. A demonstration of your deep affection for mathematics! I asked you somewhere whether you run from milestone 1 to 5 without passing 4 or lose weight from 100kg to 70kg without passing 80kg, but you didnt answer. But in your paper I see orderly numbering so I can guess your answer. Therefore, if I my guess s correct, the universe cannot increase from 1m size to 10

Akinbo

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Hi Akinbo,

Sorry, there's too much going on right now for me to try and answer everything in detail. You write: "I asked you somewhere whether you run from milestone 1 to 5 without passing 4 or lose weight from 100kg to 70kg without passing 80kg, but you didnt answer."

This is an important result in the mathematical field of topology, called Brouwer's fixed point theorem (every continuous map f: B^n --> B^n has a fixed point). I don't get your point, though -- the theorem means physically that all continuous functions have a fixed point; when generalized to the 3-sphere (which is like a 3-dimensional ball in 4 dimensions), the point is bounded by a simple pole at infinity. An interesting physical consequence of the theorem is that a 3-dimensional sphere like the Earth has at all times two antipodal (opposite) locations where temperature and barometric pressure are identical (Borsuk-Ulam theorem). Meterologists find this fact useful.

"But in your paper I see orderly numbering so I can guess your answer. Therefore, if I my guess s correct, the universe cannot increase from 1m size to 1025m without passing through 1010m size. If you follow this you find the universe can only be really, really big but not infinite."

Well, IF the universe is closed (we don't know the answer to that yet) the fixed point theorem might be used to show that continuous functions (field theories) are more fundamental than particle theories, regardless of how big the universe is. This was Einstein's quest -- to have a foundational field theory that unifies all forces, i.e., describes everything in terms of continuous functions, without having to specify boundary conditions.

Best,

Tom

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Sorry, there's too much going on right now for me to try and answer everything in detail. You write: "I asked you somewhere whether you run from milestone 1 to 5 without passing 4 or lose weight from 100kg to 70kg without passing 80kg, but you didnt answer."

This is an important result in the mathematical field of topology, called Brouwer's fixed point theorem (every continuous map f: B^n --> B^n has a fixed point). I don't get your point, though -- the theorem means physically that all continuous functions have a fixed point; when generalized to the 3-sphere (which is like a 3-dimensional ball in 4 dimensions), the point is bounded by a simple pole at infinity. An interesting physical consequence of the theorem is that a 3-dimensional sphere like the Earth has at all times two antipodal (opposite) locations where temperature and barometric pressure are identical (Borsuk-Ulam theorem). Meterologists find this fact useful.

"But in your paper I see orderly numbering so I can guess your answer. Therefore, if I my guess s correct, the universe cannot increase from 1m size to 1025m without passing through 1010m size. If you follow this you find the universe can only be really, really big but not infinite."

Well, IF the universe is closed (we don't know the answer to that yet) the fixed point theorem might be used to show that continuous functions (field theories) are more fundamental than particle theories, regardless of how big the universe is. This was Einstein's quest -- to have a foundational field theory that unifies all forces, i.e., describes everything in terms of continuous functions, without having to specify boundary conditions.

Best,

Tom

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Here are some thoughts from my most recent FQXi essay (http://fqxi.org/community/forum/topic/1977). First, about "Is the universe infinite or just really really big?" Simultaneously, about "bits (in some fashion) constitute the basic building blocks of the universe."

The inverse-square law states that the force between two particles becomes infinite if the distance of separation between...

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The inverse-square law states that the force between two particles becomes infinite if the distance of separation between...

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Rudiment considerations.

Mathematics is fundamentally based upon causality. Axiom of Choice. Assume a preliminary model of causality to support scalar causality.

Axiom of Choice extended to include Relativity

http://vixra.org/pdf/1402.0041v1.pdf

Based upon this model of quantum causality, infinity is used as a form of probability in math models, but in physics infinity does not exist.

Consider any closed system of causality. No matter how complex it will always repeat unless their is an outside influence.

Aliased systems that create standing nodes (constants/singularities) moderate all other causality; physics constants.

These cause distributed systems of aliasing; subatomic particles.

Entropy is an indication of evolving systems of causality transitioning from one dimensional system (physics constants) to another. Some evidence to support this comes from the speed of light changing over many years (see related reference).

Therefore Big Bangs repeat as dimensional states morph from one system to another; systems of relativity.

Infinity is relativistic and only exists within a confined relativistic causal environment. Outside of relativity, infinity does not make sense, it is unrelated to the physics involved.

Sleepy, I'll come back later when my head is clear.

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Mathematics is fundamentally based upon causality. Axiom of Choice. Assume a preliminary model of causality to support scalar causality.

Axiom of Choice extended to include Relativity

http://vixra.org/pdf/1402.0041v1.pdf

Based upon this model of quantum causality, infinity is used as a form of probability in math models, but in physics infinity does not exist.

Consider any closed system of causality. No matter how complex it will always repeat unless their is an outside influence.

Aliased systems that create standing nodes (constants/singularities) moderate all other causality; physics constants.

These cause distributed systems of aliasing; subatomic particles.

Entropy is an indication of evolving systems of causality transitioning from one dimensional system (physics constants) to another. Some evidence to support this comes from the speed of light changing over many years (see related reference).

Therefore Big Bangs repeat as dimensional states morph from one system to another; systems of relativity.

Infinity is relativistic and only exists within a confined relativistic causal environment. Outside of relativity, infinity does not make sense, it is unrelated to the physics involved.

Sleepy, I'll come back later when my head is clear.

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I think it may be possible to present a much simpler proof. I've not completed this train of thoughts and would really like feedback to check if it is sound. It goes like this:

To describe all possible universes we take an infinitely long set of binary values and treat every value with a random function. This produces a set that represent a snapshot of all possible universes regardless of whether these universes are smooth or discrete. (As long as it is possible to represent any value as a set of binary values.) Smooth universes are infinite, discrete universes are finite.

This will only represent a snapshot of all universes, but even as every possible configuration is represented it is totally static and we need to introduce a regime where all or some of the values does some interaction. A value that has a non zero chance of interacting with infinitely many other values has a problem - it will interact with infinitely many other values. If one, for example sets up a loop with three values. Value 1 interacts with value 2 which again interacts with value 3 and goes around again. If these values are part of an infinite set with a nonzero chance to interact there will be infinitely many values interacting with value 1 before value 3 gets to interact with it. Making it difficult to make structures. The only way of making it interact with finitely other values is to let it be part of a finite - and therefore discrete - subset of every possible configuration. Therfore the universe is both finite and discrete.

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To describe all possible universes we take an infinitely long set of binary values and treat every value with a random function. This produces a set that represent a snapshot of all possible universes regardless of whether these universes are smooth or discrete. (As long as it is possible to represent any value as a set of binary values.) Smooth universes are infinite, discrete universes are finite.

This will only represent a snapshot of all universes, but even as every possible configuration is represented it is totally static and we need to introduce a regime where all or some of the values does some interaction. A value that has a non zero chance of interacting with infinitely many other values has a problem - it will interact with infinitely many other values. If one, for example sets up a loop with three values. Value 1 interacts with value 2 which again interacts with value 3 and goes around again. If these values are part of an infinite set with a nonzero chance to interact there will be infinitely many values interacting with value 1 before value 3 gets to interact with it. Making it difficult to make structures. The only way of making it interact with finitely other values is to let it be part of a finite - and therefore discrete - subset of every possible configuration. Therfore the universe is both finite and discrete.

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