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Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Set-like principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, CZF with Subset Collection replaced by Exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.

Constructive ZF + full Separation is shown to be equiconsistent with Second Order Arithmetic. * Thanks are due to the referee for the careful reading and helpful suggestions received.Among Aczel's accomplishments was an interpretation of CZF, and various extensions thereof, in Martin-Löf type theory, which established CZF as a predicative theory. As it turns out, CZF is proof-theoretically equivalent with Martin-Löf type theory ML1V, as well as KP (Kripke-Platek admissible set theory), ID (inductive definability), and a host of other identified theories. Such theories can arguably be considered to be the limit of predicative mathematics, allow for a philosophical justification as being constructive, and are in any case very much weaker prooftheoretically than ZF. For an overview, references, and some proofs, see [11].It is a natural enough question to ask about the strength of variants of CZF. Michael Rathjen conjectured that adding full Separation to CZF elevates the theory's strength from ID, which is a small fragment of Second Order Arithmetic, to full Second Order Arithmetic. In this note, this conjecture will be shown to be correct.What is the value of such a result? It is certainly nicer if a theory is shown to be weak, in that one can then use that theory with fewer philosophical scruples. This is the case for instance in [2], see also [11], where it is shown that adding certain choice principles to CZF does not change the proof-theoretic strength. Still, knowing that CZF + full Separation has the strength of Second Order Arithmetic tells us at least that the former theory is not predicative, providing a warning to the constructivelyminded mathematician not to work within it. In addition, the exact determination of this strength provides independence results. For instance, Rathjen ([12], prop. 7.12 (ii)) shows that CZF + Power Set is prooftheoretically stronger than n th Order Arithmetic for all n, from which it follows that CZF, being much weaker, does not prove Power Set. Similarly, it follows from the result in this paper that even with the addition of full Separation Power Set does not follow. (The latter result is re-proven model-theoretically in [8].) None of this should be surprising. Both the main result of this paper and the consequences just cited were anticipated by Rathjen (and perhaps others), and the ideas contained in the proof to follow are themselves mostly a reworking of those found in [1] and [11]. Still, at least now the results are formally established.By way of additional background, recent work on CZF and other intuitionistic set theories, often involving categorical models, has been done