Infinity - a brief introduction

This is a brief introduction to the modern mathematical views on infinity. I've tried to make it accessible to everybody; you don't need any mathematical knowledge to follow it. I discuss the Hilbert Hotel thought experiment, some very basic set theory, one-to-one correspondence, and Cantor's Diagonal Argument.
I mention a couple times in the video, e.g. at 9:02, that we can take things away from infinity, and still have infinity. This is true, unless to take infinity from infinity. Infinity minus infinity is undefined. It's easy to see why. Consider the hotel again. Suppose it's full, and now consider what happens when: (1) everybody in the even-numbered rooms leave, but everybody in the odd-numbered rooms stay; (2) everybody but those in the first million rooms leave; (3) everybody but the person in the first room leaves. In each of these cases, an infinite number of people have left, but how many remain? In the first case, infinity -- infinity = infinity. In the second, infinity -- infinity = one million. In the third, infinity -- infinity = one.
Re the points around 27:30: the hypothesis that the cardinality of the reals is aleph-one (that is, that there is no infinity in between the naturals or the reals) is known as the "continuum hypothesis". It has been shown that, on the axioms of standard set theory, it's impossible for the continuum hypothesis to be either proved or disproved. It's completely independent of the standard axioms.

[@TheNaturalLawInstitute]

Kane, Mathematics consists in a deflationary vocabulary, grammar, and syntax, with some conflationary vocabulary for the purposes of verbal convenience. The content of that vocabulary consists of names of positions (Nouns), and Operations (verbs). The grammar provides a very limited means of organizing those nouns and verbs. The syntax provides hints for organizing operations and vocabulary within the grammar. We use glyphs to represent a positional names. We use decimal systems (or other bases) to generate positional names. All numbers(positional names) consist entirely of names of positions with constant relations. Using names for positions to pair off any item of any category, creates categorical independence. Using names for positions forces constant relations, and scale independence,. Using positional name then yields correspondence under categorical independence, and scale independence while preserving constant relations. Positional names provide perfect commensurability. All operations on numbers (positional names) are reducible to addition or subtraction of positions. All positional names other than the natural numbers (base positional names) must be produced through functions. We use inflationary grammar (conflation) to label reducible and non-reducible functions to numbers - a verbal convenience. We use the deflationary grammar of mathematics to remove scale dependence - thereby creating the requirement for limits. We use the deflationary grammar of mathematics removes time-to-perform any operation (Function) - thereby creating the requirement for infinity. We restore scale dependence and eliminate infinity in any and every application of mathematics. By restoring pairing off (context) we eliminate both limits (minimums) and infinity (maximums) In other words, as Babbage demonstrated, all computation can be produced through gears. If you were to use gears to discuss infinity, you would find that different gear ratios produce new positional names at different rates. All mathematical platonism is false (magic). If mathematics were taught operationally, and as a sequence of technical problems of measurement that we needed to solve as we increased the scales of our perception and action, we would not lose so many people who become confused at the apparent 'magic' of the discipline. This is the curse of mathematics profession. It is still operating with 'magical' or 'priestly' language. When its a terribly simple discipline. The art of composing sentences (expressions) that describe phenomenon in the language of constant relations (mathematics), should be no more difficult than learning any other language. Most of it is learning nuance. Just as learning all other languages requires a bit of nuance. Curt Doolittle The Propertarian Institute Kiev, Ukraine

[@jakecarlo9950]

Curt - Thank you for this. Can you suggest a starting point (book, article, video, website, anything) by which to enter into the grammatical (if I can call it that) understanding of mathematics you’ve outlined here? I was on the edge of commenting something like “Infinity appears to be the point at which Time reenters the a-temporal language of math,” when I read your comment; I’ve also been beating my head against limits as a kind of metaphysical sleight of hand. But all of this from an intuitive/philosophical starting point. This comment is, it seems, what I’ve been looking for! Any suggestions you can make would be very sincerely appreciated. Best, J.

[@lukecronquist6003]

An easy counter example to c = aleph-1is the power set of the reals, because it is unclear if it = aleph-2, etc. It seems these days the continuum hypothesis is not too widely accepted.

[@snatchngrab8262]

This always gets me because there is no difference between an infinite number of guests showing up, regardless of the number of buses they arrive on. As if "infinite number" was actually a thing anyway.

[@td904587]

@28:00 the claim that the cardinality of the real numbers is aleph_1 is what's known as the continuum hypothesis (CH). I wouldn't really classify this as controversial in the same way that the axiom of choice was considered controversial when it was first proposed as an axiom for ZFC. and although Cantor spend a good chunk of his life trying to prove the continuum hypothesis, the solution isn't undecided, but you can in fact prove that it is consistent with the modern axioms of set theory to have CH true, and also consistent to have CH false (ie, that CH is independent of ZFC)

[@lukecronquist6003]

the axiom of choice is still considered controversial

I'm not certain that I understand why the irrationals and the naturals cannot be paired up... Intuitively speaking, (which is where my difficulty may lie... XD ) no matter how many irrational numbers you can muster, there is always another real number to pair it with. Is it because all the irrational numbers cannot be defined or known that we consider that set larger than the naturals? Is this a problem of human ignorance? Even so, were it possible to have complete knowledge of all the irrationals, why can we not pair them up with the naturals? I mean, even though we can go on creating new unique irrationals infinitely, are there not an infinite # of naturals to always find another pairing? Perhaps I'm not understanding what is meant by a "one to one correspondence"... Is this a definitional thing having to do with the meaning of a 'proper subset' as mentioned in the beginning; and if so, then is this a semantics thing where *definitionally* some infinities can be larger or smaller, but in practice cannot? Thank you in advance!

[@alquinn8576]

The problem reveals itself if you imagine applying Cantor's argument for just real numbers between 0 and 1. By the time you've enumerated every possible real value in that interval, you've already used up all of your natural numbers, so you can't get from 1 to 2, etc. after that.

[@snatchngrab8262]

2eelShmeal Cantor was a mess. Bijection of infinite sets proves contradictory, and his diagonal proof is a fail. This is because as you said, regardless of his many reals keep being generated, you can also generate a new natural. If we assume the infinity is contained in a set, then it is. If Cantor's proof was held valid to prove anything, it would have to be that putting infinities into sets is not proper.

[@email9731]

Suppose the irrationals are countably infinite (that is, they can be put in 1-1 correspondence with the rationals). The reals are the union of the irrationals with the rationals. But the union of two countably infinite sets is also countably infinite. And we know the rationals are countably infinite. Therefore the reals are countably infinite, contradicting Cantor's theorem.

[@edbell1246]

Sorry their is no vacancy at The Grand Hotel. No finite object can be infinite in number, because anything infinite would require an infinite amount space to exist in, thus leaving no room for anything else. The only thing that can be infinite is infinity itself. The all of the all is the only thing infinite. Everything else is a sub division of it. And that's it. There is nothing more because that is everything. No extra anything can exist because everything that can exist is contained in the all of the all and it is never ending and ever increasing in size if you were going to try and quantify it. You can't. Numbers are a way in which we compare one thing to another to determine length, width, height, volume, speed, velocity, etc, etc & etc. What could you possibly compare infinity too?

[@torosalvajebcn]

Food for thought: what would happen if there was a fire and and this infinitely occupied hotel had to be evacuated?

[@SendyTheEndless]

I'm guessing it depends if it was an infinite fire or not :) If it were only finite, all of the hotel would be oblivious to it, as a finite object is lost in the vastness of infinity.

[@derekg5563]

SquareWaveHeaven All of the hotel except that finite part. One might say it could be treated as if it didn't exist, but I mean, it's not like if my cousin was in room 2 then suddenly it doesn't make sense for me to go to her funeral.

[@daemonCaptrix]

Hilbert's Hotel fails in that the set of room numbers and the set of guest numbers are already defined. They do not spontaneously change. When exclusively the same numbers are shared between both the set of room numbers and the set of guest numbers, then adding another guest to the guest set will not magically produce another room in the room set. You're just confusing yourself by getting hung up on "but it's infinite!". An infinite set contains an infinite amount of numbers, but it does not contain all numbers. And it does not spontaneously grow simply because you can't wrap your head around limited infinity.

[@derekg5563]

daemonCaptrix Infinity +1 is not a spontaneous growth. Infinity is still being infinity when it "lets someone new come in." It causes a chain reaction within the infinity, and you can look at any part of it you want, room 56901, room 4587390, it doesn't matter, and the properties of the hotel are no more or less infinite than they were previously. I mean, of course it's changed in _who_ occupies what room, but its infinite nature remains completely unchanged.

[@daemonCaptrix]

Derek G Infinity isn't a number. There is no "Infinity +1". The set of rooms is an infinite set, and the set of guests is also an infinite set. If you define these infinite sets as containing identical counts of members, then adding another meber to the guest set means that set is now larger than the room set.

[@derekg5563]

Infinity doesn't have to be a number for infinity +1 to exist. It literally means, there is an infinity, and then you add one to it. So in the hotel, there is an infinite amount of people there, and someone new asks to stay, and they move into the hotel. Infinity +1 isn't meant to be a number, it's describing what happens when you take infinity and add one to it, like in the hotel example. And no, adding to infinity doesn't make it larger. Adding one to it doesn't change its nature. Each set will spread out as long as you want regardless.

[@daemonCaptrix]

Derek G Half the members in an infinite set is still half as many as the full set. It's also infinite. That doesn't mean it's equal.

[@derekg5563]

Well sure, I would agree that just because two sets are infinite doesn't mean they are equal, or exactly the same in every conceivable way. Not sure what that has to do with anything, though. The infinite hotel is a different place after the change in the sense that each person is now in a different room than they were originally, due to the new person coming in. And, that you can point to a new person that was not there originally. It's hard to see what other difference there could be. Even these ideas of infinities being bigger than others... it all depends on what you mean with your terms. Two infinite streams might be bigger than one infinite stream in the sense that its infinity stretches in two different ways. It would also be equal to one infinite stream in terms of how well it can go on forever. It just depends on what differences you care about.

[@ostihpem]

I do not buy in both arguments here: 1. Hilbert's Hotel. It is already inconsistent to say that an hotel with infinite rooms is full. If you have infnite many rooms then you have always free rooms by definition and thus it's impossible for the hotel to be "full". So the Hilbert problem seems to be a pseudo problem. 2. Cardinality. Ok, so we have an infinite list of real numbers. Then we construct a diagonal real number, different from any number in the list. But because the list is infinite - never ending!!! - how can you ever claim that your constructed diagonal number is not in the list? That'd require a final point where you can judge from, but this final point doesn't exist in infinity. At best we have a challenge between two infinities: our infinite list with real numbers vs. our infinite long diagonal number. But it is impossible to say who wins between those infinities.

[@SendyTheEndless]

I have these revservations (no pun intended) as well, only you worded them better than I could have. To be honest, the first one is a bit like saying infinity doesn't exist because we can't count to it. If there can be an infinite amount of rooms, it's no stretch that each room has a bed, suitable plumbing, and a person inside it. "Full yet not full" is just part of the weirdness of infinity. As for the diagonalization thing, I am stuck at the same point you are, though I respect the truth of it, my mind flickers between understanding it as true and wondering if it may be false.

[@Tensoren]

These arguments are just informal examples. In formal set theory the diagonal argument is valid. The same goes for Hilbert's hotel.

[@edemsauce7742]

ostihpem I think you will find my 2 minutes long video interesting. I tried to find the real meaning of the diagonal argument. /watch?v=1tMNjALG55g

[@ostihpem]

@Edem: Yes, I did find your video interesting. Thx for sharing it. As to the uncountability proof I came to this conclusion I can live with: 1. First we have a list of all reals. 2. We construct the diagonal number like Cantor did. 3. Now let's assume that the diagonal number is somewhere in the list. That means the number must be at some line of the list of reals, let's call that line: N. But by the way we constructed the diagonal number we know that it must be different from the number in N. That's a contradiction, so the assumption is wrong, thus is follows that the diagonal number is nowhere on the list. As for Hilbert's Hotel, I still find it a rather confusing analogy to talk about a full-occupied hotel with infinitely many rooms and still being able to get a vacant room for a new guest. It doesn't make sense: If a hotel is fully occupied and no guest leaves the hotel then a new guest doesn't fit in, period. It doesn't matter how many rooms the hotel has.

[@edemsauce7742]

ostihpem I accept that the new number is not on the list, therefore the set can not be listed _that way_ . However this not means necessarily that the list is uncountable. If a list contains infinitely many undefinable (uncomputable) elements, they can not be proved to be countable or uncountable. The new number in this case will be always a number with infinite amount of information, therefore undefined. This happens only if the list also have not-completely-defined elements (as themselfs). So these elements can not be defined one by one, and this is the conclusion of the diagonal argument. I think that the "uncountable" set is a self-fulfilling property: exists only if you think so, this is why something about it can neither be proved to be true or false.