I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.'s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians).

Many philosophers take set-theoretic discourse to be about objects of a special sort, namely sets; correlatively, they regard truth in such discourse as quite like truth in discourse about nonmathematical objects. There is a thin “disquotational” way of construing this construal; but that may candy-coat a philosophically substantive semantic theory: the Mathematical-Object theory of the basis for the distribution of truth and falsehood to sentences containing set-theoretic expressions. This theory asserts that truth and falsity for sentences containing set-theoretic expressions are grounded in semantic facts (about the relation between language and the world) of the sort modelled by the usual model-theoretic semantics for an uninterpreted formal first-order language. For example, it would maintain that ‘{ } ∈ {{ }}” is true in virtue of the set-theoretic fact that the empty set is a member of its singleton, and the semantic facts that ‘{ }’ designates the empty set,‘{{ }}’ designates its singleton, and ‘∈’ applies to an ordered pair of objects iff that pair's first component is a member of its second component.

Now this theory may come so naturally as to seem trivial. My purpose here is to loosen its grip by “modelling” an alternative account of the alethic underpinnings of set-theoretic discourse. According to the Alternative theory, the point of having set-theoretic expressions (‘set’ and ‘∈’ will do) in a language is not to permit its speakers to talk about some special objects under a special relation; rather it is to clothe a higher-order language in lower-order garments.

Much of the literature on the model theory of modal logics suffers from two weaknesses. Firstly, there is a lack of generality; theorems are proved piecemeal about this or that modal logic, or at best small classes of logics. Much of the literature, e.g. [1], [2], and [3], confines attention to structures with the expanding domain property (i.e., if wRu then Ā(w) ⊆ Ā(u)); the syntactic counterpart of this restriction is assumption of the converse Barcan scheme, a move which offers (in Russell's phrase) “all the advantages of theft over honest toil”. Secondly, I think there has been a failure to hit on the best ways of extending classical model theoretic notions to modal contexts. This weakness makes the literature boring, since a large part of the interest of modal model theory resides in the way in which classical model theoretic notions extend, and in some cases divide, in the modal setting. (The relation between α-recursion theory and classical recursion theory is analogous to that between modal model theory and classical model theory. Much of the work in α-recursion theory involved finding the right definitions (e.g., of recursive-in) and separating concepts which collapse in the classical case (e.g. of finiteness and boundedness).)

The notion of a well-behaved modal logic is introduced in §3 to make possible rather general results; of course our attention will not be restricted to structures with the expanding domain property. Rather than prove piecemeal that familiar modal logics are well-behaved, in §4 we shall consider a class of “special” modal logics, which obviously includes many familiar logics and which is included in the class of well-behaved modal logics.

The jump hierarchy of Turing degrees assigns to each ξ < (ℵ1)L the degree 0(ξ); we presuppose familiarity with its definition and with the basic terminology of [5]. Let λ be a limit ordinal, λ < (ℵ1)L. The central result of [5] concerns the relation between 0(λ) and exact pairs on Iλ = {0(ξ) ∣ ξ < λ}. In [6] this question is raised: Where a is an upper bound on Iλ, how far apart are a and 0(λ)? It is there shown that if λ is locally countable and admissible, they may be very far apart: 0(λ) = the least member of {a(Ind(λ))∣, a is an upper bound on Iλ}; this is rather pathological, for Ind(λ) may be larger than λ. If λ is locally countable but neither admissible nor a limit of admissibles, we are essentially in the case of λ < ; by results of Sacks [12] and Enderton and Putnam [2], 0(λ) = the least member of {a(2) ∣ a is an upper bound on Iλ}. If λ is not locally countable, Ind(λ) is neither admissible nor a limit of admissibles, so we are again in a case like that of λ < . But what if λ is locally countable and nonadmissible, but is a limit of admissibles? For the rest of this paper let λ be such an ordinal. The central result of this paper answers this question for some such λ.

Let I be a countable jump ideal in = 〈The Turing degrees, ≤〉. The central theorem of this paper is:

a is a uniform upper bound on I iff a computes the join of an I-exact pair whose double jump a(1) computes.

We may replace “the join of an I-exact pair” in the above theorem by “a weak uniform upper bound on I”.

We also answer two minimality questions: the class of uniform upper bounds on I never has a minimal member; if ⋃I = Lα[A] ⋂ ωω for α admissible or a limit of admissibles, the same holds for nice uniform upper bounds.

The central technique used in proving these theorems consists in this: by trial and error construct a generic sequence approximating the desired object; simultaneously settle definitely on finite pieces of that object; make sure that the guessing settles down to the object determined by the limit of these finite pieces.

Where AR is the set of arithmetic Turing degrees, 0(ω) is the least member of {a(2) ∣ a is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's . This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on results of Jensen and is detailed in [3] and [4]. In § 1 we review the basic definitions from [3] which are needed to state the general results.

Where a is a Turing degree and ξ is an ordinal < (ℵ1)L1, the result of performing ξ jumps on a, a(ξ), is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through (ℵ1)L1 of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.

Given I, a reasonable countable set of Turing degrees, can we find some sort of canonical strict upper bound on I? If I = {a ∣ a ≤ b}, the upper bound on I which springs to mind is b′. But what if I is closed under jump? This question arises naturally out of the question which motivates a large part of hierarchy theory: Is there a canonical increasing function from a countable ordinal, preferably a large one, into D, the set of Turing degrees? If d is to be such a function, it is natural to require that d(α + 1) = d(α)′; but how should d(λ) depend on d ↾ λ, where λ is a limit ordinal?

For any I ⊆ D, let MI, = ⋃I. Towards making the above questions precise, we introduce ideals of Turing degrees.

Definition 1. I ⊆ D is an ideal iff I is closed under jump and join, and I is downward-closed, i.e., if a ≤ b & b ϵ I then a ϵ I.

The following definition reflects the hierarchy-theoretic motivation for this paper.

Definition 2. For I ⊆ D and A ⊆ ω, I is an A-hierarchy ideal iff for some countable ordinal α, MI = Lα[A]∩ ωω.

All hierarchy ideals are ideals, but not conversely.

Early in the game Spector knocked out the best sort of canonicity for upper bounds on ideals, proving that no set of degrees closed under jump has a least upper bound.