The axiom of infinity cannot be derived from the rest of the axioms of ZFC, if these other axioms are consistent. Nor can it be refuted, if all of ZFC is consistent.ZFC is the minimal set of axioms for modern math to work out. So basically a few things:
- Modern math is just as consistent with or without Infinity. Every finite result in mathematics could have been derived with or without needing to refer to infinity. (Sometimes people do like when then claim things like they are “integrating to infinity”, but technically a reference to infinity isn’t needed to get the same answer)
- No matter how hard you try, Infinity "cannot be derived...Nor can it be refuted" by ZFC meaning you cannot ever infer the existence or non-existence of Infinity from the rest of mathematics.
It would ruin mathematics for a lot of people, especially the mathematicians themselves, since infinity is required to produce all the most interesting stuff. Furthermore, what if infinity does exist and you are removing it just because you can't know either way given everything else? Think of all the intellectual damage done to mathematics if infinity is falsely removed! (There are entire branches of good math if infinity is real!) But then by that same token, I guess you also have to ask if myriads of mathematicians are just wasting their time studying an entity that you can only be axiomatized and no more.
As for me and my house. I for one double majored in mathematics and so am quite aware of the amazing intellectual structures that can be built with the assumption of infinity and to be honest... they are pretty amazing! At some point, the beauty, wonder, and to be quite honest *practical utility* (integrating to infinity is darn helpful sometimes!) of assuming the existence of Infinity outweighs any pros that seem to come from getting rid of armed by nothing more than philosophical arguments.
The reality is Infinity has been around for enough centuries to have proven it's utility, beauty and ability to make many magnificent contributions to mathematics. I can't say there is any evidence beyond philosophical wishful thinking to the claim that math would be better off without Infinity... even if you were to believe Infinity was false! I for one have tasted math containing Infinity and love it, am inclined to believe it, and am pretty sure I will be engaging in math containing/refereeing to infinity for the rest of my life as it seems like the most wise, lovely, productive and correct thing to do.
You can get infinite sequences -- in the sense of unending sequences -- without the axiom of infinity. They show up in Peano Arithmetic, for example. You only need the axiom of infinity if you want to talk about an actually infinite set, like the set of all natural numbers. As it turns out, a huge chunk of ordinary mathematics can be done without committing to infinite sets. There are, however, results that cannot be proved without the axiom of infinity -- for example, the Paris-Harrington theorem. But I'm not exactly sure what practical value that theorem has. Anyway, there is a really nice review article on this stuff here.
ReplyDeleteOh, I should have asked: is someone picking on the axiom of infinity? I thought it was usually the axiom of choice that came in for all the hard times.
ReplyDeleteJonathan,
ReplyDeleteThank you for your links. Now I have a question: "You can get infinite sequences... without the axiom of infinity... You only need the axiom of infinity if you want to talk about an actually infinite set"
But how is an infinite sequence not equivalent to an infinite set? Isn't the set containing the elements of the sequence infinite?
What I mean is something like this. Define zero as the empty set. Define the natural number n+1 as the power set of n. So, 1 = P(0) = {0}; 2 = P(1) = {0, {0}}; 3 = P(2) = {0,{0},{{0}},{0,{0}}}; etc. Define the successor function S(.) to be the power set operation. So, the function takes in a natural number i and gives the natural number i+1. What we have here is an unending sequence. For any natural defined in this way, I can construct one greater. (Greater gets the obvious definition in terms of the successor function.)
ReplyDeleteWhat I *can't* do (without the axiom of infinity) is collect all of the elements of this sequence and talk about that aggregate (set) as a whole. So, I can prove things like, "There is no greatest prime number," without the axiom of infinity, because I don't need to talk about the *set* of natural numbers (or even the set of prime numbers) in order to do it. I don't need for there to be a completed set of natural numbers or a completed set of prime numbers.
So, yes, the set containing the elements of the sequence defined above is indeed infinite. But in order to do ordinary arithmetic and anything (which is a lot) that can be coded into ordinary arithmetic, you don't need to gather up all of those infinitely many elements into a definite collection. The result is that I get infinity in some sense without the axiom of infinity. But in another sense, I don't get infinity without the axiom of infinity.
Okay, I think I see. You can have all the machinery that goes into the natural numbers without having to make reference to the full structure avoiding the idea you have an infinite set. Fine.
ReplyDeleteBut still, it to me sounds funny to say: I have a sequence, and the empty set is in my sequence, and given x in my sequence x+1 is also in my sequence, etc... (where x+1 is defined in the power set sense you describe), and yet since you used the word sequence and not the word set you are okay. :)
But joking aside I think I see what you mean. You can define a procedure that leads to an unending sequence without having to refer to the sequence as a whole and thus never have to talk about an actual infinite set existing.
This is actually more interesting the more I think about it...
ReplyDeleteJS,
ReplyDeleteInfinity begins where we either can not know beyond a certain value (numerical or otherwise), or we do not care to know beyond that value.
JS,
ReplyDeleteBy the way, what is all that greek stuff mean? In words may be?
Am I the only one that sees the obvious parallel to a question Joseph has discussed in other posts that has a lot more gravitas?
ReplyDeleteI think it's obvious, so I'm gonna respond to that. You are arguing from the postmodern standpoint of practicality. If it's useful, don't get rid of it even if it might not be true. That's a fair argument in my mind as an engineer. The problem is that something's utility is entirely a function of perspective (something science seeks DESPERATELY to remove)! You're arguing that infinity is useful. But what if infinity, or the belief in infinity causes us to limit our progression in other areas of mathematical research? What if it is a stumbling block of sorts? Even worse, what if the use of infinity has been responsible for the death of millions of people because someone implemented it improperly on the controller of a nuclear weapon (hint: this HAS happened though not on a nuclear weapon but on a missile)?
Now in the previous paragraph, I'm arguing from the perspective of someone who doesn't believe in infinity. In actuality, I do agree with you. I think the utility of something is a good reason to "believe" in it and keep it around. But I fully acknowledge that this is 100% a function of my perception of that utility, and the fact that I think overall, infinity does more good than harm. But there are many things the people of our world have believed in that appeared to be useful (demonic possession anyone?) but ended up being responsible for untold destruction and mayhem. In the end, we are better off NOT putting our FULL confidence in the things we cannot reliably demonstrate by tried and true means. We should keep them in the realm of belief/hope and treat them with the level of uncertainty they deserve.
And finally, as an engineer, I'll just say y'all are ridiculous for even worrying about this. To the engineer, infinity is whatever you want it to be (usually just a really big number) and we don't worry about it ;-) .
Also, for further food for thought, Newton assumed an inertial frame existed. Guess what, it's not provable that one does (and in fact, all the evidence seems to suggest there likely isn't one at all). So the same questions apply to all of Newtonian mechanics (which have shaped engineering for hundreds of years).
I agree that there is something funny about saying there is a such a sequence and then saying that the sequence as a whole does not exist. I think mostly that is just a product of figurative language. In this case, when I talk about the sequence, I am really only referring to a finite head of a sequence. It just turns out that I can always write down new members at the tail of that finite sequence!
ReplyDeleteThe weird thing about the axiom of infinity -- and what it gives you that is new -- is that it postulates the ability to actually write down the whole of an infinite sequence. Anyway, yes, I think you are correctly describing the scenario: "You can define a procedure that leads to an unending sequence without having to refer to the sequence as a whole and thus never have to talk about an actual infinite set existing."
I see you have discovered the metaphor. However, let me just state I am trying to argue beyond just practicality.
ReplyDeleteSeriously, I think Infinity is a good metaphor, both because it is an obvious in that it is a common metaphor, and because I of this second parellel which I want to comment on again:
Not everyone likes math. And many who have not studied anything beyond required math can probably care less about whether Infinity exists. And then there are those who have studied a bit more math and are fairly good at it but, like the engineers above, but for whatever reason can't decide if infinity is real or just some large finite number people are getting overly excited about and are thus pretty agonistic to the issue. Then there are those who really have studied a lot of math but because of the philosophical technicalities which I mentioned have instead opted for the extreme stance of ultrafinitism where even things like the existence of exponentials and natural numbers are thrown into question.
But there are many mathematicians, who have studied and have learned to love pure math. And to them, Infinity makes a lot of sense. Math seems so much more complete, without gapping holes if and only if Infinity is included. (Like where Jonathan points out you can have an unending sequence where you can just never reference the whole thing as a set if you don't want infinity to exist. Completely unsatisfying and unhelpful!!!!) They are they who see infinity for all it's beauty, wonder and yes... down right practicality too... (But more then just practicality mind you.) To them, infinity seems is a necessary piece for the whole complex system we call math to to be the full and majestic structure it so often hints at being at to those who love it.
Can such mathematicians prove their case? No, but they are not going to lose any sleep over it because they have one of the most intellectually satisfying structures to be explored at their fingertips and all the nay-sayers have no more that philosophical angst.
The Greek stuff is the axiom itself and means that a set exists with properties Jonathan describes above. Namely, I is the set. The funny looking zero is the empty set. x is an element in the set, the upside-down A means for every and the epsilon means element of and U mans union and the backwards E means there exists.
ReplyDeleteSo it reads, "There exists a set I such that the empty set is in I and for every x in I the element xU{x} is also in I." This creates the unending structure: {0} = 1, {0, {0}} = 2; 3 = {0,{0},{{0}},{0,{0}}}, etc... forever and I is a reference to this whole infinite set meaning assuming I exists assumes and infinite set exists.
Seems silly I know, but it turns out this is the most simple way to define infinity. All that needs to exist in this definition is the concept of a set, something called 0, and the ability to take any element, including 0, and make a new element by doing {x,{x}}. That is all that is required and so it is deemed simple.
cool metaphor you sly dog
ReplyDeleteJoseph:
ReplyDelete"And then there are those who have studied a bit more math and are fairly good at it but, like the engineers above, but for whatever reason can't decide if infinity is real or just some large finite number people are getting overly excited about and are thus pretty agonistic to the issue."
DISCLAIMER: I'm mostly arguing here from the "other side" or at least from a neutral standpoint. I've already said I mostly agree with you.
Hmm, I think this misses the point. You have set up a dichotomy that infinity is either real or not real. My point in the "engineer" comment is that it doesn't matter that much because we currently don't know (not that we shouldn't try to find out though). I'm not agnostic to the issue, I choose to use infinity when it suits my problem appropriately. When I'm implementing something in a computer, well, I don't (ever tried to integrate to infinity in a computer?). In other words, infinity is a tool in a toolbox. Believing in it has practical application at times, and at other times is impractical.
And ultimately, this is why I think the argument only kind of works (though as I said before, it works well enough for me). To the mathematicians, you argue, it "makes a lot of sense" (read: makes my life easier, but might be wrong). And the mathematician INFERS from his "makes sense" perspective that something exists in actuality. But, the reality is, as far as we can tell, we don't know if it exists. So why not acknowledge this? My argument isn't that people are unreasonable for believing in infinity, but that their certainty in their inference is unwarranted.
As an example, if you paraded around the physics world, declaring you "know" the Higgs Boson to truly exist, you would be laughed at. Rather, you provide a telling story, demonstrate why you are inclined to hypothesize its existence and you await increasing confirming evidence. At some point, perhaps the evidence is so overwhelming, everybody starts to agree that they "know" it's true. Even then, you know that logically you haven't proven anything, only that you haven't been disproven. The difference is subtle, but important especially in the scientific world as you know.
Besides that, as far as intellectually satisfying goes, I'm not so sure. Perfect circles don't seem to exist, nor perfect triangles, nor inertial frames. These are abstractions made to allow us to do things. And it may turn out that they exist in actuality. But until that is proven, why don't we treat them as abstractions?
Suppose that each mathematician had a different view of what infinity meant, what it could do, and how you had to use it? Would infinity still be as useful as you claim? The starting point for the utility of infinity at all, as an abstraction, is an agreement upon a definition of some kind (this is what gives math its power). Each group of mathematicians who use infinity, agree upon some baseline definition to satisfactorily prove their point du jour.
Finally, I claim that you would view infinity differently if you believed it was destroying people's lives. Even if it appeared to complete your abstraction of reality, I doubt you would advocate it as good and right if it was obviously doing wrong and bad.
At the end of the day, this comes back to a discussion you and I have had before. You infer from an abstraction of reality (physics), that the abstraction IS reality and we just don't know all the pieces yet. I claim the abstraction only MIGHT be true. I claim the abstraction is nothing more than a useful approximation for reality and that it will only represent true reality when it can PERFECTLY predict it, model it, and describe it. And I think I'm right because my job exists. Until physics can solve engineering problems I claim it is ONLY an abstraction that may or may not be true.
Fun topic though.
i am enjoying reading this discussion though as a layperson i have to admit some of it is a little above me... but i like what i understand about it.. interesting stuff... thanks...
ReplyDeleteAnd thank your for your compliment and always being very supportive. I will try harder to explain things in more user friendly ways.
ReplyDeleteactually i google terms i don't know and often am referred to wikipedia or other sources that explain them more simply... you guys are having fun, keep at it.. i just enjoy "listening in"... i like being stimulated to think further about things of this nature and learn more...
ReplyDeleteHere is how I was introduced to infinity. Take a bucket holding infinite marbles (somehow!) and then take one marble put it in a new bucket marked 1, then take two marbles and put in a bucket marked 2, and so on... Repeat this operation for infinite times and now you an infinite series of buckets containing infinite marbles each, but the bucket 2 contains an infinity that is twice as big as bucket one. No greeks here!
ReplyDeleteI think my favorite thing along these lines is the Hilbert Hotel. Or the fact that you can show there are as many integers as rational numbers.
ReplyDelete