# Does the set of natural numbers contain infinity? [duplicate]

Let $N=\{1,2....\}$ be the set of naturals numbers. Is $\infty \in N$, as the list is never ending ?

• What is $\infty$? – nombre Oct 16 '15 at 12:04
• Is this some kind of trend? ;-) Almost the same question was asked earlier today: math.stackexchange.com/questions/1482450/… – Hans Lundmark Oct 16 '15 at 12:15
• Short answer: No. $\infty\notin\Bbb N$. – Akiva Weinberger Oct 16 '15 at 12:46
• @BLAZE: Yes, of course. But why would the pedal curve of the hyperbola.be in $N$? (I was asking that because asking if $\infty$ is in the set of natural numbers suggests one wonders if $\infty$ is a natural number) – nombre Oct 16 '15 at 13:35
• @BLAZE You can think of it this way. It doesn't make sense to speak about "John" in general, because there are many people called John. But you can consider a particular person called John, and say "From now on when I say John I mean this person." If you want you can even take someone called "Amy" for that, but people will look at you funny. It's similar here: there are many different objects in math that some people like to call "$\infty$", and so if you don't say which one you're talking about it's meaningless. If I ask "Is John in the room right now", it's meaningless, it could be any John. – Najib Idrissi Oct 16 '15 at 13:53

One property about the set $\Bbb N$ of natural numbers is that it does not have a largest element, so if you defined $\infty$ as some element which is larger than every natural number, it wouldn't be in $\Bbb N$ anyway because of this property.

Having said that, I recommend you look up the ordinal numbers and what it means for a set to be well-ordered (it means every subset has a least element). We could add an element into the set $\{1,2,3, \dots \}$ which we define to be larger than every element in $\{1,2,3,\dots \}$. Usually, instead of $\infty$, we call the element $\omega$. But the set $\{1,2,3,\dots, \omega \}$ is a different set than $\Bbb N$ (it contains $\Bbb N$ as a proper subset). $\omega$ is the largest element in this set, and you could think about $\omega$ as "infinity" in some sense.

But then we can add an element defined to be larger than every element in this set. Let's call this element $\omega + 1$. So $\{1, 2, 3, \dots, \omega, \omega + 1 \}$ is the set where $\omega$ is larger than every natural number, and $\omega + 1$ is larger than every natural number and larger than $\omega$. So, it's not useful to call $\omega$ "$\infty$" because there is something larger than it. (Although, Georg Cantor proved that, as far as set cardinalities/sizes go, there are infinitely many distinct infinites...).

• Is $\omega$ the same as "aleph_0", which we were told was the highest countable number? – holroy Oct 16 '15 at 15:12
• @holroy Sort of. In set theory, everything is a set, and $\aleph_0=\omega=\Bbb N=\{0,1,2,\dots\}$. However, since everything is a set, we have weird things like $2=\{0,1\}=\{\{\},\{\{\}\}\}$ in set theory. Outside of set theory, $\aleph_0$ refers to the amount (cardinality) of elements in $\Bbb N$, and $\omega$ refers to their order type (sort of), and there's no reason to think of these as sets. – Akiva Weinberger Oct 16 '15 at 17:07
• @holroy The ordinals have the property that, for any set $S$ of ordinals, there is a smallest ordinal bigger than anything in $S$. The ordinals include $0$, $1$, $2$, etc. (No negatives of fractions, though.) By the above property, there is a smallest ordinal bigger than anything in $\{0,1,2,\dots\}$, and it's called $\omega$. ($\omega-1$ is not an ordinal.) – Akiva Weinberger Oct 16 '15 at 17:09
• @holroy A big difference between $\aleph_0$ and $\omega$ is that $\omega+1\ne\omega$, but $\aleph_0+1=\aleph_0$. In set theory, where $\omega$ and $\aleph_0$ refer to the same set, they just say that the symbol $+$ means different things in those two equations (the former being ordinal addition, the latter being cardinal addition.) Fun fact: ordinal addition isn't commutative! It's defined in such a way that $\omega=1+\omega\ne\omega+1$. – Akiva Weinberger Oct 16 '15 at 17:12

No, if $\infty \in N$, then the list would have an end, namely $\infty$. Of course, you would still need to go infinitely far up from $1$ (or indeed any other number) to get to $\infty$.

• To add to this answer, you can define the extended Natural numbers to be the union of the Natural numbers AND infinity. – amcalde Oct 16 '15 at 12:10

Infinity is not well-defined as it has different meanings depending on the context so placing it in a set that is well-defined (the set of natural numbers $\mathbb{N}$ for instance) is clearly meaningless.

• Sure. This is an example of the general idea of "Compactification" which is studied in topology. Though the $\overline{\mathbb C}$ is a very elegant and intuitive example, as it is a sterographic projection of Riemann sphere. – Ranc Oct 16 '15 at 13:01
• @Ranc Really? What your definition of "$\infty$"? There are many things that can conceivably be called "$\infty$". One compactification of $\mathbb{R}$ is $S^1$, another one is $[-\infty,+\infty]$, etc. And this is a completely different beast than, say the ordinal $\omega$ mentioned in another answer. – Najib Idrissi Oct 16 '15 at 13:49
• @NajibIdrissi I agree. The definition (or somewhat - aspect) of infinity is context-wise. That being said: it is always (as in mathematics) well-defined. I was only referring to the "definability" issue, and attempted to show more aspects of the term (infinity). – Ranc Oct 16 '15 at 14:02