how do we assume there is infinity?

Question

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 (including subatomic particles), so how do we assume there is infinity?

How do we know that our universe doesn't contain infinite number of elements? It clearly does: ${}$, ${{}}$, … but you need an axiom to replace the dots. — lhf, Feb 05 '14 at 17:34
There have been explorations of "set theories" in which there are no infinite sets. — André Nicolas, Feb 05 '14 at 17:43
I know about infinite set theories, I am asking while it is physically not possible, how do we assume it exists? — kenn, Feb 05 '14 at 17:48
I think there are some scientific theories where you can't really assign a value to the "number of objects in the universe", so it is not finite. — user76568, Feb 05 '14 at 17:48
@kenn physicists use infinity on many occasions in their theories. — user76568, Feb 05 '14 at 17:52
Our universe doesn't contain the number $2$ either, so your concern seems equally valid for assuming that $2$ exists. — Dave L. Renfro, Feb 05 '14 at 18:16
I meant particle level countable physical objects, not abstract mathematical objects — kenn, Feb 05 '14 at 18:20
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 05 '14 at 19:00
Without infinity it would impossible to pull apart a solid sphere into a finite number of pieces, move them around and assemble them again into a sphere twice as big. The real world is far too limiting. — Disintegrating By Parts, Feb 12 '14 at 19:38
Your addition is an assumption as well. How do you know that there is a definitive smallest particle? You don't. You just know that currently, a theory that doesn't predict infinitely many particles is giving good predictions. But you don't know what the future might hold, and the sort of paradigm shifts physics might go through in two, ten or thousands of millennia from now. — Asaf Karagila, Nov 29 '14 at 11:25
@AsafKaragila Good point! Unfortunately, the theories of the universe are always sold as truths. — Peter, Jun 29 '16 at 10:29
The notions "finite" and "infinite" used in mathematics apply only to mathematical objects, they do not apply to physical things. There are lots of words that have both a mathematical and non-mathematical meanings, such as "positive", "set", "entire", "function", "group", "field", "universe", etc. Their meaning in everyday language need not have anything to do with their technical meaning as used mathematically. — Tommy R. Jensen, Aug 11 '19 at 13:18

score 17 · Accepted Answer · edited Jun 12 '20 at 10:38

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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.

edited Jun 12 '20 at 10:38

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answered Feb 05 '14 at 17:51

Zev Chonoles

129,973

your answer is almost satisfactory, you say human being can create objects, it's rather against religious doctrines but in that way infinite sets are possible – kenn Feb 05 '14 at 17:57
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@kenn: I do not follow your comment. – RghtHndSd Feb 05 '14 at 19:10
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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. http://plato.stanford.edu/entries/hilbert-program/ – Peter Smith Feb 05 '14 at 19:15
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Meanwhile the mathematical realists would probably just ask "what does the physical universe have to do with it?" – Malice Vidrine Feb 05 '14 at 19:25
@rghthndsd I don't understand what you mean by I do not follow your comment would you please be more precise? – kenn Feb 06 '14 at 12:20
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@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 06 '14 at 13:37
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@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. – zwol Feb 06 '14 at 14:53
@rghthndsd i)with "it" I meant "creation of something", it's quite clear if you follow the context of the sentence. ii) where do I assert that "infinite sets are possible" ? I just asked a question that I want to learn something, is it that funny? – kenn Feb 06 '14 at 15:19
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@kenn: I do apologize if my previous comment came off as rude, it was certainly not my intention. I believe I understand your comment now: you need a comma before "but". As it stands, it sounds like you are saying that being against religious doctrines makes infinite sets possible. With the comma, I see now you are saying that the fact that human beings can create objects is what makes infinite sets possible. Sorry it took me so long to figure out! – RghtHndSd Feb 06 '14 at 20:30
@rghthndsd No problem, I should have been more precise. I was hard pushed for time then. I am glad that my point got understood. – kenn Feb 07 '14 at 20:25
Readers shouldn't get the impression that mathematicians are in the habit of perpetually inventing fantasy worlds based on new and ever more bizarre axiom systems. In some sense, the majority of work done by mathematicians today remains the working out of the consequences of axioms first introduced in the 19th and early 20th centuries. – Dan Christensen Feb 09 '14 at 06:19
If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. —Sir Arthur Stanley Eddington So I must have referred to first law of thermodynamics instead of religious doctrines? I got misunderstood. – kenn Feb 15 '14 at 14:17

score 15 · Answer 2 · edited Apr 13 '17 at 12:21

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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).

Some threads of interest.

edited Apr 13 '17 at 12:21

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answered Feb 05 '14 at 19:00

Asaf Karagila

393,674

Thank you for the answer and detail info you provide, according to your profile you must be a math genious. – kenn Feb 05 '14 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 05 '14 at 21:25
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$10^{100}$ 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 06 '14 at 09:25
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@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 06 '14 at 09:28
@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 06 '14 at 09:29
The ancient Greeks also believed there was a dude on top of mount olympus who throws around lightning bolts. While the argument they believed makes sense, that they believed it is not quite convincing (appeal to antiquity: http://www.logicalfallacies.info/relevance/appeals/appeal-to-tradition/ ) – Martijn Feb 06 '14 at 10:07
@Martijn: I don't see anywhere in my answer that says that someone should be convinced that infinite sets exists in a physical way. Not because of the Greeks thought that there are infinitely many numbers, nor otherwise. Don't put words in my mouth, you're not my brain. – Asaf Karagila Feb 06 '14 at 10:19
wow, no need to get so defensive. What exactly are you trying to convey when you say "And yet, even the ancient Greek believed that if n is an integer, then n+1 exists." if not an appeal to tradition that because they believed it this is a valid notion? – Martijn Feb 06 '14 at 10:38
@Martijn: (1) Mathematics, like anything else in the history of civilization, is based on tradition. Even paradigm shifts are based on tradition. Even today Archimedean property is used wildly, perhaps in a different universe, where the Greek worked modulo $10^{100}$ things look different. (2) Your comment reads to me as somewhat offensive in its tone. Maybe it's just me, but that's how I read it. – Asaf Karagila Feb 06 '14 at 10:52
@AsafKaragila If your desk is that big, you should revise your estimate of the number of particles in the universe. :-P – David Richerby Feb 06 '14 at 12:32
@David: It's not my estimate; and what does my desk have to do with it? – Asaf Karagila Feb 06 '14 at 15:52
+1 for the formulation "visible universe" instead of simply "universe". – Peter Jun 29 '16 at 10:38

score 5 · Answer 3 · answered Feb 05 '14 at 17:33

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

answered Feb 05 '14 at 17:33

Karolis Juodelė

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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 06 '14 at 10:59
@MarcvanLeeuwen, are you assuming something other than Peano arithmetic? If so, you should specify. – Karolis Juodelė Feb 06 '14 at 16:54
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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 06 '14 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 06 '14 at 18:05
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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 06 '14 at 18:27
@MarcvanLeeuwen, can you give definitions of $\mathbb N$ and "finite", from which it is not obvious that $\mathbb N$ is not finite? – Karolis Juodelė Feb 06 '14 at 19:06
$5$ upvotes ? Probably because it is a good joke ... – Peter Jun 29 '16 at 10:39

Ilmari Karonen · Answer 4 · 2014-02-06T12:58:16.637

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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).

edited Feb 06 '14 at 12:58

answered Feb 06 '14 at 11:44

Ilmari Karonen

26,117

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 06 '14 at 12:08
@kenn: So what? Also, how do you know? – Asaf Karagila Feb 06 '14 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 06 '14 at 16:18
@AsafKaragila Is there any example of infinite sets in Universe physically? – kenn Feb 06 '14 at 16:37
@kenn: Is there a physical proof that there aren't any? – Asaf Karagila Feb 06 '14 at 18:15
@AsafKaragila question to question, we are going nowhere. – kenn Feb 06 '14 at 18:36
@kenn: Are there any physical examples of sets, finite or not, in the Universe? That's actually a serious question -- if you can show me what you'd consider a "physical example of a set", I might be able to tell whether there's anything like that which isn't finite. But one could also argue that sets are purely logical groupings of concepts that have no physical existence: sure, I can take any collection of things or concepts, like "my fingers" or "Greek deities" and call it a "set", but that set will have no physical existence apart from its members (which may or may not themselves exist). – Ilmari Karonen Feb 06 '14 at 18:40
@kenn: I agree. But was the point. – Asaf Karagila Feb 06 '14 at 18:49
@IlmariKaronen A set consists of elements such as, {apple, banana, cherry, lemon} or {Kenn, Karonen, Karagila}, they are abstract representation of countable physical objects. Isn't math an abstract reflection of our universe in human mind? – kenn Feb 06 '14 at 18:53
@kenn: OK, so let's refine your concept a bit. Do you think {Zeus, Athene, Poseidon} is a set (given the assumption that none of those deities actually exists)? What about the set of notes playable on a piano? Or on a violin? – Ilmari Karonen Feb 06 '14 at 19:04
@IlmariKaronen It's the subject of ontology, my assumption here is not important indeed. My point was similar, sutle answer of my question lies in if human being can create objects as Zev Chonoles posted answer above, if so, then we can say yes, infinite sets are possible. I hope I make my point understood. – kenn Feb 06 '14 at 19:21

how do we assume there is infinity?

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