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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".

Chris

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 Gödel'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.2011-05-18
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    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 statements2011-05-18
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    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.2011-05-18
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    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.2011-05-18
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    If I might be so bold, perhaps some of what I wrote [in this answer](http://math.stackexchange.com/questions/16358/understanding-godels-incompleteness-theorem/16383#16383) 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".2011-05-19
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    @xtiansimon: I'm not sure I understand what you mean by "formulas both provable and unprovable" (or how Goedel's Theorem "requires" them, but if I don't know what you are refering to, it would be hard for me to understand the latter, I guess...), or what you mean by "admitting a statement" or a "proper theory". I may try my hand at this tomorrow, but perhaps I'll wait for one of our logic gurus to try first.2011-05-19

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