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Would I be right in saying that $\aleph_0 \in \mathbb N$?

Or would it be a wrong thing to do?

Rodrigo de Azevedo
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    "Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall" (repeat). ([source](http://mathworld.wolfram.com/Aleph-0.html)) – mvw Oct 16 '15 at 01:38
  • @mvw Thank you for that! – BrianO Oct 16 '15 at 01:57
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    @mvw You're so slow! `:D` "Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take aleph-null down, and pass it around, Aleph-null bottles of beer on the wall." – yo' Oct 16 '15 at 19:12
  • @yo': That's not-a-number bottles of beer on the wall after taking aleph-null down ;-) – Deduplicator Oct 16 '15 at 20:19
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    @Deduplicator It depends *which* $\aleph_0$-many bottles you take down. :) – Noah Schweber Oct 16 '15 at 20:23
  • @Deduplicator Exactly what Noah says. Cleverly taking down aleph-null bottles can be done forever :-) – yo' Oct 16 '15 at 20:40

2 Answers2

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$\aleph_0$ is not a natural number. It is the cardinality of the set of natural numbers - each individual natural number is finite, but the set of all natural numbers is infinite.


Based on the comments below, let me share an anecdote: I was once part of a medical survey on Things Neurological. One of the tasks they had us do was come up with short definitions of common words, on the spot. Beforehand, I (talk about tempting fate) joked about how easy this would be. The very first word they gave me: "Number." For the life of me, I couldn't come up with anything. The doctors gave me very weird looks. Finally I sputtered something like "An element of a formal system," which didn't help with the weird looks but did let us move on to other, simpler words.

The Moral: Man, number is weird.

Noah Schweber
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  • Ok, so we can't really categorize it in any set of numbers then ? –  Oct 16 '15 at 01:43
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    @Astroman I'm not sure what that means. It is certainly not a: natural number, real number, complex number, quaternion, surreal number (the ordinal $\omega$ *is*, but the cardinal $\aleph_0$ *isn't* depending on how precisely it's defined :P), fuzzy number, nimber, or dyadic rational smaller than 37. What it is, is a **cardinal number**, and these are very well-understood things - if extremely different from e.g. real numbers. So, what do you mean by "categorize it in any set of numbers"? It sounds like you have a specific notion of what "number" is in mind - can you clarify? – Noah Schweber Oct 16 '15 at 01:47
  • No, you exactly answered my question. I understand now, thank you! –  Oct 16 '15 at 01:48
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    I really have no idea how a layperson would define "number" in any simple way. Perhaps "something used to count things"? But then surely even the layperson can see the issues with that, provided they are given time to think a bit. – Will R Oct 16 '15 at 02:30
  • @Will R I would say number is a common property of sets of various objects whose elements can be put in bijectional relations. Of course this would not work for non-integer numbers though – Anixx Oct 16 '15 at 12:05
  • @WillR Being a layperson myself I would say 'number' is what we use to count and/or value things. - You must find laypersons very weird indeed. – nilo de roock Oct 16 '15 at 12:29
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    A number is any element of the algebraic completion of the unique complete ordered field. Duh. – Akiva Weinberger Oct 16 '15 at 13:07
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    @AkivaWeinberger: That doesn't cover transfinite ordinal or cardinal numbers, $p$-adic numbers, split-complex numbers etc. – hmakholm left over Monica Oct 16 '15 at 13:34
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    @HenningMakholm True. But if you argue like that, there's not much preventing you from saying that _everything_ is a number. – Akiva Weinberger Oct 16 '15 at 13:35
  • @AkivaWeinberger: Except the fact that people _don't_ actually use the word "number" about most of everything -- which is plenty good for me. – hmakholm left over Monica Oct 16 '15 at 13:37
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    Number is an abstraction of quantity. – Asaf Karagila Oct 16 '15 at 15:16
  • On my opinion, the definition of "number" should directly or indirectly include, that it _counts_ things. Well, maybe it closes out the negative or the complex numbers, but I don't think it would be a problem. If you ask somebody on the street to say a number, nearly everybody will say a natural number. And, if you are talking in a non-mathematical sense, you shouldn't apply to the mathematical formulas rigorously. – peterh Oct 16 '15 at 15:24
  • @AsafKaragila Yay! And Henning's examples just show how well we are in our ability to abstract from the concept of counting sheep – Hagen von Eitzen Oct 16 '15 at 19:37
  • @Hagen: I have spent a nontrivial number of hours and many discussions to get to this elevator pitch. – Asaf Karagila Oct 16 '15 at 19:44
  • @AsafKaragila I hope those hours spent are still in the realm of concrete and not abstracted counting ;) – Hagen von Eitzen Oct 16 '15 at 20:16
  • @ndroock1: Aha! You used the word "value"; that's what I was forgetting. I think something like "a number is a thing that is used to denote or indicate the value of something" might be close to a working definition for the layperson. – Will R Oct 16 '15 at 23:34
  • I'm curious, why the downvote? – Noah Schweber Oct 20 '15 at 21:25
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No.

  1. 1 is a natural number.
  2. A number made by taking a natural number and adding 1 to it is a natural number.
  3. There are no other natural numbers than the ones whose existence is implied by 1 and 2.

$\aleph_0$ is not equal to 1. Therefore it can only be a natural number if there is some natural number $x$ such that $x+1 = \aleph_0$. If you think it is a natural number, you have to be able to say what $x$ is.

In fact, there is no such $x$.

jwg
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    I don't think this helps. Somebody who can't tell whether $\aleph_0$ is a natural number isn't going to be able to tell whether "$\aleph_0-1$" is a natural number, either. – David Richerby Oct 16 '15 at 09:48
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    @DavidRicherby that's not how it works. Firstly there is no presumption that you can reverse the operation of adding 1. Secondly, either you *defined* your thing by saying it is something else + 1, and that thing is a natural number. Or you didn't. If you didn't then it isn't. – jwg Oct 16 '15 at 09:50
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    Sure, I know that (the quote-marks were meant to indicate that aleph-zero-minus-one isn't a real thing). But somebody who doesn't know whether or not $\aleph_0$ is a natural number doesn't know that, or they wouldn't be asking the question. What you're saying amounts to "If you believe that $\aleph_0$ is a natural number, you must believe that $\aleph_0-1$ is a natural number, too." To which the response is, "Great but I can't tell whether *either* of those things is a natural number!" – David Richerby Oct 16 '15 at 09:58
  • Comments are not for extended discussion; this conversation has been [moved to chat](http://chat.stackexchange.com/rooms/30458/discussion-on-answer-by-jwg-is-aleph-0-a-natural-number). – user642796 Oct 20 '15 at 03:51