We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(x(p) − 1)-coloring, where x(p) is the least number of colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs.

The extreme interest of set theorists in the notion of “closed unbounded set” is epitomized in the following well-known theorem:

Theorem A. For any regular cardinal κ > ω, the intersection of any two closed unbounded subsets of κ is closed and unbounded.

The proof of this theorem is easy and in fact yields a stronger result, namely that for any uncountable regular cardinal κ the intersection of fewer than κ many closed unbounded sets is closed and unbounded. Thus, if, for κ a regular uncountable cardinal, we let denote {A ⊆ κ ∣ A contains a closed unbounded subset}, then, for any such κ, is a κ-additive nonprincipal filter on κ.

Now what about the possibility of being an ultrafilterκ It is routine to see that this is impossible for κ > ℵ1. However, for κ = ℵ1 the situation is different. If were an ultrafilter, ℵ1 would be a measurable cardinal. As is well-known this is impossible if we assume the axiom of choice; however if ZF + “there exists a measurable cardinal” is consistent, then so is ZF + “ℵ1 is a measurable cardinal” [2]. Furthermore, under the assumption of certain set theoretic axioms (such as the axiom of determinateness or various infinite exponent partition relations) can be proven to be an ultrafilter. (See [3] and [5].)

The concept of an α-space was introduced and studied by Dekker in [1] and [2] from which we take our terminology and notation. The reader is assumed to be familiar with their contents. In this paper we are interested in the following conjecture which appears in [2, p. 493]:

(*) S ⊕ C = V and S ∥ C and V an α-space ⇒ both S and C are α-spaces.

It can be simply rephrased as:

(**) The two components of an α-decomposition of an α-space are both α-spaces.

In either form it is relevant to the ideas of α-subspace and α-homomorphism. Dekker [2, T5] has established (*) in the case S (or C) is an isolic α-space. In this paper we will prove (*) in two specific cases, one of which is a generalization of Dekker's result.

An ordinal in a model of set theory is truly countable if its set of predecessors is countable in the real world. We classify the order types of the sets of truly countable ordinals. Models with indiscernibles and other related results are discussed.

For the first Gödel incompleteness theorem, the existence in a formal system of arithmetic L of a sentence which is neither provable nor refutable, all that is required of the formula Th(x) of L used to express the notion ‘x is the g.n. (gödel number) of a theorem of L’ is mere numeralwise correctness, i.e. that for a numeral n, Th(n) is provable in L iff n is the g.n. of a theorem of L. It is well known that much more is needed for the second Gödel incompleteness theorem, the unprovability in L of the formula Con =df ¬(∃y, z)(Th(y) ∧ Th(z) ∧ neg(z, y)), which (if neg expresses negation) expresses the consistency of L. Conditions sufficient for this second theorem, more or less as stated by Hilbert-Bernays [1, p. 286] and elegantly formulated by Löb [2] may with a cavalier disregard for the distinction between use and mention be stated thus: The result of the first incompleteness theorem: there is a sentence G such that ⊢G ↔ ¬Th G), together with, if ⊢A then ⊢Th A, if ⊢(A → B) then ⊢(Th A → Th B), ⊢(Th A → Th Th A). On the other hand Feferman [3], Kreisel [4, p. 154] and Jeroslow [9] have given examples of systems and consistency formulae, based on numeralwise correct formulae Th(x), which are provable within the system.

One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {Rn }n∈ω called requirements. If n < m, requirement Rn is given priority over Rm and action taken for Rm at some stage s may at a later stage t > s be undone for the sake of Rn thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Post's problem and is characterized by the fact that each requirement is injured at most finitely often.

Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity.

The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own.

The notions of recursively saturated and resplendent models grew out of the study of admissible sets with urelements and admissible fragments of L ω1ω, but, when applied to ordinary first order model theory, give us new tools for research and exposition. We will discuss their history in §3.

The notion of saturated model has proven to be important in model theory. Its most important property for applications is that if , are saturated and of the same cardinality then = iff ≅ . See, e.g., Chang-Keisler [3]. The main drawback is that saturated models exist only under unusual assumptions of set theory. For example, if 2κ = κ+ then every theory T of L has a saturated model of power κ+. (Similarly, if κ is strongly inaccessible, then every T has a saturated model of power κ.) On the other hand, a theory T like Peano arithmetic, with types, cannot have a saturated model in any power κ with ω ≤ κ ≤ .

One method for circumventing these problems of existence (or rather non-existence) is the use of “special” models (cf. [3]). If κ = Σλ<κ2λ, κ < ω, then every theory T of L has a special model of power κ. Such cardinals are large and, themselves, rather special. There are definite aesthetic objections to the use of these large, singular models to prove results about first order logic.

Since the late 1940's model theory has found numerous applications to algebra. I would like to indicate some of the points of contact between model theoretic methods and strictly algebraic concerns by means of a few concrete examples and typical applications.

§1. The Lefschetz principle. Algebraic geometry has proved to be a fruitful source of model theoretic ideas. What exactly is algebraic geometry? We consider a field K, and let Kn be the set of n-tuples (a 1 … an ) with coordinates ai in K. Kn is called affine n-space over K. Fix polynomials p 1 …, pk in K[x 1, …, xn ] and define

that is V(p 1 …, pk ) is the locus of common zeroes of the pi in Kn . We call V(Pi …, Pk ) the algebraic variety determined by p 1, …, pk . With this terminology we may say:

Algebraic geometry is the study of algebraic varieties defined over an arbitrary field K. This definition lacks both rigor and accuracy, and we will indicate below how it may be improved.

So far we have placed no restrictions on the base field K. Following Weil [4] it is convenient to start with a so-called “universal domain”; in other words take K to be algebraically closed and of infinite transcendence degree over the prime field. Any particular field can of course be embedded in such a universal domain.