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

Dear William,

I have read your essay quite a while ago, but only realized that I forgot to comment yesterday, when I wanted to post something in reply to Michel's question. My apologies for this; I realize the rating itself is the utmost expression of appreciation but I also know it's very satisfying when people interact with your work.

You're making an unexpected and original analysis for the infinite hotel and I admire your argumentation when drawing parallels to physics as you are taking into consideration possible objections. I think Fitzgerald was saying that "the test of a first-rate intelligence is the ability to hold two opposed ideas in mind at the same time and still retain the ability to function". Here I mean of course how you develop the point about the laws of physics and the operation of the hotel in finite time, two competing points brought together in short sequence. You are making a profound analysis about the placement of the rooms and in general about the topographical properties that may or may not impact how and iff the hotel works, thus developing your idea completely. Not a single thread is left out of place because you explain how Boltzmann brains come into picture. Thank you for a good read and wish you best of luck in your work and in the contest!

Warm regards,

Alma

Gentlemen, If I may, I might have a hint to a possible answer.

This sounds very much like a problem of ordering. The guests can be assigned to the empty rooms by a pairing function, but in this case the guests would need to be numbered uniquely. Alternatively (and more interestingly now) it can be done without numbering the guests, if the axiom of countable choice is used. AoCC allows to arbitrarily extract one element from each set and then pair them, which results in creating a sequence, therefore ordering. It works for countable infinities but I think that using the axiom of choice it can be generalized to uncountable infinities as well.

I think this is the equivalence you are looking for because choice is equivalent to ordering and in this respect the permutation group is similar to the AoCC possibility to distribute guests inside the hotel.

Dear Bill,

Infinity, set theory and relativity are some of the most difficult concepts in math and physics. Not difficult per se, but difficult to grasp to the level where you can work with them properly. You are going at the heart of the problem when you are distinguishing infinity as the biggest plague of physics. It is both clever and brave to bring these together in your essay and also a strong proof of very original thinking. You are also making a strong point about the difference between physical and mathematical infinity. I particularly enjoyed your analysis of the FLRW metric and the cosmological implications and possibilities to measure physical infinity. This metric is an old friend of mine. I enjoyed reading your well written and well argued essay because you take a point and follow it to the end of the line, not unlike a mathematical proof.

Cheers,

Cristi Stoica

Dear William T. Parsons,

I like the topic and the style of your essay and I agree with your main stance (mathematical infinity is fine, physical infinity is something else). However, I do think there are a few flaws as well.

One minor thing is that you state that the lemniscate is the common and correct symbol for mathematical infinity. You use it both for infinity in the sense of calculus (you mention limit and series) and set theory. However, I don't think the use of the lemniscate symbol is appropiate in the latter case, or in the section on Hilbert's paradox of the Grand Hotel, which is about countably infinite sets. For infinite sets it is possible to rigorously define operations on the infinite cardinalities or ordinals. This is not "improper". (Later on, you mention lemniscate lemniscate among examples of meaningless use of infinity, but this seems fine, even within calculus; the example makes more sense with a minus sign, which does lead to an indeterminacy within the context of calculus.)

There are some sloppy phrasings: you write "subsets of infinity", rather than "subsets of infinite sets" and you state Dedekind's definition of infinity in terms of equality of a set with a subset, but this has to be equality _of size_ of those sets. I agree that the standard of rigour can be a bit lower in these kind of essays, but in these examples, it requires little effort (and no loss of accessibility) to make it more accurate.

You write yourself that "the mathematics of infinity is tricky". I would like to add: especially if you mix it with probabilities! Indeed, my main complaint is about the probabilistic argument on p. 7: as it is written here, you suggest that it is possible to use (infinite) cardinalities as a basis for a probability measure, but such a measure simply isn't defined. There are ways to define probability measures in such sitations, for instance using natural density. (Or using non-standard measures, something I work on, but which is not even needed for the case at hand.) On none of the approaches I known of, it will follow that all subsets of the same cardinality have the same probability measure. For instance, the natural density of the set of even natural number is 1/2, although it has the same cardinality as the entire set of natural numbers (namely, countably infinite): this is easy to interpret as the probability that a random natural number is even. And this better be the case, otherwise you come into conflict with the basic (finite) additivity axiom of probability theory. (Consider the subsets of even and odd numbers to see this point.) In other words, I don't buy your 50-50 argument for the Boltzmann brains vs Universe odds. ;-) (I can send you references if you would be interested in the details of mixing infinity and probability.)

I do like your use of the Bravo card (actually, this idea is quite close to that of a non-standard natural number, which is an alternative way of dealing with infinity in the context of calculus) and I do think most of this text would be enthusing for a lay audience, so my vote is a 7/10.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

Dear Bill,

Thank you very much for your visit and for your words! I just dropped by to tell you that I answered your question.

My sincerest appreciation!

Alma

Dear Bill,

I am very grateful for your reply, which demonstrates that our opinions on these matters are even closer than I had realized initially!

You are right, getting these essays right is difficult: they are supposed to be for a general audience, but in these forums there are mainly people with highly specialized backgrounds and it remains impossible to please everyone. :) As I indicated before, I do think your piece is exciting for the intended audience.

So, you actually considered mentioning non-standard numbers? How cool. :)

I did notice that you were not really defending the 50-50 argument, but my response was that there are better options available for this. Whether the alternatives are applicable to physics is indeed a different matter. I would say that the natural numbers are a helpful concept for modelling physical situations, but not literally applicable to the physical world. Yet, currently, they are really entrenched in physics: the natural numbers occur in the construction of the real numbers, they occur in the definition of limit and series,... Similarly, also natural density and related probability measures are approximations at best.

So, I agree that "In the physical world, infinity has got nothing to offer." Yet, in physical reasoning, it is considered to be a helpful abstraction for unspecified large quanties. Trouble arises when we forget that we are working with an unphysical idealization. The Bravo card can be an alternative for the idealization step. Like you, I would prefer this option. So, indeed, we are on the same page here.

O, and I will send you some references by e-mail!

Best wishes,

Sylvia