Are Boltzmann Brains running Hilbert\'s Hotel? by William T. Parsons

Dear William,

From my essay

"The relative size of a subgroup H of G is called the index which means that there are n inequivalent copies (called cosets) of H that fill up G. The action of generators on these cosets creates the permutation group P by the Todd-Coxeter algorithm " that is there are n rooms in the hotel with infinitely many slots.

I have been asking myself if this is related to Cantor's trick.

If you know something, please let me know.

Your text is unique in the sense that it introduces the discrepancy for the infinity in maths and physics.

I intend to read it in more details.

Best.

Michel

Bill,

"Bravo". This was a very entertaining essay. It is very easy to mistakenly equate mathematical infinity with physical infinity. You show this clearly and present a method of recognizing and avoiding it.

I'm curious, is Tegmark's nearest twin within our Hubble Bubble? That distance looks pretty big to me.

Many of the essays deal with variations of what you present. If you have the infinite time needed, you should have some interesting reading.

Best Regards and Good Luck,

Gary Simpson

Dear William Parsons,

As Gary said, very entertaining. I don't believe in physical infinity, and therefore don't spend much time worrying about mathematical infinity, but I can certainly enjoy the originality and verve of your essay, and can appreciate your arguments and your conclusions.

It is time to dump physical infinity, and although you're not the first to suggest it, you certainly are the first to suggest Bravo, which has a whiff of Bell's FAPP.

Thank you for providing much more fun than most discussions of infinity provide, without in any way sacrificing rigor.

My very best regards,

Edwin Eugene Klingman

Dear William T. Parsons,

I agree with your view on infinity. By adding finite numbers we can never reach infinity. We can introduce an infinite loop: go on adding infinitely, and it becomes a never ending process. Regarding the relation between physics and mathematics, I invite your attention to my essay: A physicalist interpretation of the relation between Physics and Mathematics

Hilbert's Hotel is based on wrong assumptions or axioms. (i). "Hilbert's Hotel consists of an infinite set of rooms". Never. Hilbert can start from zero, constructing rooms one by one but can never complete infinite rooms; here, construction is a never ending process. (ii). "an infinite set of buses arrives". Never. Buses keep coming; it is a never ending process. Hilbert's Hotel is actually an infinite loop (of finite processes). Thus in fact, there is no paradox.

I agree with you that infinity has no role in physics. However, I think the best method to avoid infinity is to 'finitise' one by one all the concepts and equations by suitably modifying the existing ones. Newton's straight-line motion, force- acceleration relation and equation for gravity, all lead us to infinity. Thus infinity crept into physics from the time of Newton (his third law is an exception). I have brought out a model devoid of infinity (refer: finitenesstheory.com).

Hi Bill,

I thoroughly concur with the idea of replacing infinity with Bravo, and suggest that the definitive enumeration of B is the number of Planck volumes in the observable universe, with B increasing as that which we are able to observe increases. I like to equate Planck volumes with Leibnitz's 'monad', and to have them capable of replicating like von Neumann's automata. We can always make Hilbert's physical hotel arbitrarily larger, but it always remains finite (the monads are countable). The infinite capacity resides in the mathematical (computational) composition of the monad itself.

Cheers

Rowan

Dear William T. Parsons,

In your essay you wrote, "Currently, the best theoretical extension of the ΛCDM model is "cosmological inflation". It was originally formulated by Alan Guth and others to iron out problems concerning the "Hot Big Bang", i.e., the early moments surrounding physical creation of our Observable Universe. Word count constraints prohibit churning through all the details. We cut to the bottom-line. Inflation postulates the existence of an infinite number of universes, such as ours, all of which may be physically infinite in one way or another." Guth's theory of inflation, WHICH IS MERELY ONE VERSION OF NEWTONIAN-EINSTEINIAN INFLATION, postulates the existence of an infinite number of universes. However, the EMPIRICAL FINDINGS of Milgrom, McGaugh, Kroupa, and Pawlowski suggest that Newtonian-Einsteinian inflation is incorrect and Milgromian inflation is correct (whatever Milgromian inflation might be). According to Kroupa the ΛCDM concordance cosmological model has been ruled out by empirical observations. Google "kroupa dark matter". The space roar and the photon underproduction crisis also indicate that something is seriously wrong with the ΛCDM model (despite its successes). I MAKE THE FOLLOWING CLAIM: NASA's space roar science team affirms that 3 independent empirical data sets confirm the space roar. Do you agree or disagree with my claim?

Bill -

Thanks for a great essay (one of the most entertaining this year!) and an excellent exposition of the conceptual difficulties with infinity. There are parallels to other difficulties (e.g. Godel and Turing) that raise similar problems. In one way or another all deal with self-referential properties - in the case of infinity, the fact that whatever arithmetic operations you perform, you still end up with the same cardinal infinity.....

It is reassuring to believe that the physical universe is finite, and I hold that belief myself. But I wonder at the usefulness of the concept of infinity - why is it so damn helpful in physics? Would you agree that mathematical infinity has a form of existence? Perhaps we can only see it in the formulas or think it in our minds, but it is real!

Many thanks - George Gantz

Bill,

Congratulations, this essay has everything needed for a great essay: accessible to a general audience, humor, enlightening figures and new ideas.

Infinite and infinitesimal are useful non-numbers that can be placed in functions to show limits and relationships between functions (as used in calculus). The philosophical implications of these convenient non-numbers are normally ignored. It should come as no surprise that relationships first found with amber, rabbit's fur and pith balls have trouble at very small scales and relationships first found by observing the planets in our solar system, would not find the expansion rate of the universe. Infinite and infinitesimal push equations far beyond the limits of our current understanding of physics. The idea of a "Bravo", placed right at the observable limit, is a great idea. Maybe you should have a "speculative" a place where there might be dragons, just beyond our current observable limit where we can push things just a little bit. The nature of "real" infinite (if it exists) is unimportant, expanding what we got a little bit farther is important.

One final note, a "speculative" would be larger than a "Bravo" most of the time, but thinking of the three body problem, the trillions of particles in Saturn's ring would be the "Bravo" and the "speculative" would be three.

Jeff Schmitz

William -

Clever title, congrats. I can see that infinity causes lots of trouble, but as for banishing from physics: despite possible alternatives as you note, the simplest interpretation of current curvature ranges and expansion is that space is probably flat and not closed, and "goes on forever" in spatial extent (and perhaps temporal as well)? Although "for all practical purposes" we can neglect the existence of that unending range of space-time, it would still be a foundational aspect of our reality, in principle, and thus presumably "meaningful" as part of why things are the way they are. Sure, it doesn't have to be true, but neither can the lack of proof that it is, be called a banishment or grounds for positive doubt either. Your thoughts?

Dear William Parson,

You convincingly explained that Hilbert's hotel has been improperly based on oo*20=oo. Why do you hide this criticism behind pretended lightheartedness? Those who fabricated or defended the mathematics of Hilbert's hotel did it very emotionally. Dedekind hesitated for a decade. Cantor even claimed having got CH immediately from God and got insane. Hilbert behaved rude toward Brouwer.

I quoted Galileo Galilei, D. Spalt, and the ultra-finitist W. Mückenheim. Russians made me aware of Zenkin too. Meanwhile, I understood that Fraenkel 1923 is sufficient for a critical reader who perfectly understands German as to grasp what went wrong, and I see the main necessity in correcting the notion of number, in particular correcting the interpretation of aleph_1.

By the way, C. S. Peirce just failed publishing the definition of the "mathematical" infinity you mentioned which has therefore been ascribed to Dedekind. Because I see infinity an ideal property, not a number, I would hesitate speaking of a subset of infinity.

You wrote: "Meaningless use of infinity includes invoking computational set-ups such as oo+oo, oo*0, and oo/oo." Only the latter two invite to use Bernoulli/Hospital.

While I will feel honored if you are ready to substantially criticize my essay(s), I already confirm you to support and possibly extend the suggestion that your essay offers for the sake of physics.

With my best regards,

Eckard