Elements of Mathematics: Theory of Sets by Nicolas Bourbaki

By Nicolas Bourbaki

Ever because the time of the Greeks, arithmetic has concerned facts; and it's even doubted by means of a few no matter if facts, within the exact and rigorous experience which the Greeks gave to this note, is to be came upon outdoor mathe- matics. W e may well particularly say that this feeling has now not replaced, simply because what constituted a prooffor Euclid continues to be a prooffor us; and in occasions whilst the concept that has been at risk of oblivion, and hence arithmetic itself has been threatened, it's to the Greeks that males have became back for modds of facts. yet this venerable bequest has been enlarged in past times hundred years through very important acquisitions.
By research of the mechanism of proofS in certainly selected mathematical texts, it's been attainable to figure the constitution underlying either voca- bulary and syntax. This research has ended in the realization suffi- ciently specific mathematical textual content might be expressed in a traditional language containing just a small variety of mounted "words", assembled in keeping with a syntax together with a small variety of unbreakable principles : one of these textual content is related to be formalized. the outline of a video game of chess, within the ordinary notation, and a desk oflogarithms, are examples offormalized texts. The formulae of normal algebraic calculation will be one other instance, if the principles governing using brackets have been to be completdy codified and strictly adhered to; in perform, a few of these principles are by no means made particular, and sure derogations of them are allowed.

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For A ~ (A or A) and (A or A) Now use 06. 09. to to. ~ is a relation and B a theorem in A are axioms, by S2 and Sl. to, then A ~ B is a theorem in to. For B~ «not A) or B) is a theorem by 07, and therefore "(not A) or B", that is to say A ~ B, is a theorem by 01. 010. if A is a relation in 'CO, then "A or (not A)" is a theorem in 'iV. For "(not A) or A" is a theorem by OS; now use S3 and 01. 011. If A is a relation in 'CO, "A ~ (not not A)" is a theorem in 'iV. For this relation is "(not A) or (not not A)", and the result follows fromOlO.

For the rest of this section, 'i9 will denote such a theory, and 'i90 will denote the theory without explicit axioms which has the same signs as 'i9 and whose only schemes are S 1 through 85. 'i9 is stronger than 'i90 • C29. Let R be a relation in 'i9 and let x be a letter. "not (3x)R" and (Vx)(not R) are equivalent in 'i9. Then the relations It is sufficient to prove the criterion in the theory 'i90 in which x is not a constant. The theorem R -+=> (not not R) gives us, by C3 (§2, no. 3), the theorems (3x)R =+- ('tx(R)lx)(not not R) and (3x) (not not R) =+- ('tx(not not R)IX)R.

In particular, by 010, if "A => C" and "(not A) theorems in 'l5, then C is a theorem in 'lO. IV. Method of the auxiliary constant. 019. => C" are both This is founded on the following rule : Let X be a letter and let A and B be relations in 'l5 such that: (1) the letter X is not a constant of 'lO and does not appear in B; (2) there is a term T in 'l5 such that (Tlx)A is a theorem in 'lO. Let 'lO' be the theory obtained by adjoining A to the axioms of to. if B is a theorem in 'l5', then B is a theorem in 'lO.

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