In the comments section of Bob Murphy’s blog, I was asked (in effect) why I insist on the objective reality of the natural numbers (that is, the counting numbers 0,1,2,3…) but not of, say, the real numbers (that is, the numbers we use to represent lengths — and that are themselves represented by possibly infinite decimal expansions).
There seem to be two kinds of people in the world: Those with enough techncal backgroud that they already know the answer, and those with less technical background, who have no hope — at least without a lot of work — of grasping the answer. I’m going to attempt to bridge that gap here. That means I’m going to throw a certain amount of precision to the winds, in hopes of being comprehensible to a wider audience.
As you might have heard, the set of real numbers is larger than the set of natural numbers, in the sense that there is no one-to-one correspondence between the two sets, and any attempt to construct such a correspondence will leave out some (in fact infinitely many) of the real numbers. We express this by saying that the reals are uncountable.
Now let’s be fanciful. Suppose the Devil comes along and prunes our mathematical universe. He throws away a whole lot of numbers, a whole lot of sets, and a whole lot of functions. When I say he “throws these away”, I mean he somehow arranges that these things will be erased from our mathematics; they will play no role in any calculations or proofs, and, though we will go on merrily doing mathematics as always, we will never notice they are missing.
In fact, he throws away so many real numbers that the remaining ones — the ones we have to work with — form a countable set.
The Devil’s problem is that we might notice that the reals are coutable, say by discovering an explicit one-to-one correspondence with the natural numbers. He solves this problem by insuring that, while he’s busy throwing away all those real numbers, he also throws away all the one-to-one correspondences between the natural numbers and the remaining reals. Those one-to-one correspondences exist, but they’ve been removed from our mathematics so we can’t discover them.
Then we’ll go on merrily believing that the reals are uncountable (in fact, we’ll still have our perfectly good proof of that fact) and there’s a sense in which we will be right. Indeed, when we say “the reals are uncountable”, what that means is: “There is no one-to-one correspondence between the natural numbers and the reals”. And in fact there is no such one-to-one correspondence. There used to be, but the Devil destroyed it.
But although we believe correctly that our set of reals is uncountable, the Devil believes correctly that it’s countable — because he kept copies of all those one-to-one correspondences he threw away. And though we believe correctly that our “set of all real numbers” is the set of all real numbers, the Devil believes correctly otherwise — because he kept copies of all those real numbers that he erased from our Universe.
Now comes the theorem: No matter what mathematical Universe you live in — that is, no matter what sets, functions, numbers, etc. you have access to — there is always a larger mathematical Universe, with more real numbers, and more one-to-one correspondences, whose denizens will believe correctly that our Universe was constructed by a Devil who started with the “true” real numbers, threw most of them away, left us only a countable set of them, and threw away enough one-to-one correspondences to keep us oblivious to that fact. (Of course, the Devil must worry that we’ll somehow discover that those one-to-one correspondences are missing, which means he’ll have to throw away a bunch of other stuff too, to prevent that discovery. And so on. But it turns out that he can always complete this herculean task.)
As we pass to larger and larger mathematical Universes, the set of natural numbers remains the same, while the set of real numbers keeps growing. No matter what Universe you inhabit, the denizens of the “next Universe up” will always agree that you’ve got the natural numbers exactly right, but they’ll sneer at your set of real numbers, which looks to them like a product of the Devil’s work — a mere countable set that doesn’t begin to account for all the “true” reals. Of course, there’s always another Universe where people are saying exactly the same things about them.
In that sense, there is only one true set of natural numbers, but there is no such thing as the “one true set” of real numbers. That makes it easy for me to believe that the natural numbers have a more solid kind of existience than those slippery reals.
That’s the main post. Now let me say a few things by way of partial penance for my (intentional) misprecision. What follows will be (slightly) less imprecise, but if you want true precision you should of course turn to the textbooks. (Try scouring the indexes for terms like forcing and Levy collapse.)
The standard axioms for arithmetic have many models — that is, there are many “number systems” that satisfy those axioms. The standard axioms for set theory also have many models — that is, there are many “mathematical Universes” that satisfy those axioms. The usual tools of first-order logic don’t allow us to distinguish among these models. On the other hand, almost all mathematicians believe that among the many models for arithmetic, there is exactly one, the so-called “standard model” that we’re actually talking about when we talk about arithmetic. (You can use the tools of second-order logic to specify this model uniquely, but I’d argue that that’s cheating. But while we might disagree about how to specify it, we pretty much all agree that the standard model exists.) On the other hand, when it comes to set theory, there is no clear way to point to one of the many models and say “that’s the one I’m talking about”. They all — or at least many of them — seem equally good.
In that sense, when we do arithemetic, we know exactly what we’re talking about. We’re talking about the good old standard natural numbers, and we pretty much all agree on exactly what those are (though we might disagree on how we know what those are). But when we do set theory, we’re in far murkier territory. I might be talking about one Universe, you might be talking about another, and we’d never know it. I’d say “the reals are uncountable” and you’d say “I agree”. But for all we know, we’re talking about different sets of reals, and either of us, if we knew what set the other was talking about, might say “but those aren’t the reals — and dammit, your set is countable”.
If you’ve got a solid advanced undergraduate math background, you might object that the reals are constructed from the naturals. We start with the naturals, form quotients, to get the rational numbers, then proceed to the reals by a process that involves taking limits (or something of the sort). Therefore, you might say, if I know what the naturals are, then I know what the reals are a fortiori. But the reals you construct depend not just on the rational numbers you’ve got available (which are the same in every Universe); they depend also on the sets of rationals you’ve got available — and the available sets differ from one Universe to another. No matter what Universe you live in, your construction of the reals at some point invokes the notion of “all” subsets of the rationals. And somewhere, in a higher Universe, the Devil is chortling — because he knows, and you don’t, that you’re missing a whole lot of subsets, and hence missing a whole lot of reals. But he too is missing a whole lot of reals, according to the next higher Devil.
I’m tempted to sum this up by saying that the natural numbers are real, but the real numbers are imaginary. Or, as Kronecker put it, “God created the natural numbers; all else is the work of Man”. (Note to Bob Murphy: My endorsement of Kronecker is contingent on a sufficiently metaphorical interpretation of the word “God”.)
I don’t get it. You have posted many times about how the Peano axioms can have many models, besides the usual natural numbers. Now you are saying that axiomatizations of the real numbers also have multiple models. So why are the naturals more objective? It seems to me that your arguments just shows that both have the same paradoxes.
“I’m tempted to sum this up by saying that the natural numbers are real, but the real numbers are imaginary.” That’s too complex.
@Roger: Steve will say that N is the same but that the subsets of N and Q you talk about differ.
Steve is saying mathematical statements are like giant multi-lingual puns. They have subtly different meanings in each language. A language is determined by the set theory it uses. So Steve only has to make one ‘choice’ in getting N, and he appeals to intuition here. But he says there is no such simple ‘obvious’ choice for R or C.
@Steve: In non-standard analysis Robinson style you have non-standard integers. They are ‘infinite’.
Roger: It seems to me that your arguments just shows that both have the same paradoxes.
The difference is that, before we ever start making advanced mathematical constructions, we know what the natural numbers are.
The real numbers depend on advanced constructions, and therefore have some ambiguity to them. If the natural numbers depended on similar constructions, they’d have the same ambiguity. But for most of us, the standard natural numbers *precede* set theory.
So roughly: At Stage One we have the standard natural numbers. At Stage Two, we have formal logic, which depends on some aspects of the natural numbers (like being able to count). At Stage Three, we have models of our logical systems, which include non-standard models of the natural numbers, along with multiple models of set theory. There is no Stage at which there is a “standard” version of the real numbers.
Ken B: Your reply to Roger is exactly right except for this:
@Steve: In non-standard analysis Robinson style you have non-standard integers. They are ‘infinite’.
No. From outside the model they might “look” infinite, but within the model all integers are finite, even the non-standard ones.
I count; therefore … Sum!
J Storrs Hall:
That’s brilliant.
@Steve: I meant that if S is a non standard integer S > x for all standard integers x (speaking within the model).
So infiite in the sense of the ordering <. I concede there is some ambiguity here about infinite.
I’d say your intuitive model of N omits these beasts.
Yes, from within the model, the infinite integers look finite, but the same could be said of the reals. Within the model, the reals look like the usual reals. So what’s the difference?
Steve, your stage 1-2-3 philosophy seems to repudiate everything you’ve said about the Peano axioms. What stage are they? Why bother with them?
Roger: The Peano axioms are Stage Two.
Mathematics starts with (at least some fragment of) arithmetic. From arithmetic we build up logical systems, including the theory of Peano arithmetic and Zermelo Fraenkel set theory. Those logical systems have models. In the case of the Peano axioms, one of those models is the arithmetic we started with. There are also other models.
We know what arithmetic is because we started with it. We don’t know what the real numbers are, because “sets of real numbers” popped up in multiple models, and there’s no satisfying way to pick one of them out as the “true” set of real numbers.
I have no idea what you think this conflicts with.
Prof. Landsburg, this is an interesting post. It sort of reminds me of Euclid’s three definitions of parallelism; they’re all true, depending on the assumptions you make about the universe.
I think that’s one of the things I’ve always loved about Set Theory and Vector Spaces – you can pretty much construct an alternative universe that is absolutely real and concrete within the confines of the characteristics you’ve given it.
Now you add an element of depth: No matter which universe I like to pretend I’m in, the Natural Numbers are always the same. That is a powerful revelation indeed.
However, I’m still scratching my head over the “existence” of Natural Numbers. Do concepts ever “exist?” Is logic a “thing?” You’ve given me a lot to think about here, and it’s wonderful. Thanks for that!
@RPLong:”However, I’m still scratching my head over the “existence” of Natural Numbers. ” As well you should. Steve’s argument turns on this and I think he’s way off base. Saying something ‘exists’ mathematically is not the same as saying it exists.
“it’s wonderful. Thanks for that!” Don’t feed the beast! :)
An interesting bit of history. The Greeks thought of ‘length’ numbers as quite different things from ‘counting’ numbers. The idea of unifying length numbers and counting numbers with ‘real’ numbers comes much, much later. I wonder if an ancient Greek mathematician would say we just have an intuition about length …
My logic sucks so apologies for anything idiotic I say here.
Steve I think Roger’s question goes something like. Your objection to the existence of the reals is that when we talk about them it’s ambigous what we mean. Hang on when we talk about naturals it’s we’re talking about things satisfying the peano axioms. But there are other models that satisfy the peano axioms. Why do the naturals exist if we could choose 2 different models satisfying the peano axioms talk about them “agree” on he truth of alot of statements and be talking about different things the whole time.
Professor Landsburg, I hope you’re preparing a book on this topic.
Jonathan Kariv:
when we talk about naturals it’s we’re talking about things satisfying the peano axioms.
The naturals do satisfy the peano axioms, but you don’t need to know about the peano axioms to know about, and talk about, the natural numbers. Nobody had any trouble knowing what the natural numbers were for thousands of years before peano was born.
“I insist on the objective reality of the natural numbers (that is, the counting numbers 0,1,2,3…)”
I think you are on philosophically shaky ground when you insist on the “objective reality,” whatever that means, of an infinite set. As you clearly know, the moment you grant me the infinite set of counting numbers, and the standard (ZF) rules of set manipulation, I can immediately conjure up the reals. Of course you made the same point.
You also acknowledged that there exist nonstandard models of the Peano axioms.
So I don’t think you’ve solved this problem. What you’ve done is to take a deep philosophical mystery: that “everybody knows” what the counting numbers are, yet we can not axiomatize them in such a way as to preclude nonstandard models; and answer thusly: “Oh yeah? Well there IS TOO only one objective set of counting numbers no matter WHAT your silly logic says.”
I am afraid that I can’t distinguish your post from the usual anti-Cantor crankery common on the Internet, other than that you clearly know what you’re talking about. But in that case, why make such an unsupportable statement?
The “objective reality” of the infinite set of natural numbers? What on earth can that possibly mean? The infinite set of natural numbers is a fictitious mental construct, like a novel or a video game; that gives us intellectual pleasure and perhaps insight.
Objective reality is a strong phrase and it most definitely does not apply to the set of natural numbers. But if it does … then you have to grant the same ontological status to the reals, as you yourself pointed out.
Steve, you draw a big distinction between N and R, but the difference is not really based on properties of naturals or reals. The difference is that you have to chosen to take a non-axiomatic approach to N, and an axiomatic approach to R. You could have taken an axiomatic approach to both, or neither.
This is interesting but it has been awhile since I thought about these things so I have two questions. I will intentionally avoid terms like “exist” and “objectively real”, etc. and hopefully will not distort the spirit of the discussion too much.
1. Do you feel that the standard model of arithmetic is “given to our direction intuition” more so than the continuum? If so, why? And does this matter for your point?
2. Suppose I, in a fit of religious dogma, point at the cumulative hierarchy as the “true” class of sets. Does this declaration affect your reasoning?
Steve, as others have said it seems to be an equally respectable position to have a preformal conception of the real numbers. Just as our intuition concerning the natural number is connected to the question “how many?”, we also have quite a bit of intuition concerning the question “how big?”, and I invite you to consider an alternate history in which quantitative reasoning emerged from the continuous quantities of space rather than the discrete quantities of counting. Number, then, would be understood as magnitude rather than multitude. Here is how I think things might play out:
1. Attempts to make our preformal intuition rigorous would perhaps start with the informal axioms of Euclid and then proceed to the formal first-order theory of real closed fields. Everyone would agree that they had the same standard model in mind for this theory, even if they couldn’t specify it precisely. (Attempts to make the theory categorical would result in the second-order theory of complete ordered fields, but people like you would say that that’s cheating, because the unique model for this second-order theory depends on the background set theory, just like the unique model of second-order arithmetic depends on the background set theory).
2. What about the natural numbers? Well, they require work. When people would face the challenge of counting, they would fall back on the system of real numbers they had already developed. They would label the first object with the real number 1, the second one with 1+1, the third with 1+1+1, and so on. But how do you make the “and so on” precise? Well, you could attach the Peano axioms to the first-order theory of real closed fields, but then you’d get nonstandard models with infinitely large natural numbers. (We would know that they’re infinitely large, but to the denizens of the model they wouldn’t know that the devil had added some extra elements to the natural numbers but had changed the definition of “finite” so that natural numbers that were “really” infinite seemed finite to them.)
3. Finally, mathematicians would settle on an advanced construction which gives us a seemingly unique set of natural numbers. We define a set X of real numbers to be “hereditary” if x+1 is an element of X whenever x is an element of X. We then define the set of natural numbers N to be the intersection of all hereditary sets containing 1 (or 0, depending on where you want to start the natural numbers). So the real numbers emerge from a preformal intuition, and the natural numbers emerge as a complicated set-theoretic construction. And of course, since there are a variety of set theoretic models to choose from, people may disagree on the question of what sets of real numbers there ARE, so they may have different conceptions of the natural number system.
I look forward to hearing your thoughts on this. How does the approach I lay out differ from your viewpoint?
I’m with @Keshav here. If the majority of mathematicians think they “know” which N is the “real” one, I suspect they’re mistaken. After all, there’s nothing within any formal system of N that can be used to distinguish the “true” N from the “non-standard” N that arise from applying Godel’s construction to it and adding unproveable statements as axioms.
Mathematician’s intuition about N is shaped by whatever formal system they work within, so even if it’s working well, it can’t single out the “true” N. And I don’t see any reason to assume it would always work well. Why promote intuition to the position of “Arbiter of mathematical truth”?
Perhaps all these non-stadard N are just as “true” as the “standard” N (whichever one that is), and the term “non-standard” is just as slanderous as terms like “complex”, “imaginary”, “irrational” and “negative”?
Dear Steve,
That’s a wonderful post! I never imagined I would see the essence of Skolem’s paradox and the absoluteness of the set of natural numbers discussed in a popular blog! Along with other gems like Tennenbaum’s Theorem.
The Big Questions has to be the only blog in the Universe for libertarian logicians, I suppose! May you entertain and educate us for long.
Ok. So I thought integers were infinite. When I tell my son that infinity plys one is infinity, am I wrong?
Robert:
You’re basically wrong, with a hint of right. You’re wrong because ‘plus’, in this context, is an operation that we perform on two natural numbers. Infinity is not a natural number. However, if you take the union of a countably infinite set A and a set B containing only one element, the ‘number’ of elements in this new set A union B will be equal to the ‘number’ of elements in A, which is basically what you mean when you tell your son that infinity plus 1 is infinity.
@Robert, as Colin says, infinity is not a “natural number”. However, as he hints, we can define number systems that include infinity. A number is the size of a set. Infinity is the size of a set that is not finite. Note, importantly, that not all infinite sets have the same size – so there’s more than one infinite number*.
The smallest infinity is called aleph0. The next smallest is called aleph1, and the next smallest is aleph2, and so on.
In the standard version of this system (called Zermelo-Fraenkel Set theory),
~ aleph0 + 1 = aleph0
~ aleph0 + 1000 = aleph0
~ aleph0 + aleph0 = aleph0
~ aleph0 x aleph0 = aleph0, but
~ in fact, the sum or product of two infinite numbers is just the larger of them.
If that doesn’t sound interesting, the next fact and the footnote make up for it.
~ 2^aleph0 > aleph0, ther’s no way to work out exactly what it is. Whether 2^aleph0 = aleph1 or not is an undecideable proposition in ZF. There can be no proof either way.
Footnote :
* there are, in fact, an infinitely many infinite numbers. Exactly how many there are is an interesting question. It turns out that the “number” of infinite numbers is bigger than any particular infinite number. This sounds like a contradiction, but for technical reasons, isn’t.
@Mike,
When you refer to 2^aleph0, you’re referring to the cardinality of the power set of the natural numbers, yes? What do you mean when you say there’s no way to work out what it is? It is equal to the cardinality of the rean numbers, no?
er, real numbers
I know what natural numbers are. I know what rational numbers are. I know what the Dedekind cut is. Ergo, I know what real numbers are. Where am I wrong?
MR: Dedekind cuts require you to consider “all” subsets of the rational numbers of a certain sort. This in turn pretty much requires you to know what are “all” the subsets of the rational numbers. That turns out to be a much more slippery concept than it at first appears.
This whole argument becomes a lot more believable and intuitive if instead of the natural numbers vs the real numbers it’s framed as the computable numbers vs the real numbers.
Most must agree on the objective reality of computable numbers, which are indeed countable. The set difference between the computables and the reals are the uncomputable numbers. It seems very strange for uncomputable numbers to have an objective reality.
@Colin, yes, 2^|N|=|R|. Let’s say |R|=c. What I mean is, there’s no way to work out which of aleph1, aleph2, aleph3, … (if any) this c is. ZF leaves that undecided and undecideable.
Here’s an example which I hope will clarify things for a few readers here. A concrete way to think of some of this ferocious abstractness.
The ‘usual’ mental model for the natural numbers, N, is an endless sequences of steps
1,2,3,…
The … means and so on forever.
But this isn’t the only easily visualized one. Call it N.
Consider first the usual image of all the integers …, -3, -2, -1, 0, 1, 2, 3, … with … on both ends. Call it Z.
Now mentally form a new thing consisting of N, in black, followed by Z in blue:
1,2,3,4, … , …, -3, -2, -1, 0, 1, 2, 3, …
That comma between the first and second … indicates that the second one comes *after* the first. We have counted past infinity.
Blue 4 and black 4 are different numbers.
Now, as long as you don’t talk about sets, this provides a workable mental model of ‘non standard intgers’. In particular it models PA.
Note two things. First it isn’t the model Steve asserts is the right, true one. And it’s not too difficult to picture.
You need to be a little careful using this mental model but it does convey the basic idea.
For geeks here is a bit more detail on formalities.
Add a predicate St(x) for “x is standard” to the lan Peano
Arithmetic to get NPA.
* Usual axioms for =
* Usual axioms for successor, addition, and multiplication.
* The successor of every standard integer is standard.
* Induction for all formulas with respect to the Standard integers only.
* The “nothing special” axiom: x1,…,xk standard & (therexists y)(A(x1,…,xk,y), implies (therexists
standard y)(A(x1,…,xk,y)).
* There exists a nonstandard integer.
Then you can show that any theorem without St that is provable in NPA is provable in PA.
Keshav Srinivasan: That is an extremely insightful comment (one of many). Thanks. Do you do any writing anywhere else that I can also read?
As Steve’s post demonstrates, it’s a little hard to be a hard line Platonist about the real numbers. There’s just more to them than meets the eye. But as several of us have argued, the same is true about intgers. Steve talked in an earlier post about bizarre Tannebaum models. But the one I present above is pretty intuitive.
It’s perhaps more intuitive than you might think. If you use infinitesimals in calculus, which is par for the course in engineering and physics, then you are essentially committed to a model of natural numbers like the one I gave and NOT Steve’s alleged ‘one true vine’. Otherwise you get contradictions.
So which view of the integers is ‘intuitively obvious’ is perhaps dependent on history. Had model theory developed in 1800 we might all use non standard analysis and find the NPA model intuitive.
Aside from that, as I remarked in passing, and Keshav argued in some detail, length also provides a natural intuitive idea of number, and had history been otherwise our ideas of what’s ‘intuitivel obvious’ might be different.
Let me add one final observation. Steve’s ‘nature is math’ theory suffers from a multiplicity problem. There are just so many things that are independent of any axioms we use that you get multiple universes from his argument. What Steve shows in this post is that this multiplicity enters early, even before he gets R and C squared away. So the multiplicity is even more serious than it first looks. To me that’s an argument *against* Steve’s platonism-run-rampant. But I admire the hutzpah in presenting that as an argument for his side!
Per @SteveL’s comment, how is infinity more real than real numbers? Natural numbers have application in our universe. If I take a snapshot of the universe at some point in time, (very) theoretically I could count the particles (bosons and fermions) in the universe. The result would be a very large number, but not infinite. So, is infinity anything other than a useful mathematical construct?
Similary, are there “real world” examples of real numbers? Even if I draw a very large and precise circle, my measurement of pi cannot exactly match pi, but would only be a rational approximation. I am hard pressed to think of where we could find a “real” real number.
Al V., there is no consensus that there are only a finite number of bosons in the universe. And a measurement of pi might not match pi, but it would still be a measurement of a real number.
@Al V: to second Roger, I don’t see why your measurement would necessarily be a rational.
If I measure pi by dividing the measured circumference of a circle by the measured diameter, the result is a rational. Let’s say I draw a circle exactly 1000 meters across (as accurately as I can determine), and then measure the circumference as 3141.593 meters, measuring to the nearest millimeter. The measurement of pi is 3141593 / 1000000, which is a rational. You can never determine a real number by measurement, only by inference, or via a series.
@Al V
This is because your measurements are rounded off to fractions, which happens because we use the natural numbers to define the reals. This may be a mere cultural thing or may be ingrained into our DNA, but it is a human thing, not a fundamental feature of measurement.
As pointed out earlier, we can, theoretically, use the reals as a basis, and define the natural numbers in terms of them. Therefore it’s imaginable that some species somewhere would (through culture or natural inclination) actually do math that way. For them, prime numbers, far from being (as imagined in “Contact”) a sure way to begin communicating with aliens (like us), might be some esoteric concept taught only in advanced mathematics courses at Sqrx University.
The inhabitants of Sqrx might even now be beaming signal to us based on “obviously foundational” mathematics such as chaotic solutions to Schroedinger’s equation, and we’re missing their transmissions completely because we insist on thinking in terms of discretized countable lumps.
@Mike H ‘The inhabitants of Sqrx might even now be beaming signal to us based on “obviously foundational” mathematics such as chaotic solutions to Schroedinger’s equation, and we’re missing their transmissions completely because we insist on thinking in terms of discretized countable lumps.’
How would they be beaming that signal exactly?
Doug’s perceptive to say that at least the computable numbers are as objective as the natural ones. We can’t get rid of the square root of 2 or pi (Is pi computable? my definition maybe wrong). “e” also is a consequence of everyday geometry. Leaving them out is like leaving out “4”. It’s just that we don’t need the great majority of irrational numbers that are defined only to close in gaps.
Is the concept of mathematics intelligible without the canvas to host these reifications? By ‘canvas’ I mean, of course, the universe. We know that time is implicit in the computability of space’s horizons. In other words, time isn’t part of an independent platonic ‘timeless’ world, it is woven into the physics of spacetime, and I think that might naturally apply to numbers too. Nature and mind seem to have a hand in glove relationship.