Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics » Logic

The Big Surprise

Steve Landsburg — Thu, 15 Dec 2011 07:01:29 +0000

Back in the 1930’s, Kurt Godel proved two amazing facts about arithmetic: First, there are true statements in arithmetic that can’t be proven. Second, the consistency of arithmetic can’t be proven (at least not without recourse to logical methods that are on shakier ground than arithmetic itself).

Yesterday, I showed you Gregory Chaitin’s remarkably simple proof, of Godel’s first theorem. Today, I’ll show you Shira Kritchman and Ron Raz’s remarkably simple (and very recent) proof of Godel’s second theorem. If you work through this argument, you will, I think, have no trouble seeing how it was inspired by the paradox of the surprise examination.

Start with this list of statements:

It takes more than 10000 characters to specify the number 1.
It takes more than 10000 characters to specify the number 2.
It takes more than 10000 characters to specify the number 3.

and so forth.

Obviously, statements 1, 2 and 3 are all false, since it only takes a single character (namely “1″) to specify the number 1, and a different single character (namely “2″) to specify the number 2. But if you continue this list long enough, you’ll eventually get to some true statements. Yesterday we saw a proof that none of those true statements is provable.

But suppose that, undeterred by the proof, we are determined to identify a specific true statement on the list. Here’s our strategy:

First we write down the first gazillion statements on the list, where a gazillion is some number so big that we’re sure to have included some truths.
Next we go down the list, eliminating false statements. Notice that if we’re willing to work long enough, every false statement eventually gets eliminated — we just keep trying shorter-than-10000-character prescriptions and crossing off the numbers they describe. This leaves us a list of candidates.
Suppose there’s just one true statement on the list. Then eventually we cross all the others off, and learn that the one remaining candidate is true (because the list was so long there had to be at least one true statement). In fact, we’ve just proved (by process of elimination) that this statement is true. But we learned yesterday that no such proof is possible. Conclusion: There can’t be just one true statement on this list.
Suppose there are just two true statements on the list. Then eventually we’ll cross all the others off, leaving two candidates, at least one of which is true. But we’ve already proved that there can’t be just one true statement, so these must both be true. In fact, we’ve just proved they’re true. Which is impossible. Conclusion: There can’t be just two true statements on the list.
Do you see where this is going? There also can’t be just three true statements on the list, or four, or five, or any other number. Yet we know there are true statements on the list! Something’s wrong here!!!!!

Okay, what went wrong? Answer:

Our argument relies on the fact that every statement on our list must be unprovable. Why should we believe that? Well, we proved it yesterday; that’s why!
Actually, our argument relies on a bit more — it relies not just on the fact that every statement on our list is unprovable, but that we can prove their unprovability. But again, we did that yesterday.
The only way out of this is to conclude that there’s some gap in yesterday’s proof.
But there’s only one thing we used yesterday without proving it: We began from the assumption that our arithmetical reasoning is consistent.
So that must be where the gap is — and it must be impossible to fill that gap! In other words, it must be impossible to prove that arithmetic is consistent.
Tada!

There’s just one other way out: If arithmetic actually is inconsistent, then all bets are off and we can prove its consistency — but in that case, of course, our conclusion will be wrong. In any event, there are very few people who think this is a contingency worth worrying about.

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Berry Interesting

Steve Landsburg — Wed, 14 Dec 2011 07:01:46 +0000

Today, I’m going to give you a short, simple proof of Godel’s First Incompleteness Theorem — the one that says there are true statements in arithmetic that can’t be proven. The proof is due to Gregory Chaitin, and it is far far simpler than Godel’s original proof. A bright high-schooler can grasp it instantly. And it’s wonderfully concrete. At the end, we’ll have an infinite list of statements, all easy to understand, and none of them provable — but almost all of them true (though we won’t know which ones).

Chaitin’s proof, like Godel’s, is inspired by a classical paradox — in this case, Berry’s Paradox as opposed to the Liar Paradox (both of which I described yesterday).

We start by observing that some numbers are more complicated than others. The 10,000 digit number that starts off 10101010101010101….. and continues the same way is somehow less complicated than a random 10,000 digit number.

Kolmogorov formalized this notion by defining the “complexity” of a number as the length of the shortest prescription (in some fixed language) for writing it down. The number above has a 48-character prescription: “Write down a 1, then a 0, then repeat 5000 times”. That makes it pretty simple.

Now suppose we want to find a more complicated number — one that requires at least, say, 60 characters to prescribe. We know there must be many such numbers, but we’d like to find a specific example — together with a proof that our example can’t be prescribed in less than 60 characters. So we write a computer program, called Finder60, which searches for examples-with-proofs. Ideally, the program outputs something like: “I have found a proof that the number 2834932709472398472328923478902342903848927189374901742309842398742 cannot be prescribed with fewer than 60 characters.”. Finder60 searches systematically, so if there’s such a number/proof combination to be found, Finder60 will find it. Otherwise, it keeps on running forever.

We can also, of course, write programs called Finder90, Finder120, Finder10000 and so on, to find ever-more-complicated numbers together with proofs that they are complicated.

Once you’ve written the code for Finder60, you only have to tweak it slightly to get the code for Finder10000 — and the code gets only slightly longer in the process. Basically, you just change all the 60’s to 10000’s, replacing two-digit strings with five-digit strings. Not much difference.

So if the code for Finder60 is, say, 5000 characters long, then the code for Finder10000 is just a bit more than that — say 5200 characters.

Now suppose M is any number that provably requires more than 10000 characters in its prescription. Then Finder10000 will find it and print it out. Which means we have the following less-than-10000-character prescription for M:

Run the following program: [insert the 5200-character code for Finder10000 here]

Uh oh! If prescrbing M provably requires more than 10000 characters, then M can be prescribed in fewer than 10000 characters. Contradiction!

Conclusion: There must be no such M. That is, no number can be proved to require more than 10000 characters in its prescription.

But surely some numbers do require more than 10000 characters; after all there are only finitely many ways to prescribe a number with fewer than 10000 characters, which leaves infinitely many numbers left over.

So what’s a true statement that can’t be proven? Answer: Any true statement of the form “The number M cannot be prescribed in fewer than 10000 characters”. No matter what M you plug in , this statement must be unprovable. If you plug in M=1 or 2, you’ll get a false statement. But for most values of M (though we don’t know which values!) you’ll get a statement that’s true, but unprovable.

Isn’t that about the coolest trick ever? Well, it turns out that you can add another twist to make it even cooler. That’s where the surprise examination paradox comes in. I’ll explain it all one day later this week, though you won’t know which day till you log in that morning.

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A Tale of Three Paradoxes

Steve Landsburg — Tue, 13 Dec 2011 07:00:09 +0000

This is a tale of three paradoxes and why they matter.

First, the ancient Liar Paradox: “This sentence is false”. If this sentence is true, it must be false. If it’s false, it must be true.
Next, the century-old Berry Paradox: Call a phrase “short” if it contains fewer than 13 words. The English language contains a finite number of words, and hence a finite number of short phrases. Hence there must be some natural numbers that can’t be described by any short phrase. Among these natural numbers, there must be a smallest. What is that natural number? Why, it’s the smallest natural number that can’t be described by any short phrase, of course. Except that this number is in fact described by the short phrase in boldface.
Finally, the more modern Paradox of the Surprise Examination (or the Unexpected Hanging), which we discussed yesterday.

The paradoxes are slippery, because they are stated in the imprecise language of English. But each of them has inspired a precise mathematical counterpart that is central to a brilliant argument in mathematical logic.

Start with the liar: “This sentence is false” can’t be true, or it would be false — and can’t be false, or it would be true. This tells us that there’s such a thing as an English sentence that’s neither true nor false, which comes at first as a considerable surprise, but isn’t devastating.

One of Kurt Godel’s great insights was that you can go a lot deeper by considering a slightly different sentence: “This sentence is not provable”. If that statement is false, then it’s provable. But surely no false statement should be provable! So maybe the statement is true. In that case, it’s true but not provable, which says something about the limits of logic. It says that not every true statement can be proved.

At one level, this is still just wordplay. What makes it profound is Godel’s discovery of a code that converts certain English sentences into statements of pure arithmetic (that is, statements of the form “Every number is the sum of four squares” or “Every prime number is divisible by 2″) in such a way that true statements are matched with true statements and false statements are matched with false statements. The code is cleverly constructed so that there’s a statement in pure arithmetic (say, for illustration, that it’s the statement “every even number is the sum of two primes”) that corresponds to the English sentence “The statement that every even number is the sum of two primes cannot be proven.” These statements are either both false, in which case it’s possible to prove a false statement, which we believe (and hope to God!) is not the case — or they’re both true, in which case we’ve found a true statement in pure arithmetic that can’t be proven.

Much brilliant work goes into constructing the code, but the brilliant idea is to adapt the Liar Paradox to a context where you can’t just say “Well, I suppose it’s neither true nor false” — because statements like “Every even number is the sum of two primes” must be either true or false.

(The above intentionally sacrifices a little precision in the interest of readability; the linked post is more carefully worded.)

So that’s why mathematicians care about the Liar Paradox. More on the Berry Paradox and the Surprise Exam (and how all three tie together) as the week goes on.

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The Surprise Exam, and More Surprises

Steve Landsburg — Mon, 12 Dec 2011 06:01:13 +0000

If you’re the sort of person who reads this blog, you’re likely to be familiar with the paradox of the unexpected hanging, which has been floating around since 1943 but achieved popular notoriety around 1969 through the writing of Martin Gardner. But you’re less likely to be aware that the unexpected hanging plays a central role in a wonderful new piece of serious mathematics related to algorithmic complexity, Godel’s theorems, and the gap between truth and provability.

The unexpected hanging might as well be a surprise examination, and that’s the form in which I present this paradox to my students every year: In a class that meets every weekday morning, the professor announces that there will be an exam one day next week, but that students won’t know exactly which day until the exams are handed out.

The students, of course, immediately start trying to guess the day of the exam. One student (call him Bob) observes that the quiz can’t be on Friday — because if it is, the students will know that by Thursday afternoon. After all, if Monday, Tuesday, Wednesday and Thursday mornings have all passed by, only Friday remains. A Friday exam can’t be a surprise exam.

A more thoughtful student (call her Carol) observes that this means the quiz must be on one of Monday, Tuesday, Wednesday or Thursday — and that if it’s on Thursday, they’ll know that by Wednesday night. After all, Friday’s ruled out, so if Monday, Tuesday and Wednesday have passed by, then only Thursday remains. That rules out a surprise exam on Thursday.

Another student (call him Ted) observes that thanks to Bob and Carol, we know the exam must be on one of the first three days of the week — which means that if it’s not on Monday or Tuesday, it must be on Wednesday. Therefore if it’s on Wednesday, they’ll know this by Tuesday night. Scratch Wednesday from the list of possibilities.

Now Ted’s girlfriend Alice points out that the exam can’t be on Tuesday either. Whereupon Bob concludes that the exam must be on Monday. But wait a minute! Carol points out that if they know the exam will be on Monday, it can’t be a surprise. Therefore no surprise exam is possible.

The students, relieved, decide not to study. But they’re awfully surprised when they show up in class the following Tuesday and the professor hands out an exam.

Where did the students go wrong? There is no consensus among the many philosophers and logicians who have considered this problem. The great Willard Van Orman Quine believed that Bob went wrong at the very beginning when he ruled out Friday. (According to Quine, Bob’s argument fails to distinguish between a proof that the exam can’t be on Friday and a proof that the students will know that the exam can’t be on Friday.) Other deep thinkers have accepted Bob’s argument (agreeing that the exam can’t be on Friday) but refused to accept Carol’s (thus refusing to rule out Thursday). You can, if you wish, read a pretty comprehensive survey of this literature here. But even among those who think Bob (or Carol) is mistaken, there is little agreement about exactly why they are mistaken.

Now, I happen to think the surprise examination paradox is pretty interesting as a pure intellectual exercise. But it’s also got important applications. I use it in the classroom to illuminate our discussion of the underlying “backward induction” technique, which economists (and especially game theorists) use all the time in serious arguments. Much more recently, the surprise examination has been used to illuminate some key concepts in mathematical logic, which I alluded to back in the first paragraph of this post. That’s the coolest part of all, and I’ll tell you all about it later in the week.

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Big News

Steve Landsburg — Tue, 04 Oct 2011 06:01:18 +0000

Last week, the highly distinguished Princeton Professor Ed Nelson announced a proof that the Peano axioms for arithmetic are inconsistent — and hence so is arithmetic itself. If true, this would be much bigger news than faster-than-light neutrinos. It would be bigger news than a discovery that the South had won the American Civil War. It would be far, far bigger news than a discovery that all life on Earth was intelligently designed.

There are, after all, multiple proofs that Peano Arithmetic (that is, the fragment of arithmetic described by the Peano axioms) is consistent. Among those, the simplest and most convincing (to the overwhelming majority of mathematicians) is this: The axioms of Peano Arithmetic, and therefore the theorems of Peano Arithmetic, are all true statements about the natural numbers — and a set of true statements cannot contradict itself.

Ed Nelson rejects that argument because (exempting himself from that overwhelming majority) he doesn’t believe in the set of natural numbers — or perhaps even in individual numbers when those numbers are very large. (How do you know that 8¹⁰⁰⁰⁰ exists? Have you ever counted to it?)

Needless to say, this announcement — and the announcement of a forthcoming book providing details — generated more than a flurry of excitement on the math blogs — including one of my very favorite blogs, the n-Category Cafe. After Fields Medalist Terry Tao raised a specific technical objection to Nelson’s argument, Nelson showed up in the comments section to defend himself — and then Tao showed up to expand on his objections. Nelson responded, Tao re-responded, and then Nelson posted:

You are quite right, and my original response was wrong. Thank you for spotting my error.

I withdraw my claim.

Just to be clear, here: That’s Ed Nelson cheerfully acknowledging that the book-length argument he’s been painstakingly constructing for (probably) years, and which was intended to shake the mathematical world to its foundations, doesn’t work. This says so many good things about the culture of mathematics, and so many good things about the Internet, and so many good things about the way they interact (see here and here for more examples), and it says those things so eloquently, that I see no further need for comment.

(On the other hand, if you’re hungry for additional comments, the philosopher Catarina Dutilh Novaes provides some good ones here.)

The Internet’s impact on mathematics is a huge huge thing. Not quite as huge as an inconsistency in Peano Arithmetic, but huge enough to count as a marvel.

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Friday Humor

Steve Landsburg — Fri, 05 Aug 2011 07:01:34 +0000

Three logicians walk into a bar.

The bartender says: “Would any of you guys like a drink?”

The first logician says: “I don’t know.”

The second logician says: “I don’t know.”

The third logician says: “No.”

Hat tip to Adam Merberg, who isn’t sure of the source.

Click here to comment or read others’ comments.

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Inconsistency

Steve Landsburg — Wed, 08 Jun 2011 06:01:57 +0000

Vladimir Voevodsky, one of the world’s best and most influential mathematicians, has stirred up a bit of a hornet’s nest with a video lecture suggesting the possibility that the Peano Axioms — the standard axioms for arithmetic — might be inconsistent.

Since the Peano Axioms are known to be consistent, it’s tempting to dismiss the whole lecture as either a prank or a shocking display of ignorance. The latter temptation is buttressed somewhat by Voevodsky’s bold misstatement of Godel’s Incompleteness Theorem, which plays a central role in the lecture. On the other hand, Voevodsky is smarter than almost anyone else on earth, which earns him the benefit of the doubt — maybe what he’s saying is subtler than it seems. On the other hand, some of those in the “shocking display of ignorance” camp are among the few people in the world who might be as smart as Voevodsky.

To believe that the Peano Axioms are inconsistent, Voevodsky must reject all of the known proofs that they are consistent. In particular, he must reject the simplest and most convincing of all those proofs, which goes like this:

The Peano Axioms, and therefore all of their logical consequences, are true statements about the natural numbers,
A collection of true statements cannot contradict itself.
QED.

If you were going to reject that argument, you’d pretty much have to reject part a). There are multiple ways to do that. You could deny that there is any such thing as “the natural numbers”, or you could deny that the Peano axioms are true statements about the natural numbers, or you could deny that all of the logical consequences of the Peano axioms are true statements about the natural numbers.

Denying the existence of the natural numbers is pretty much a non-starter. There is a class of people (mostly college sophomores) who refuse to admit the existence of the natural numbers but are nevertheless willing to debate the consistency of Peano Arithmetic (that is, the Peano Axioms and their consequences). But for a mathematician to deny the existence of the natural numbers would be as rare and fruitless as for a psychologist to deny the existence of conscious beings or for a physicist to deny the existence of physical objects. We can’t get by without them.

Besides, it is quite impossible to study Peano Arithmetic unless you know about the natural numbers in advance. For example, a “proof” in Peano arithmetic is a list of statements, each of which is either an axiom or follows from preceding statements. To understand that concept, you must know what a list is. To understand what a list is, you must have the concepts of “first”, “second”, “third”, etc. In other words, you need to know about the natural numbers! Anyone who claims to understand Peano arithmetic has already implicitly admitted that the natural numbers exist — and that you’ve got to be familiar with them before you can axiomatize them.

It would be a bit of an exaggeration, but not much of one, to call this assessment 100% noncontroversial among mathematicians. Almost (but not quite) everyone who studies math, including, I am almost sure, Voevodsky, accepts this account of what we do.

But if Voevodsky, like the rest of us, believes in the natural numbers, then it’s hard to see how he can doubt the consistency of Peano arithmetic. Surely the axioms are obviously true, but Voevodsky seems to be doubting that every logical consequence of the axioms is true. How can this be? If I understand him right (and I’m not sure I do), it’s because he thinks that some logical consequences of the axioms might be so complicated as to be meaningless, and are therefore neither true nor false.

This makes no sense to me at all, because complication does not by itself imply meaninglessness. More to the point, if some consequences of the axioms are meaningless, there must be a least complicated meaningless example. That least complicated example is only slightly more complicated than some other consequence which is meaningful. How can a slight increase in complexity introduce meaninglessness?

Again, I might have missed his point completely. If so, I’m in good company. There have been several threads in the May and June editions of the FOM mailing list (”FOM” stands for “Foundations of Mathematics”) where a mighty distinguished crew have expressed much the same confusion, and, in several cases, dismay.

For those who like this sort of thing, this is the sort of thing they’ll like. You can click on May or June above and select pretty much any thread with the word “Voevodsky” or “consistency” in it. I’m sure there’s content there I’ve failed to digest in full.

Edit: I should have said that in my (possibly flawed) understanding of Voevodsky, it’s not just the consequences of the axioms that can be too complicated to be meaningful, but some of the axioms themselves. For any property that can be expressed in Peano Arithmetic, there is an axiom saying (essentially) that if there are any numbers with that property, then there is a smallest one. V. seems to be saying (and again I want to stress that I’m not sure I understand him) that some of these properties are too complex to be meaningful, and therefore so are the corresponding axioms.

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Lord Russell’s Nightmare

Steve Landsburg — Mon, 20 Dec 2010 07:01:55 +0000

Bertrand Russell, that most rational of men, was nonetheless plagued by intermittent depression and the occasional nightmare. Including this one, as reported by Russell’s confidante, the mathematician G.H. Hardy:

[Russell] was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated….

Principia Mathematica, to which Russell had devoted ten years of his life, was his (and co-author Alfred North Whitehead’s) audacious and ultimately futile attempt to reduce all of mathematics to pure logic. It is a failure that enabled some of the great successes of 20th century mathematics. And — the first volume having been published in December, 1910 — this is its 100th birthday.

Having determined to write the Principia ten years earlier in 1900, Russell was at first stymied by his discovery of the famous paradox that now bears his name: Consider the set of all those sets that don’t contain themselves. Call this set R. Does R contain itself? If so, it belongs to the set of all sets that don’t contain themselves, and therefore does not contain itself. Does it fail to contain itself? If so, it fails to belong to the set of all sets that don’t contain themselves, and therefore contains itself. Either way, something’s screwy.

The Russell Paradox suggested that set theory was too shaky a foundation on which to build the edifice of mathematics, and Russell realized that the first thing he needed to do was shore up that foundation — a task that led to months, and then years, of staring at blank pages with no idea of how to proceed:

At first I supposed that I should be able to overcome the contradiction quite easily, and that probably there was some trivial error in the reasoning. Gradually, however, it became clear that this was not the case…It seemed unworthy of a grown man to spend his time on such trivialities, but what was I to do?

Russell’s ultimate solution was his Theory of Types, which (some would say artificially) limits the ways in which you’re allowed to define a set — and in particular prohibits you from defining the set R in the first place. It was clunky and ad hoc and many found it profoundly unsatisfactory, but it worked.

There were other problems, too. Russell wanted to derive all of mathematics from pure logic, but there was one mathematical fact that defied his every effort — namely the fact that there are infinitely many natural numbers. Surely this is a mathematical fact, but does it follow from purely logical considerations — or is it just something that happens to be true without being logically necessary? In the end, Russell punted, taking the infinitude of the natural numbers as an (essentially non-logical) axiom. Here was the first hint that maybe mathematics is more than just logic after all.

Once he had solved these problems — or, on a less charitable interpretation, decided how he was going to weasel his way around them — Russell spent the years from 1907 through 1910 working on the manuscript, ten to twelve hours a day, eight months a year. He worked, of course, with the technology of the time, with all of its attendant dangers and inconveniences:

The manuscript became more and more vast, and every time that I went out for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up. It was not, of course, the sort of manuscript that could be typed, or even copied.

When we finally took it to the University Press, it was so large that we had to hire an old four-wheeler for the purpose. Even then our difficulties were not at an end.

The University Press estimated that there would be a loss of £600 on the book, and while the syndics were willing to bear a loss of £300, they did not feel that they could go above this figure. The Royal Society very generously contributed £200, and the remaining £100 we had to find ourselves. We thus earned minus £50 each for ten years’ work. This beats the record of Paradise Lost.

The first edition of Volume I ran to 750 copies. Volumes II and III, published in 1912 and 1913, ran to 500 copies each.

Aside from dissatisfaction with the Theory of Types and the Axiom of Infinity, there were a couple of other nagging questions left unsettled, though. First: Could all of mathematics be derived from Russell and Whitehead’s logical system? Surely some of it could (though not always easily — R and W notoriously required hundreds of pages to reach the conclusion that 1+1=2) — but could all? And second: Could the Russell/Whitehead system be proven to be free of logical contradictions? The Russell Paradox had been excised by the Theory of Types, but could one exclude the possibility of other paradoxes lurking in the background?

Russell was surely hopeful on both counts. Kurt Godel, the logician of the millennium and the man who would dash those hopes, was four years old in 1910.

I venture to guess that nobody has ever read the three large volumes of Principia Mathematica, a typical page of which looks about like this:

But, for having set the agenda for (at least) a century of research into the foundations of mathematics, I am sure it will still be celebrated in the A.D. 2100 of Lord Russell’s nightmare.

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Eighty Years of Incompleteness

Steve Landsburg — Mon, 25 Oct 2010 06:01:07 +0000

This weekend marked the 80th anniversary of the most significant event in the history of logic since the days of Aristotle. On October 23, 1930, the 24-year-old Kurt Godel presented his incompleteness theorems to the Vienna Academy of Sciences.

The first incompleteness theorem says this: No matter what axioms you start with, there will always be statements in arithmetic that you can neither prove nor disprove. (I am glossing over some technicalities here, but in this context they are not important.) Some of those statements take the form “Such-and-such an equation has (or does not have) any solutions”. You tell me your axioms, and I’ll produce an equation that you can neither prove solvable nor prove insolvable.

When we’re doing arithmetic, we often start with the Peano Axioms, which codify the basic facts about addition and multiplication and the ordering of the natural numbers. (When I say “the ordering of the natural numbers”, I mean things like “every number has a successor”, “no two numbers have the same successor” and “starting from zero, if you keep taking successors, you’ll eventually get to any number you want”.) Sometimes, if we want a more powerful system, we begin with the Zermelo-Frankel Axioms, which allow us to talk not just about numbers, but about sets of numbers. Among mathematicians, these axiom systems are the industry standards.

Now Godel’s first incompleteness theorem tells us that there is an equation whose solvability/insolvability cannot be determined from the standard axioms. This means that if you like, you can simply assume this equation to be either solvable or unsolvable, and you will never risk contradicting yourself.

Nevertheless, mathematicians usually want to do more than just avoid contradicting themselves. They want to restrict themselves to proving things that are true. One interpretation of Godel’s theorem is that, regardless of your axiom system, there are true statements in arithmetic that you won’t be able to deduce from your axioms. That’s unsettling.

Godel’s second incompleteness theorem says that not only are there true statements in arithmetic you can’t derive from your axioms; there are also true statements about the axioms themselves that you can’t derive from the axioms — and one of those statements is that your axioms are consistent. More precisely: A consistent set of axioms cannot prove its own consistency. (An inconsistent set of axioms can prove its own consistency, but of course it will be lying to you.)

(Once again I’m glossing over technicalities that would be important in other contexts but are not important here.)

The second incompleteness theorem is sometimes misstated in something like the following form: “It is impossible to know that the axioms we use for arithmetic are consistent” or even “It is impossible to know that the things we prove about arithmetic are true”. That’s not what the theorem says at all. Godel’s theorem rules out the possiblity of deducing the consistency of the axioms from the axioms themselves. This is in fact earth-shattering (or was in the context of what other people were trying to accomplish in 1930), but it still doesn’t rule out the possibility that we can know the consistency of the axioms in some other way.

Indeed, most mathematicians are perfectly comfortable with the following argument: The Peano axioms must be consistent because the Peano axioms are true — and a set of true statements cannot be inconsistent. An extreme skeptic might reject this argument on the grounds that we can’t know the Peano axioms to be true — just as an extreme skeptic might insist we can’t know that anyone other than ourselves experiences consciousness, or that we weren’t all created five minutes ago with a lifetime’s worth of false memories built in. But as the late lamented logician Torkel Franzen never tired of pointing out — if that’s your position, then you must be skeptical not just of the consistency of the Peano axioms, but of pretty much everything else the rest of us think we know about arithmetic. This just doesn’t seem like a very productive way to go.

Godel is famous not just for his incompleteness theorems, but for his completeness theorem, which says that if a set of axioms is consistent, then there must be some mathematical structure it describes. So we can run this argument in both directions: If the natural numbers exist, then the Peano axioms are true statements about them, and therefore the Peano axioms are consistent. In the other direction, if the Peano axioms are consistent, then the completeness theorem says that the structure they describe — that is, the natural numbers — must exist.

Of course I’ve blogged about much of this before, but the anniversary seemed like a good occasion to reiterate it. Here are few relevant past posts:

Godel in a Nutshell gives the essence of Godel’s argument.

Basic Arithmetic, Part I, Part II, Part III and Part IV, on the relationship among truth, provability, consistency and existence.

First Things and Second Things and Godel, Fermat, Hercules, answering a readers’ questions about the applicability of Godel’s theorem and what it means for working mathematicians.

Just the Facts — more on the distinction between truth and provability.

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Basic Arithmetic, Part III: The Map is Not the Territory

Steve Landsburg — Thu, 19 Aug 2010 06:01:58 +0000

Today let’s talk about consistency.

Suppose I show you a map of Nebraska, with as-the-crow-flies distances marked between the major cities. Omaha to Lincoln, 100 miles. Lincoln to Grand Island, 100 miles. Omaha to Grand Island, 400 miles.

You are entitled to say “Hey, wait a minute! This map is inconsistent. The numbers don’t add up. If it’s 400 miles straight from Omaha to Grand Island, then there can’t be a 200 mile route that goes through Lincoln!”

So a map can be inconsistent. (It can also be consistent but wrong.) Nebraska itself, however, can no more be inconsistent than the color red can be made of terrycloth. (Red things can be made of terrycloth, but the color red certainly can’t.)

With that in mind, suppose I give you a theory of the natural numbers — that is, a list of axioms about them. You might examine my axioms and say “Hey! These axioms are inconsistent. I can use them to prove that 0 equals 1 and I can also use them to prove that 0 does not equal 1!” And, depending on the theory I gave you, you might be right. So a theory can be inconsistent. But the intended model of that theory — the natural numbers themselves — can no more be inconsistent than Nebraska can. Inconsistency in this context applies to theories, like the Peano axioms for arithmetic, not to structures, like the natural numbers themselves.

(See yesterday’s post for more on theories and models.)

Philosopher Alfred Korzybski admonishes us to remember that the map is not the territory. The theory is the map. The model is the territory. The hallmark of Internet crankery in this area is the refusal to distinguish them.

Whenever someone drones on at length about “the consistency (or inconsistency, or possible consistency, or possible inconsistency) of the natural numbers”, you’ll know he’s blathering. The concept simply doesn’t apply. Nebraska can’t be inconsistent. Only a description of it can be inconsistent.

(This brief snarky detour is brought to you by the small but determined band of commenters who consistently and vocally ignore this distinction in order to spout nonsense both here and on other blogs. I am not talking about anyone who’s commented here lately.)

It can be easy — and therefore entirely excusable — to get confused about this issue because in informal discussions of this subject — as in every other informal discussion of every subject in the English language — a single word can have multiple meanings. That sometimes happens with the word “arithmetic”. The phrase “Peano arithmetic” is the name of a theory — a list of axioms. On the other hand, some of us (me, for example) sometimes use the word “arithmetic” (a bit sloppily) to refer to a structure, namely the natural numbers themselves, which form a model of Peano arithmetic. Fortunately, the meaning is usually clear from context. If someone talks about “the consistency of arithmetic” you know that he’s talking about the theory (unless of course you have reason to suspect that he’s badly confused).

Now then. Let’s start with a theory. There are (at least) two sorts of questions you could ask about this theory. First: Is this theory consistent? In other words, is the theory free of self-contradiction? Second: Does this theory have a model? In other words, is there actually some structure that this theory describes?

If you’re given a map, the first question is like asking whether all the distances add up. The second question asks whether this is a map of someplace that actually exists or just a figment of the mapmaker’s imagination.

Inconsistent theories, obviously, have no models. A map that makes no sense cannot be a map of anything.

What about consistent theories? A consistent theory might a priori have either no models, or just one model, or many models.

The first possiblity is ruled out by Godel’s Completeness Theorem, not to be confused with the far more famous Godel’s Incompleteness Theorem. According to Godel’s Completeness Theorem, every consistent theory has at least one model. This is like saying that if you draw a map, and if nothing about the map is self-contradictory, then somewhere there is a territory that corresponds to the map. You should find this at least mildly surprising, but there it is.

If your theory is a theory of the natural numbers — in other words, if the natural numbers constitute a model for your theory — then the Lowenheim/Skolem Theorem says that your theory has a jillion other models as well. In other words, your map applies equally well to a jillion different territories. And there is no way, just by looking at the map, to tell those territories apart.

In other words, no theory — no list of axioms — can be a complete description of the natural numbers. It will always be a partial description, which applies equally well to a jillion other mathematical structures that look a little bit like the natural numbers but mostly a whole lot different.

Today’s moral: The map is not the territory. The map — the set of axioms — is either consistent or it’s not. If it’s inconsistent, there’s no corresponding territory. If it’s consistent, there are many corresponding territories and the map can’t tell you which one you’re in. That’s a fundamental limiitation on the power of the axiomatic method to describe a mathematical structure such as the natural numbers. It means there’s more to the natural numbers than any set of axioms can possibly know about.

Still to come: Are the Peano axioms consistent? Do the natural numbers really exist? And just how much about the natural numbers is any axiomatic system doomed not to know?

And finally: Thanks to those of you who encouraged me to continue this series. Let me know if you want still more.

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