# how do we assume there is infinity?

Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets.

We know that our universe doesn't contain infinite number of elements, so how do we assume there is infinity?

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Axiom of Infinity –  GPerez Feb 5 at 17:33
What is this Universe people talk about so much recently? –  Hagen von Eitzen Feb 5 at 17:36
Our universe doesn't contain the number $2$ either, so your concern seems equally valid for assuming that $2$ exists. –  Dave L. Renfro Feb 5 at 18:16
kenn, you may represent the number $2$ by using physical objects, but that doesn't mean that the number $2$ exists in a physical way. –  Asaf Karagila Feb 5 at 19:00
Things$\ne$ideas. –  Martín-Blas Pérez Pinilla Feb 6 at 11:48

Mathematics, being a human creation, doesn't necessarily have anything to do with the universe (besides which, I would hardly say we know the universe is not infinite).

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." - J. W. N. Sullivan

"Mathematics is a game played according to certain simple rules with meaningless marks on paper." - David Hilbert

We can take anything as an axiom that we want, though of course we tend to focus on the collections of axioms that are interesting to us. In particular, there is nothing preventing us from taking as axioms statements that do not describe what (we think) we know about reality, and different people may well decide to study the consequences of different collections of axioms - even if those collections contradict each other! You are entirely welcome to study set theory where it is taken as an axiom that infinite sets do not exist, as I'm sure people already do.

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@kenn: I do not follow your comment. –  RghtHndSd Feb 5 at 19:10
It is worth remarking that the remark in that Hilbert quote doesn't actually represent Hilbert's more nuanced view about the content of mathematics. (The thought is that, when doing proof-theory, we can/should proceed as if maths is done with meaningless marks.) See e.g. plato.stanford.edu/entries/hilbert-program –  Peter Smith Feb 5 at 19:15
Meanwhile the mathematical realists would probably just ask "what does the physical universe have to do with it?" –  Malice Vidrine Feb 5 at 19:25
@kenn: If that was meant as a joke, I really found it quite funny. But it would help if you could (i) identify the pronoun "it's" in "it's rather against religious doctrines" and (ii) explain how "infinite sets are possible" follows. –  RghtHndSd Feb 6 at 13:37
@kenn Mathematical constructs are ideas in human minds. Your ideas are not constrained by the rules of the physical universe, nor by any doctrine. You can choose to only think things that are consistent with physics-as-we-know-it or your preferred religion, but you may then have difficulty getting any mathematics done. –  Zack Feb 6 at 14:53

Note that while it's tempting to think that mathematics is only used to model our physical reality, this is not true.

If this was true, then what sense does the number $10^{100}$ make? It's larger than the number of particles in the visible universe, so surely we can't represent it physically.

And yet, even the ancient Greek believed that if $n$ is an integer, then $n+1$ exists. So if $10^{100}$ doesn't exist, but for every $n$ which exists, $n+1$ does exist... something goes wrong.

Even if you don't talk about infinite sets, infinity is inherent into the natural numbers as we are used to thinking about them. Sets were created to allow collections of mathematical objects (like numbers) to be mathematical objects on their own accord. So naturally, we are inclined to talk about the set of natural numbers which is infinite.

Some people do reject this approach to mathematics, they may believe that infinite sets do not exist, but there are infinitely many natural numbers nonetheless; or sometimes that there is a largest number (even though we don't know what it is). These philosophical (and mathematical) schools of thought are joined under the term "finitism" (and ultrafinitism in the latter case).

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Thank you for the answer and detail info you provide, according to your profile you must be a math genious. –  kenn Feb 5 at 21:22
Heh, thanks, but that can't be right. I'm not good at anything except set theory, and even at that I'm not as great as I wish I could be... –  Asaf Karagila Feb 5 at 21:25
$10^{100}$ makes perfectly good physical sense. OK, you can't show me a set of $10^{100}$ baryons but you can just show me 333 objects and ask me to enumerate the subsets. –  David Richerby Feb 6 at 9:25
@David: Go ahead. Here are 333 objects, $\{0,\ldots,332\}$. I want a full enumeration of their subsets. If it's not on my desk by tomorrow morning there will be hell to pay. :-) –  Asaf Karagila Feb 6 at 9:28
@David, I can do even better, and I can give you just $10$ objects and ask you to enumerate the sets of subsets of these objects. You still cannot do it physically. Moreover, if there is a number $n$ which is the least which doesn't make physical sense I can find two reasonable numbers $k,n$ and ask you to enumerate the sets of sets of ... ($k$ times) of sets of the $n$ objects. Conclusion: either every number can be represented physically, or mathematics has nothing to do with physical reality. –  Asaf Karagila Feb 6 at 9:29

There is $1$ and there is $n+1$. That's infinity.

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No, it is not. At best if for every $n\geq1$ there is $n+1\neq 1$ (which is quite a project to ensure), and if $n+1=m+1$ implies $n=m$, then it implies that every nonzero natural number exists. It does not ensure there is a set of all nonzero natural numbers, and even less it gives us infinity (whatever that might be). –  Marc van Leeuwen Feb 6 at 10:59
@MarcvanLeeuwen, are you assuming something other than Peano arithmetic? If so, you should specify. –  Karolis Juodelė Feb 6 at 16:54
I'm not assuming anything. I'm just saying that more is needed than having $1$ and (some? which?) $n+1$ in order to get "infinity". But now that you mention it, note that in Peano's arithmetic there is no infinity (and fortunately so). –  Marc van Leeuwen Feb 6 at 17:27
@MarcvanLeeuwen, if we define $\mathbb N$ with Peano axioms, define a finite set as one which has a bijection with $\{1 \dots n\}, n \in \mathbb N$ and define an infinite set as one that is not finite, infinite set is obvious. If you thin some of those definitions are unreasonable, feel free to give alternatives. –  Karolis Juodelė Feb 6 at 18:05
One doesn't define $\Bbb N$ with Peano's axioms. Peano's axioms define a theory where the universe of discourse is intended to model the natural numbers, but it is not any form of set theory, and in particular does not construct a set of natural numbers. –  Marc van Leeuwen Feb 6 at 18:27

If you really mean to ask "how" we assume that there is an infinite set, that's simple: we include it as an explicit assumption (that is to say, an axiom) in the foundations of set theory. So far, that seems to work — no-one's been able to find a contradiction between that assumption and the other usual axioms of set theory.

The reason why we need to do that is that the other standard axioms of set theory are not strong enough to let us prove it — it's possible to construct models that satisfy all the other axioms of ZF set theory, but which don't have any infinite sets.

So why do we want to assume the existence of infinite sets, then? Well, the reason set theory was developed in the first place was so that we would have a formal language for talking about collections of numbers. In particular, we would like to have a formal way of saying "all numbers" or, say, "all even numbers" or "all odd numbers" or "all numbers greater than 5". In set theory, we call all those things "sets" of numbers.

Unfortunately, it's easy to show that, if a "set of all even numbers" exists, then it cannot be finite: it's possible to define a one-to-one correspondence between the set of even numbers and some proper subset of it, like, say, the even numbers greater than 10. (Try to do it yourself! It's not hard.) Thus, if we want to be able to talk about "the even numbers" (or "the odd numbers", etc.) as a set, then we have to construct our set theory in such a way that it includes some infinite sets.

The alternative, of course, is to work in a theory without infinite sets, but that gets kind of awkward pretty fast, because you won't be able to formalize statements like "this property holds for all even numbers", since "all even numbers" is not a well defined set in a finitistic set theory. You can work around such limitations in various ways, e.g. by saying "if $x$ satisfies the definition of an even number, then this property holds for $x$" instead (which formally avoids using "even numbers" as a set), but most mathematicians would prefer to avoid such logical contortions and just work in a theory in which "even numbers" is a thing (specifically, a set).

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it's easy to show that, if a "set of all even numbers" exists, then it cannot be finite There is nothing wrong regarding definition, it's not selfcontradictory but in our universe there is no example of infinite sets. –  kenn Feb 6 at 12:08
@kenn: So what? Also, how do you know? –  Asaf Karagila Feb 6 at 15:51
Assuming the existence of a set on which an injective, but non-surjective function is defined is a somewhat smaller "leap of faith" that just starting with Peano's axioms. The "natural" place for such an assumption is probably within axioms of set theory. From this axiom, you can prove the existence of a infinite subset $N$ that satisfies Peano's axioms. –  Dan Christensen Feb 6 at 16:18
@AsafKaragila Is there any example of infinite sets in Universe physically? –  kenn Feb 6 at 16:37
@kenn: Is there a physical proof that there aren't any? –  Asaf Karagila Feb 6 at 18:15

In mathematics, anyway, you want an Axiom of Infinity that postulates the existence of a relatively simple set that can be shown to be infinite. There is such an axiom in the ZFC axioms of set theory, though I don't find it all that intuitive. A roughly equivalent axiom is to assume the existence of at least one set $S$ on which is defined a function $f:S\to S$ that is injective, but not surjective. (A slight variation of Dedekind-infinite.)

A set is Dedekind-infinite if and only if there exists a bijective mapping from S to a proper subset of S.

It can be shown that a set if Dedekind-infinite iff and only if a subset of it has defined on it a successor function that satisfies the Peano Axioms, i.e. a subset that is itself a self-contained number system that supports proof by induction. Sketching the proof:

Suppose $S$ is a Dedekind-infinite set, i.e. there exists a $T\subsetneq S$ and a bijection $f : S \to T.$ Let $n_0$ be an element of $S$ not in $T.$ We can then easily prove that there exists $N\subset S$ such that the Peano axioms hold on $N$ with $n_0$ being the "zero" and $f$ the required successor function. The key is the selection criteria for subset $N$:

$\forall a:[a\in N\iff a\in S \land \forall b:[n_0\in b \land \forall c:[c\in b \land c\in S \implies f(c)\in b]\implies a\in b]]$

Going the other way, suppose $N \subset S$ such that the Peano axioms hold on N. We can easily prove that $N$ is Dedekind-infinite. We can also prove that, in general, any superset of a Dedekind-infinite set is also Dedekind-infinite. So, S must also be Dedekind-infinite.

Nearly everyone agrees that the set of natural numbers $N$ is infinite. So, you could define a set $S$ to be infinite if and only there exists an injective mapping $f:N\to S$.

This definition can also be shown to be equivalent to Dededkind-infiniteness.

In mathematics, to assume there is infinity, we simply define it (as above) and construct or postulate the existence of a set that can be shown to be infinite.

Whether or not there are finite elements in the physical universe is immaterial to mathematics. That does not mean that, say, the integers inhabit some fantastic fantasy land concocted by bored mathematicians. Even though I will only ever have a finite amount of money in my bank account, I still model it's rapid swings using the infinite set of integers. It's just convenient to have a theoretically limitless bank account balance. Infinite sets are a practical necessity.

Follow-Up

See "Infinity: The Story So Far" at my math blog.

There I present an informal development of the notion of infinity beginning with a novel, non-numeric approach to the finite set (equivalent to Dedekind) along with accompanying formal proofs.

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The last paragraph is not "too provocative". It is entirely false. Moreover "the structure of The Natural Numbers" implies addition, etc. are defined, and your statement can somehow be interpreted as if every set carries such structure with it. What is true, however, is that a set is Dedekind-infinite if and only if there is an injection from the natural numbers into it. –  Asaf Karagila Feb 7 at 4:09
Ugh. Nobody argues that there is no injection from the set of natural numbers into Dedekind-infinite sets. But you wrote, and I quote (in case you have troubles reading your own posts) "a set is Dedekind-infinite iff the natural numbers are a subset of it". That is false. And I furthered remarked that the phrasing in which you correct yourself can be interpreted in a very misleading way. –  Asaf Karagila Feb 7 at 5:12
Dan. You are using the word subset wrong. Mathematical language has a precise meaning, and not just anything that you feel fine with. –  Asaf Karagila Feb 7 at 6:13
Why does it have to take an infinite string of comments before you are even willing to concede that you might have done something wrong, and even then you are pussyfooting around it?? The results may have been translated to natural language, but this is a mathematical site and the word "subset" is a part of the mathematical language. –  Asaf Karagila Feb 7 at 6:59
@DanChristensen So by "a set identical in the structure to The Natural Numbers," you mean a countably infinite subset? –  Braindead Feb 7 at 18:34