ALBERT

Notes: 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.

Notes: Abstract Corresponding to a graduation of the comprehension predicateC there arise two systems PC1 and PC2. PC1 represents constructive mathematics whereas PC2 as a restricted enlargement of PC1 enables one to deal with full set theory. Unnatural definitions likeKuratowski's ordered pair and identification of 0 and $$\not 0$$ are avoided. In PC1 the postulate introduced for ordered pairs holds not only for sets but for classes generally. The comprehension condition in PC1 implicitly contains a set axiom for at most denumerable calsses and for ordered pairs. By introduction of the concept “ordered self-pair” the real numbers are interpreted as self-pairs of (non denumerable) equivalence classes of concentrated ratioal sequences and they can be shown to be sets in PC1.

Notes: Abstract It is well known that equivalence holds between the weak axiom of choice (AC) and the well ordering principle (WOP) for sets, resp. between strong AC and WOP for classes. It will be shown that in a theory PC* with inpredicative classes (i. e. with no restriction of quantification in the defining formula) the super-strong AC used by the informally working mathematician is equivalent to a superstrong WOP. The equivalence between strong AC and super-strong AC is implied by a conditionC refutable in PC* but provable in PC which is PC* with predicative classes only and with the general ordered pair axiom. PC* [super-strong AC] is inconsistent because the super-strong AC impliesC. Therefore the application of choice functions to non-empty classes generally makes a predicative definition of these classes necessary. Connected with these problems is a statement equivalent to the conjunction of the axioms of power set and foundation based on a function which coincides with the von Neumann-function under the assumption of one of the mentioned axioms.