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My question stems from reading Swetz, 1994 (mostly excerpts from the journal Mathematics Teacher) and Berlinski, 2005 (a popular book on 10 most important mathematical breakthroughs in history).

1) I'm having difficulty understanding why Godel's theorem (if I understand what I have read) requires both formulas provable and unprovable. My naive concept of mathematics is a system containing only "positive" theorems and laws and I imagine the same for science. That the scientific method culls anything proven wrong, so as not to carry failures as baggage together with its laws. Where failure, mistakes, and problems are separated as historical study or challenge. Why do unprovable statements need be admitted to a proper system?

In anticipation of my enlightenment, I have a few follow-up Qs,

2) Its my understanding the Hilbert project meant to take the point of view that mathematics and formalism needed to take a meta-point of view for the purpose of separating formulas (as symbols) from the discussion (in natural language) about mathematics. If these unprovable formulas are necessary to prove other provable proofs, then why not segregate them to yet another meta-level?

3) What I gather about Godel's argument from these sources includes a point about mathematics' use of symbols (representing numbers, variables for formulas, variables for sets and sets of sets, etc.). Then is it his argument to suggest that where a proof can be admitted to the system which can be neither proved or disproved, the substitution of variables will propagate this error?

Answers need not be technical, only sufficient to sort my "Godel baggage".


PS. I found this January post very useful for its references and look forward to finding a volume for my naive appetite for the history of formalism:Understanding Godel's Incompleteness Theorem---

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About 1) You don't admit unprovable statements to the system. If your system is sufficiently strong (can model the natural numbers) then the statements are there! These statements can be extremely concrete (thing of statements of the form: does this polynomial have roots in $\mathbb{Z}$?), and should not be viewed as something extremely esoteric. –  Thomas Rot May 18 '11 at 22:42
As I understand the incompleteness theorem, it basically states that within a closed axiomatic system there can exists unfalsifiable premises, that is, statements that can neither be proved nor disproved. This is not equivalent to a requirement that the system must admit false statements –  ItsNotObvious May 18 '11 at 22:49
Berlinski summarizes Godel's argument in four points (p160). Point 1 "The formula R(q) is a formula from the metamathematician's [point of view]. It has a specific number which has a place on the list [of formulas in the Principia]." Point 2 "But the formula [R(q),q] names a formula of the Principia--one that defines the property of unprovable within the system." [note R(q) is, I gather, an arbitrary construct from earlier pages about symbolization within mathematics]; Its from this statement I get my idea unprovable statements are part of the system. –  xtiansimon May 18 '11 at 22:53
Point 3. "When the numeral for q replaces the variable x in the formula, the formula says of the number that that numeral denotes--q, in fact--that it corresponds to the number of an unprovable formula on the master list." Point 4: "That unprovable formula id [R(q), q] itself, which ...[states] that it is not provable." The author leads up to this summary with discussing how a system is capable of commenting on itself--the meta-mathematical commenting on the formula without using natural language. –  xtiansimon May 18 '11 at 23:00
If I might be so bold, perhaps some of what I wrote in this answer might help, though I don't address in any way the actual argument or proof of the theorem, only trying to give some context and some "take-away". –  Arturo Magidin May 19 '11 at 3:34
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4 Answers

As your quotes from Berlinski suggest, the heart of Godel's statements is arguably not in the unprovability itself, but in the fact that the system is able to 'talk about itself' - that is, that statements about provability can be cast as strictly arithmetic questions. It might help to familiarize yourself with another undecidability result which goes through roughly the same route: Matiyasevich's Theorem about the solvability of diophantine equations (i.e., the question 'does this polynomial in $x$, $y$, $z$, $w$, etc. ever evaluate to zero at some integer values of $x$, $y$, $\ldots$?'). The root of the proof lies in showing that diophantine equations are expressive enough to represent all recursively enumerable sets; what Godel does is essentially the same, showing that Peano Arithmetic (or any similar system) is expressive enough that questions about 'provability' can be cast as arithmetic questions.

As for why unprovable statements need admission, the point isn't simply that they're unprovable but that they're unprovable and true - that is, that the set of 'things we can prove' will always be a proper subset of 'things that are true'. Obviously this isn't a crippling discovery; mathematics didn't stop as of Godel! But it is arguably a profound one, and one that has substantial practical implications: for instance, as noted above there can't possibly be an algorithm for solving diophantine equations; there can't be algorithms for finding a 'minimal forbidden minor' categorization of many sets of graphs; there can't be algorithms for solving the Word Problem for groups; etc, etc.

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Arturo Magidin has a cogent comment about "true but not provable" (and many other topics) in his answer to math.stackexchange.com/questions/16358/… –  Ross Millikan May 19 '11 at 0:18
I think perhaps that the fact that the system is "able to talk about itself" is the heart of the argument, rather than the heart of Goedel's statements (by which I understand you to refer both to the Goedel Sentence 'G', and to the statement of Goedel's Theorem itself). –  Arturo Magidin May 19 '11 at 3:58
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When people set about formalizing Mathematics, they did it this way:

  1. List the symbols you're allowed to use to make mathematical statements,

  2. Give some rules to distinguish "well-formed formulas" (like $3+4=6$) from other strings of symbols (like $3\div\gt4=$),

  3. List the axioms (a finite list),

  4. List the "rules of inference" (the methods you are allowed to use to get from one well-formed formula to another).

The theorems are then the well-formed formulas you can get to, starting from the axioms and using the rules of inference; they are the things you can prove.

The naive hope was that for every well-formed formula $P$, there would either be a proof of $P$, or a proof of the negation of $P$ (but not both).

What Godel proved was that if your axioms and rules of inference are strong enough to do ordinary arithmetic, then the naive hope cannot be realized. Either the system is inconsistent (meaning that you can prove every statement, including the negation of statements you can prove), or it is incomplete (meaning there are well-formed formulas $P$ such that you can't prove $P$ and you can't prove the negation of $P$).

I don't understand what you mean by "admitting unprovable statements to a proper system." We don't accept them as theorems; we note that neither they nor their negations are theorems. What would you have us do with unprovable statements?

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Aren't the axioms sometimes (as in PA and ZFC) an infinite but recursive list? –  Ross Millikan May 19 '11 at 0:40
Ssh! I'm trying to keep things simple.... Technically, I suppose one can also object that what I'm calling a well-formed formula should really be a well-formed formula without free variables. But if my over-simplified account helps xtiansimon see what's what, it will have served its purpose. –  Gerry Myerson May 19 '11 at 1:49
Fair enough; perhaps as a second-level approximation, "list the axioms (you should be able to determine algorithmically if a given statement is or is not an axiom)". –  Arturo Magidin May 19 '11 at 3:31
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It is important to see the theorems in the context of Hilbert's attempt to formalize mathematics. Formalization here means reducing the whole Mathematics-making art into something as simple, in principle, as manipulating strings according to some fixed rules. In Hilbert's times there wasn't computers yet, but I'm sure he had in mind a goal that can be stated as "allowing computers to do math without thinking".

This formalization has two strengths. One is obvious - you can find theorems by letting your computer run, in an ordered way much different from current-day's guessing and searching in the dark game. The second one was no less important to Hilbert - if you can show all mathematics can be derived from extremely simple, "obviously true" rules, then our faith in the correctness of mathematics will be high. Remember, this was around the time of the "great crisis in the foundations of mathematics", raised by Russel's paradox (which requires only the extremely plausible hypothesis that you can define sets freely using mathematical predicates) and other paradoxes.

So Hilbert was looking out for a simple axiomatic system from which all mathematics could be derived. But what is "all" mathematics? It is not only what can be proven in some sort of system, but everything that is true about the objects we are interested in. One such object are the natural numbers. Now consider Goldbach's conjecture - that every even number greater than 4 can be written as the sum of two odd primes. This is surely either true or false (or is it...? But we adopt the viewpoint that believes this). The only question remaining is - can Hilbert's wonderful system prove it using no thinking at all, only manipulation of symbols? If it can't prove nor disprove it, surely it does not cover "all" mathematics.

What Godel showed is exactly that - that no matter what proof system Hilbert will give, it will have some flow: Either it won't be consistent (can prove and disprove the same thing, and from it usually follows you can prove everything), or it won't be complete (there will be some claim you won't be able to prove or disprove), or it won't be effective (in the sense you won't be able to check if a proof is legal or not - there will be no simple mechanical rules for doing it) or it won't talk about the nautral numbers (with the operations of addition and multiplication; Presburger arithmetic eludes Godel's theorems because it leaves the operation of multiplication out of the game).

What makes Godel's proof all the more interesting is that it uses the strengths of the system against it. On the other hand, the system can talk about natural numbers, and we can encode very intricate claims using natural numbers; on the other hand, the systems' rules are very simple, so they can be encoded themselves with natural numbers; and hence the system can indirectly prove things about itself, and this leads to some sort of "explosion". This is a very powerful idea, which has also given birth to the theory of Computability.

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I find some points in your response inciteful. And, I must correct the last point. Godel's theorem concerns arithmetic which my sources say is synonymous with the natural numbers. Your statement the "system can talk about natural numbers" would be incorrect... yes? ehehe –  xtiansimon May 20 '11 at 12:31
Godel's theorem concerns any (effective, consistent) system which is powerful enough to describe arithmetic. I'm not sure what you think is incorrect here... –  Gadi A May 20 '11 at 15:27
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I'm not sure how to respond to the challenge to find a place for unprovable theorems statements other than my naive notion they should be relegated to a third meta-level (first being the non-natural "proof language" of mathematics, second, provable consistent & complete, and third provable consistent, complete, or neither provable or unprovable). :)

To continue my response I can only think to answer my own question. First, dispose of any analogies between mathematical logic and experimental science. Since 'I know' they were offered only to establish the frame of the question, /I wish/ to continue only with regard to mathematics and history of mathematics.

Back to Godel's theorem, the impact of incompleteness is to make clear there are within the infinite list of functions which can be formed from a finite set of axioms, theorems, /and statements/ which are neither provable or unprovable, and which mathematics accepts as true where they produce usable results.

Only, how Godel did produced this argument is still vague. On the one hand I know his argument is aimed at the Principia. On the other hand, from my reading I perceive the /incompleteness/ theorem as an arbitrary proof of a paradox of finitary reasoning--where a finite system can be demonstrated to produce the paradox of undecidability from the very logic of recursion (is this accurate?). If the latter is true, then the question of whether mathematics does or does not suffer to admit theorems statements provable/unprovable is irrelevant.

Just to continue in good faith, my dialog here is about my observation the Hilbert project aimed to complete the theoretical/philosophical implications started by the mathematicians of the previous century, who released mathematics from deriving mathematical truth only from the natural (3-space) universe. If successful these mathematicians would have fulfilled the promise of axiomatic mathematics (again, is this accurate?).

Hilbert would have at one shot connected abstract and intuitive mathematics with rigorous methods such that an axiomatized mathematical theory in one area could with greater ease be related by logic, and this is important, to another area--wired if you will. Instead, some other mechanism must for now suffice. I can only guess statistics or shared model.

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I don't understand what you've written, but I suspect you don't either. Let's start with the phrase "unprovable theorem" in the first line - there's no such thing, as theorems are, by definition, those things we can prove. I think that rather than bloviating about a "third meta-level" you'd do well to try to engage with the answers you've received, understand what others are saying, and ask focussed questions if there are points in those answers you don't understand. Otherwise you're condemned to producing word salad forever. –  Gerry Myerson May 21 '11 at 11:33
If my jocular nature is too much, or my tenuous grasp of theoretical mathematics produces errors making you want to vomit, why respond? Its not your scene. As to the "unprovable theorem", on rereading I understand I seem to be getting my wires crossed. In my opening I asked about "formulas neither provable nor unprovable", but in my last post what I'm thinking about is the notion that axiomatic systems and their theorems don't "jive" in the way Hilbert had hoped they would. That's what, for me, makes Godel's theorem so interesting, and not just an anecdote about completeness and consistency. –  xtiansimon May 22 '11 at 23:47
I respond because I have something to say that might help you or, failing that, help someone else who is willing to take my remarks and the other answers and comments seriously. I don't object to your jocularity, only to your airs. What is "arbitrary" about Godel's proof? –  Gerry Myerson May 23 '11 at 0:25
Here is an analogue of the incompleteness theorems that comes from the natural sciences: Heisenberg uncertainty principle. What Gödel's theorems state in their core is that we cannot observe some of the properties of numbers if we want these "numbers" to follow certain rules. A proof in mathematics is a glance at the mathematical universe. Gödel's theorems state that if the "mathematical world" is complex enough it has structure that is impossible for us to perceive. –  Apostolos May 23 '11 at 7:26
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