Ali Reda's Reviews > Gödel's Proof

Gödel's Proof by Ernest Nagel
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it was amazing
bookshelves: philosophy

The Book is the best to explain Godel's Proof of the Incompleteness Theorem.

Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true arithmetical statements that cannot be derived from the set.

Godel showed that it is impossible to give a meta-mathematical proof of the consistency of a system comprehensive enough to contain the whole of arithmetic—unless the proof itself employs rules of inference in certain essential respects different from the Transformation Rules used in deriving theorems within the system. Such a proof may, to be sure, possess great value and importance. However, if the reasoning in it is based on rules of inference much more powerful than the rules of the arithmetical calculus, so that the consistency of the assumptions in the reasoning is as subject to doubt as is the consistency of arithmetic, the proof would yield only a specious victory: one dragon slain only to create another.

A few examples will help to an understanding of Hilbert’s distinction between mathematics (i.e., a system of meaningless signs) and meta-mathematics (meaningful statements about mathematics, the signs occurring in the calculus, their arrangement and relations). Consider the expression: 2 + 3 = 5 This expression belongs to mathematics (arithmetic) and is constructed entirely out of elementary arithmetical signs. On the other hand, the statement ‘2 + 3 = 5’ is an arithmetical formula asserts something about the displayed expression. The statement does not express an arithmetical fact and does not belong to the formal language of arithmetic; it belongs to meta-mathematics, because it characterizes a certain string of arithmetical signs as being a formula. if we wish to say something about a word (or other linguistic sign), it is not the word itself (or the sign) that can appear in the sentence, but only a name for the word (or sign). According to a standard convention we construct a name for a linguistic expression by placing single quotation marks around it. Our text adheres to this convention. It is correct to write: Chicago is a populous city. But it is incorrect to write: Chicago is tri-syllabic. To express what is intended by this latter sentence, one must write: ‘Chicago’ is tri-syllabic. Likewise, it is incorrect to write: x = 5 is an equation. We must, instead, formulate our intent by: ‘x = 5’ is an equation.

Godel devised a method of representation such that neither the arithmetical formula corresponding to a certain true meta-mathematical statement about the formula, nor the arithmetical formula corresponding to the denial of the statement, is demonstrable within the calculus. Since one of these arithmetical formulas must codify an arithmetical truth, yet neither is derivable from the axioms, the axioms are incomplete. Gödel’s method of representation also enabled him to construct an arithmetical formula corresponding to the meta-mathematical statement ‘The calculus is consistent’ and to show that this formula is not demonstrable within the calculus. It follows that the meta-mathematical statement cannot be established unless rules of inference are used that cannot be represented within the calculus, so that, in proving the statement, rules must be employed whose own consistency may be as questionable as the consistency of arithmetic itself. Gödel established these major conclusions by using a remarkably ingenious form of mapping. Since every expression in the calculus is associated with a (Gödel) number, a meta-mathematical statement about expressions and their relations to one another may be construed as a statement about the corresponding (Gödel) numbers and their arithmetical relations to one another. In this way meta-mathematics becomes completely “arithmetized.” Each metamathematical statement is represented by a unique formula within arithmetic; and the relations of logical dependence between meta-mathematical statements are fully reflected in the numerical relations of dependence between their corresponding arithmetical formulas which contain Godel Numbers. As if it was the mapping between Geometry and Algebra using a Cartesian system of coordinates.
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Reading Progress

September 2, 2014 – Shelved
September 2, 2014 – Shelved as: to-read
October 16, 2014 – Started Reading
October 16, 2014 – Shelved as: to-read-ebook
October 16, 2014 –
page 18
17.31%
October 16, 2014 –
page 18
17.31% "Gödel’s paper is a proof of the impossibility of demonstrating certain important propositions in arithmetic."
October 16, 2014 –
page 21
20.19% "the question whether a given set of postulates serving as foundation of a system is internally consistent, so that no mutually contradictory theorems can be deduced from the postulates. No mathematician prior to the nineteenth century ever considered this question because of the sound principle that logically incompatible statements cannot be simultaneously true."
October 17, 2014 –
page 32
30.77% "Hilbert’s distinction between mathematics (i.e., a system of meaningless signs) and meta-mathematics (meaningful statements about mathematics, the signs occurring in the calculus, their arrangement and relations)."
October 17, 2014 –
page 42
40.38% "An absolute proof of the consistency of arithmetic, if one could be constructed, would therefore show by a finitistic meta-mathematical procedure that two contradictory formulas, such as ‘0 = 0’ and its formal negation ‘~ (0 = 0)’— where the sign ‘~’ means ‘not’—cannot both be derived by stated rules of inference from the axioms (or initial formulas)."
October 18, 2014 –
page 44
42.31% "Principia Mathematica thus appeared to advance the final solution of the problem of consistency of mathematical systems by reducing the problem to that of the consistency of formal logic itself. For, if the axioms of arithmetic are simply transcriptions of theorems in logic, the question whether the axioms are consistent is equivalent to the question whether the fundamental axioms of logic are consistent."
October 18, 2014 –
page 50
48.08% "from a contradictory set of axioms any formula can be derived. But this has a converse: namely, if not every formula is a theorem (i.e., if there is at least one formula that is not derivable from the axioms), then the calculus is consistent. The task, therefore, is to show that there is at least one formula that cannot be derived from the axioms."
October 19, 2014 –
page 60
57.69% "the definition of the property of being Richardian does not belong to the series initially intended, because this definition involves metamathematical notions such as the number of signs occurring in expressions. We can outflank the Richard Paradox by distinguishing carefully between statements within arithmetic and statements about some system of notation in which arithmetic is codified."
October 19, 2014 –
page 70
67.31% "Godel is a tricky SOB :), "In the Richard Paradox, the number n is the number associated with a certain meta-mathematical expression. In the Gödel construction, the number n is associated with a certain arithmetical formula belonging to the formal calculus, though this arithmetical formula in fact represents a meta-mathematical statement because the meta-mathematics of arithmetic has been mapped onto arithmetic"."
October 19, 2014 –
page 77
74.04%
October 20, 2014 – Finished Reading
May 20, 2016 – Shelved as: philosophy

Comments Showing 1-2 of 2 (2 new)

dateDown arrow    newest »

message 1: by Matt (new)

Matt Let's see how the Google translator http://translate.google.com performs:

Translate 'The word sun has 3 letters' into German.
=> 'Das Wort Sonne hat 3 Buchstaben'

Nope. The word 'Sonne' has 5 letters.

But I forgot the quotes,
so let's try 'The word "sun" has 3 letters'.
=> 'Das Wort "Sonne" hat 3 Buchstaben'

Still wrong.
I guess this thing has no grasp of 'meta'.
Never mind :=)


message 2: by Ali (new) - rated it 5 stars

Ali Reda Interesting Matt, Google Translator also fails the Turing Test. Hence Google is stupid :) :)


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