Rudimentary and arithmetical constructive set theory

Peter Aczel
2013 Annals of Pure and Applied Logic  
Separation, Strong Collection, Subset Collection and Set Induction. The axiom system CZF is much weaker than ZF. Nevertheless when the law of excluded middle is added the resulting axiom system has the same theorems as ZF. Moreover when the powerset axiom and the full Separation scheme are added an axiom system is obtained that has the same theorems as IZF, an axiom system that has the same logical strength as ZF in virtue of a double negation interpretation of ZF into IZF due to Harvey
more » ... . The main aim of this paper is to formulate and study a weak axiom system for Arithmetical CST, ACST, that is strong enough to represent the class N at of von Neumann natural numbers and its arithmetic so that Heyting Arithmetic can be interpreted. A significant feature of CZF is the role of, possibly infinitary, class inductive definitions that define classes that may not be sets. We will see a similar role for finitary inductive definitions in ACST. A first approach to an axiom system for Arithmetical CST is the axiom system BCST+M athInd(N at). Here (i) the axiom system BCST for a basic CST is obtained by leaving out from CZF the axiom of Infinity and the axiom schemes of Strong Collection, Subset Collection and Set Induction, while adding the axiom scheme of Replacement, and (ii) M athInd(N at) is the axiom scheme of mathematical induction for a suitably defined class N at of the von Neumann natural numbers. The axiom system BCST + M athInd(N at) does not assume that N at is a set. An alternative basic axiom system for arithmetic that has been considered is ECST, which is obtained from BCST by adding the axiom of Strong Infinity, the axiom that expresses the existence of the smallest inductive set, ω. In contrast to BCST+M athInd(N at) the axiom system ECST does not have full mathematical induction, but can only derive mathematical induction for bounded formulae. The Union Axiom and the Replacement Scheme can be combined into a single scheme, the Union Replacement Scheme. The full strength of the Union Replacement Scheme seems not to be needed for our purposes. It turns out that a rule of inference, the Global Union Replacement Rule (GURR) can be used instead and we will see that this rule provides exactly enough power to enable definitions of the rudimentary functions on sets. The rudimentary functions were originally introduced by Ronald Jensen, see [Jen72] , in the context of classical set theory, in order to develop a good fine structure theory for Goedel's constructible sets. So we are led to consider the very weak axiom system, RCST, of Rudimentary CST. This axiom system has a standard system of axioms and rules for intuitionistic logic in the language L ∈ , the rule GURR, the axiom of extensionality and the set existence axioms, Emptyset, Binary Intersection and Pairing for the existence of the sets ∅, x 1 ∩ x 2 , {x 1 , x 2 } respectively, for
doi:10.1016/j.apal.2012.10.004 fatcat:yojpov5aj5cixpoun33qmek72q