Let {Wk }k ≥ 0 be a standard enumeration of all recursively enumerable (r.e.) sets. If A is any class of r.e. sets, let θ A denote the index set of A, i.e., θA = {k ∣ Wk ∈ A}. The one-one degrees of index sets form a partial order ℐ which is a proper subordering of the partial order of all one-one degrees. Denote by ⌀ the one-one degree of the empty set, and, if b is the one-one degree of θB, denote by the one-one degree of . Let . Let {Ym }m≥0 be the sequence of index sets of nonempty finite classes of finite sets (classified in [5] and independently, in [2]) and denote by a m the one-one degree of Ym . As shown in [2], these degrees are complete at each level of the difference hierarchy generated by the r.e. sets. It was proved in [3] that, for each m ≥ 0,

(a) a m+1 and ā m+1 are incomparable immediate successors of a m and ā m, and

(b) .

For m = 0, since Y 0 = θ{⌀}, it follows from (a) that

(c) .

Hence it follows that

(d) {⌀, , a o, ā 0, a 1, ā 1 is an initial segment of ℐ.

With the definition of effective operations as stated in the paper cited above (this Journal, vol. 35 (1970), p. 217), the last step of Lemma 1 does not appear to extend to types ≥ 3. This may be rectified by altering the second clause of the definition of effective operation (loc. cit. p. 217) to read:

ℱ is an effective operation of type i + 2 with Gödel number f∈E i+2, if for every g∈E i+1 there is an effective operation of type i + 1 (not depending on ℱ and f) with Gödel number g such that ℱ() = p(f, g).

With this definition the proofs go through without alteration.

There are two other variants for representing an effective operation by higher type objects.

I. We adopt the definition of effective as given in the text, and require the terms to satisfy the additional condition

II. Here we add to the definition of effective operations given in the text the condition:

If ℱ and ℱ1 are effective operations of type i + 2 and for all effective operations of type i + 1, then ℱ = ℱ1.

We now require terms s, t ∈ fi to satisfy the additional condition

The proof of Lemmas 1 and 2 are easily modified to establish I and II.

Perhaps II may be regarded as giving the most natural characterisation since in this case all terms represent continuous functionals.

A constructive version of Morse set theory is given, based on Heyting's predicate calculus and with countable rather than full choice. An elaboration of the method of [5] is used to show that the theory is combinator-realizable in the sense defined there. The proof depends on the assumption of the syntactic consistency of the theory.

The method is introduced by first treating a subtheory without countable choice of foundation.

It is intended that the work can be read either classically or constructively, though whether the word constructive is correctly used as a description of either the theory or the metatheory is of course a matter of opinion.

R. Parikh has shown that in the predicate calculus without function symbols it is possible to decide whether or not a given formula A is provable in a Hilbert-style proof of k lines. He has also shown that for any formula A(x 1, …, xn ) of arithmetic in which addition and multiplication are represented by three-place predicates, {(a 1, …, an ): A(a 1, …, an ) is provable from the axioms of arithmetic in k lines} is definable in (N,′, +,0). See [2].

In §1 of this paper it is shown that if S is a definable set of n-tuples in (N,′, +, 0), then there is a formula A(x 1, …, xn ) and a number k so that (a 1 …, a n) ∈ S if and only if A(a 1 …, an ) can be proved in a proof of complexity k from the axioms of arithmetic. The result does not depend on which formalization of arithmetic is used or which (reasonable) measure of proof complexity. In particular, this gives a converse to Parikh's result.

In §II, a measure of proof complexity is given which is based on counting the quantifier steps in semantic tableaux. The idea behind this measure is that A is k provable if in the attempt to construct a counterexample one meets insurmountable difficulties in the definition of the appropriate Skolem functions over sets of cardinality ≤ k. A method is given for deciding whether or not a sentence A in the full predicate calculus is k provable. The question for the full predicate calculus and Hilbert-style systems of proof remains open.

The purpose of this paper is to show that arithmetically minimal systems of notations can be constructed which provide notations for all ramified analytical ordinals (all the ordinals in the minimum β-model for analysis). This is a much larger section of the second number class than the Church-Kleene constructive ordinals (although still only an initial segment of the ordinals). Arithmetic minimality means that if H is an “H-set” associated with an ordinal α in our system and H′ is an H-set associated with the same ordinal α in an arbitrary system of notations S, then H is arithmetical in H′. Thus the arithmetical degrees associated with ordinals in our system are as low as possible.

In order to clarify the structure of degrees of unsolvability and, more generally, to gain a deeper insight into the power set of the integers, coarser but neater classifications than the structure of Turing degrees have been sought. Several hierarchies of sets of integers have been studied, each of which organizes a certain class of sets (or their degrees of unsolvability) into a well-ordering of levels with increasing complexity of nonrecursiveness appearing at each new level. The best known of these hierarchies is the Kleene hierarchy of arithmetical sets.

The results that follow are intended to be understood as informal counterparts to formal theorems of Zermelo-Fraenkel set theory with choice. Basic notation not explained here can usually be found in [5]. It will also be necessary to assume a knowledge of the fundamentals of boolean and generic extensions, in the style of Jech's monograph [3]. Consistency results will be stated as assertions about the existence of certain complete boolean algebras, B, C , etc., either outright or in the sense of a countable standard transitive model M of ZFC augmented by hypotheses about the existence of various large cardinals. Proofs will usually be phrased in terms of the forcing relation ⊩ over such an M, especially when they make heavy use of genericity. They are then assertions about Shoenfield-style P -generic extensions M(G), in which the ‘names’ are required without loss of generality to be elements of M B = (V B )M, B is the boolean completion of P in M (cf. [3, p. 50]: the notation there is RO(P)), the generic G is named by Ĝ ∈ M B such that (⟦p ∈ Ĝ⟧ B = p and (cf. [11, p. 361] and [3, pp. 58–59]), and for p ∈ P and c 1, …, cn ∈ M B , p ⊩ φ(c 1, …, cn ) iff ⟦φ(c 1, …, cn )⟧B ≥ p (cf. [3, pp. 61–62]).

Some prior acquaintance with large cardinal theory is also needed. At this writing no comprehensive introductory survey is yet in print, though [1], [10], [12]and [13] provide partial coverage. The scheme of definitions which follows is intended to fix notation and serve as a glossary for reference, and it is followed in turn by a description of the results of the paper. We adopt the convention that κ, λ, μ, ν, ρ and σ vary over infinite cardinals, and all other lower case Greek letters (except χ, φ, ψ, ϵ) over arbitrary ordinals.

We study in this paper the projective ordinals , where = sup{ξ: ξis the length of a Δn 1prewellordering of the continuum}. These ordinals were introduced by Moschovakis in [8] to serve as a measure of the “definable length” of the continuum. We prove first in §2 that projective determinacy implies , for all even n > 0 (the same result for odd n is due to Moschovakis). Next, in the context of full determinacy, we partly generalize (in §3) the classical fact that δ1 1 = ℵ1 and the result of Martin that δ3 1 = ℵω+1 by proving that , where λ2n+1 is a cardinal of cofinality ω. Finally we discuss in §4 the connection between the projective ordinals and Solovay's uniform indiscernibles. We prove among other things that ∀α(α # exists) implies that every δn 1 with n ≥ 3 is a fixed point of the increasing enumeration of the uniform indiscernibles.

A finite set of tiles (unit squares with colored edges) is said to tile the plane if there exists an arrangement of translated (but not rotated or reflected) copies of the squares which fill the plane in such a way that abutting edges of the squares have the same color. The problem of whether there exists a finite set of tiles which can be used to tile the plane but not in any periodic fashion was proposed by Hao Wang [9] and solved by Robert Berger [1]. Raphael Robinson [7] gives a more detailed history and a very economical solution to this and related problems; we will assume that the reader is familiar with §4 of [7]. In 1971, Dale Myers asked whether there exists a finite set of tiles which can tile the plane but not in any recursive fashion. If we make an additional restriction (called the origin constraint) that a given tile must be used at least once, then the positive answer is given by the main theorem of this paper. Using the Turing machine constructed here and a more complicated version of Berger and Robinson's construction, Myers [5] has recently solved the problem without the origin constraint.

Given a finite set of tiles T 1, …, Tn , we can describe a tiling of the plane by a function f of two variables ranging over the integers. f(i, j) = k specifies that the tile Tk is to be placed at the position in the plane with coordinates (i, j). The tiling will be said to be recursive if f is a recursive function.

We show that there is a finite set of tiles which can tile the plane but not in any recursive way. This answers a natural sequel to Hao Wang's problem of the existence of a finite set of tiles which can tile the plane but not in any periodic way. In the proof, an elaboration of Robinson's method of transforming origin-constrained problems into unconstrained problems is applied to Hanf's origin-constrained tiling of Part I. We will assume familiarity with §§2, 3, and 7 of [3].

Following Robinson, we will mark the edges of our tiles with symbols and configurations of arrow heads and tails as well as with colors. The matching condition for abutting tiles will be that the symbols on adjacent edges must be identical, every arrow head must match with a tail, every tail with a head, and the colors must be the same. This is, of course, mathematically equivalent to the original condition which involved only colors. A tiling of the plane by a set of tiles is a covering of the plane with translated copies of the tiles such that adjacent edges of abutting tiles satisfy the above matching condition. A set of tiles is consistent if the plane can be tiled by the set. A set of tiles with a designated origin tile is origin-consistent if there is a tiling of the plane with the origin tile at the center.

A square block of tiles is a tiling if every pair of abutting tiles satisfies the above matching condition. If an origin tile has been designated, a block of tiles is an origin-constrained tiling if it is a tiling with the origin tile at the center. Two blocks have the “same” center row if the blocks are of the same size and have identical center rows or if the smaller block's center row is a centered segment of the larger block's center row.

In this paper we will present an application of generalized recursion theory to (noncombinatorial) set theory. More precisely we will combine a priority argument in α-recursion theory with a forcing construction to prove a theorem about the interdefinability of certain subsets of admissible ordinals.

Our investigation was prompted by G. Sacks and S. Simpson asking [6] if it is obvious that there are, for each Σn-admissible α, Σn (over L α ) subsets of α which are Δn -incomparable. If one understands “B is Δn in C” to mean that there are Σn /L α reduction procedures which put out B and when one feeds in C, then the answer is an unqualified “yes.” In this sense “Δn in” is a direct generalization of “α-recursive in” (replace Σ1 by Σn in the definition) and so amenable to the methods of [7, §§3, 5]. Indeed one simply chooses a complete Σn−1 set A and mimics the construction of [6] as modified in [7, §5] to produce two α-A-r.e. sets B and C neither of which is α-A-recursive in the other. By the remarks on translation [7, §3] this will immediately give the desired result for this definition of “Δn in.”

There is, however, the more obvious and natural notion of “Δn in” to be considered: B is Δn in C iff there are Σn and Πn formulas of ⟨L α , C⟩ which define B.

It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable ℵ1-categorical theory.

The language of a theory T is denoted by L(T). All languages will be presumed countable. An x-type of T is a set of formulas with free variables x, which is consistent with T and which is maximal in this property. A formula with free variables x is complete if there is exactly one x-type containing it. A type is principal if it contains a complete formula. A countable model of T is prime if it realizes only principal types. Vaught has shown that a complete countable theory can have at most one prime model up to isomorphism.

If T is a decidable theory, then the decision procedure for T equips L(T) with an effective counting. Thus the formulas of L(T) correspond to integers. The integer a formula φ(x) corresponds to is generally called the Gödel number of φ(x) and is denoted by ⌜φ(x)⌝. The usual recursion theoretic notions defined on the set of integers can be transferred to L(T). In particular a type Γ is recursive with index e if {⌜φ⌝.; φ ∈ Γ} is a recursive set of integers with index e.

An operator Γ mapping P(ω) to itself is inductive if Γ(A) ⊇ A for all A. For such an inductive operator Γ we define {Γα : α ∈ ORD} by letting Γ = ⌀, Γα + 1 = Γ(Γα ) for all α, and Γβ = ⋃{Γα : α < β} for limit ordinals β. The closure ordinal ∣Γ∣ of Γ is the least ordinal α such that Γα+1 = Γα and the closure is Γ∣Γ∣.

Let be a class of operators over the natural numbers. The closure ordinal ∣∣ of is the supremum of {∣Γ∣: Γ is an inductive operator and }. The closure algebra generated by is {A ⊆ ω: A is 1-1 reducible to for some inductive operator }. The inductive algebra generated by is {Γα : Γ is an inductive operator in and α < ∣Γ∣}.

For A ∈ P(ω), is the supremum of the ordinals of well-orderings recursive in A. For , let be the supremum of . For example, it is well known that ω() = ω(∆1 0 = ω 1, ω(Π1 1) = ω 1 and ω(Δn 1) = δn 1 for n > 1 (the latter by definition).

The ω-rule in the λ-calculus (or, more exactly, the λK-β, η calculus) is

In [1] it was shown that this rule is consistent with the other rules of the λ-calculus. We will show the rule cannot be derived from the other rules; that is, we will give closed terms M and N such that MZ = NZ can be proved without using the ω-rule, for each closed term Z, but M = N cannot be so proved. This strengthens a result in [4] and answers a question of Barendregt.

The language of the λ-calculus has an alphabet containing denumerably many variables a, b, c, … (which have a standard listing e 1, e 2, …), improper symbolsλ, ( , ) and a single predicate symbol = for equality.

Terms are defined inductively by the following:

(1) A variable is a term.

(2) If M and N are terms, so is (MN); it is called a combination.

(3) If M is a term and x is a variable, (λx M) is a term; it is called an abstraction.

We use ≡ for syntactic identity of terms.

If M and N are terms, M = N is a formula.

BV(M), the set of bound variables in M, and FV(M), its free variables, are defined inductively by

A term M is closed iff FV(M) = ∅.

This paper refines some results of Barwise [1] as well as answering the open question posed at the end of [1] about the Hanf number of positively. We conclude by showing that the existence of a Hanf bound for cannot be proved in the natural formally intuitionistic set theories with bounded predicates decidable of [3], [4] and [5].

All notation not explained below is taken from [1]. In the Appendix, we give the axioms of ZF0, ZF1, and T in full. We remark that an important point about the axiom of foundation was not emphasized in [1]. This axiom was intended to be the axiom scheme (∀x)((∀y ∈ x)(A(y)) → A(x)) → (∀x)(A(x)), where y does not occur in A = A(x), instead of the more customary (∀x)(∀y)(y ∈ x → (∃z ∈ x)(∀w ∈ z) (w ∉ x)). This is of no consequence in the presence of full separation, but is vital when considering ZF0 and the T below, for with the customary form of foundation, these cannot even prove the existence of Rω+ω .

In [1], a proof of the following is sketched.