Peano-Systeme

Abstract

In a metamathematical treatment the basic requirements for the method of proof (enlarged Peano-frame or Cantor-frame) should always be laid down. An enlarged Peano-frame C[PAx] consists of the axioms and rules for quantification and identity and 3 class axioms CAx₁-CAx₃-axiom of extensionality, axiom of ordered pairs and a modified version of the comprehension scheme introduced by the author-but contains no axiom stating explicitly for any class the property of being a set, and finally containing as hypotheses the Peano axioms (A ₀–A ₁₁ relativised toN). The scheme of comprehension includes explicitly all the set axioms of elementary set theory, but the axiom of replacement only in a restricted version. In this setting a number of theorems follows. The distinction between strong and weak induction, strong and weak Peano systems, is based on the notion of being „D-arithmetical”. Besides the well known first order number-theoretic formal systemP ₁ the author introduces a first order number-theoretic formal systemP ₂ and a second order number-theoretic formal systemP ^II in which there is no quantification over predicates. The following metatheorems hold: a) becauseP ^II contains the axiom of extensionality and the scheme of comprehension,P ₁ is embedded inP ^II; b) by extending the definition of “model” to languages of second order, all models ofP ^II are strong Peano systems;P ^II is therefore categorical; because of a) and the fact thatP ^II is recursively axiomatizised, the Gödel proof for incompleteness can be transferred toP ^II,P ^II is incomplete and in addition model-theoretically incomplete; c)P ₂ which is a first order transcription ofP ^II, has not the power ofP ^II for characterising the natural numbers; because not all models ofP ₂ are weak Peano systems,P ₂ is not a pure number-theoretic formal system in the strong sense; d)P ₂ is a formal model ofP ₁ relative to theP ₂-predicate I; e) becauseP ₂ andP ^II contain the axiom of extensionality only in a restricted manner and do not hold universally,P ₂ andP ^II can be interpreted both class-theoretically and attribute-theoretically. A procedure is given for transforming the inexpansive formal theory with all formulas being true in the standard model as axioms in a consistent, complete, not recursively axiomatizised, but expansive theory of first resp. second order. To the enlarged Peano-frame C[PAx] corresponds a formal first order theory with equality with Lang=〈0, ′,+,·,<>,∈,N n〉, the definitional postulateN={x| N n(x)}, and as proper axioms the Peano axioms (A ₀–A ₁₁ relativised toN, but deletingN as index of function symbols) and the 3 class axioms CAx₁–CAx₃. Modifying in this system the comprehension clause one gets PC. In PC one could e. g. lay down the elementary theory of recursive functions. Enlarging PC with one set axiom MAx to PC′, one can develop the theory of integral, rational, real and complex numbers. In a higher order theory (PC>I) one can avoid the introduction of a set axiom for developping the theory of number systems. PC′ and PC>I reflect and confirm the old view of the mathematicians and logisticians that if the natural numbers are given all others can be constructed, even in a perfectly formal manner.