Robert M. Solovay

Extremes in the degrees of inferability

Most theories of learning consider inferring a function f from either (1) observations about f or, (2) questions about f. We consider a scenario whereby the learner observes f and asks queries to some set A. If I is a notion of learning then I[A] is the set of concept classes I-learnable by an inductive inference machine with oracle A. A and B are I-equivalent if I[A] = I[B]. The equivalence classes induced are the degrees of inferability. We prove several results about when these degrees are trivial, and when the degrees are omniscient (i.e., the set of recursive function is learnable).

Recursively Enumerable Sets Modulo Iterated Jumps and Extensions of Arslanov's Completeness Criterion

Let W e be the e th recursively enumerable (r.e.) set in a standard enumeration. The fixed point form of Kleenes recursion theorem asserts that for every recursive function f there exists e which is a fixed point of f in the sense that W e = W f ( e ) . In this paper our main concern is to study the degrees of functions with no fixed points. We consider both fixed points in the strict sense above and fixed points modulo various equivalence relations on recursively enumerable sets. Our starting point for the investigation of the degrees of functions without (strict) fixed points is the following result due to M. M. Arslanov [A1, Theorem 1] and known as the Arslanov completeness criterion. Proofs of this result may also be found in [So1, Theorem 1.3] and [So2, Chapter 12], and we will give a game version of the proof in §5 of this paper. Theorem 1.1 (Arslanov). Let A be an r.e. set. Then A is complete ( i.e. A has degree 0 ′) iff there is a function f recursive in A with no fixed point .

When oracles do not help

We study the effect of allowing an inductive inference machine access to an oracle. Specifically, we resolve the question (raised by Gasarch) of exactly which oracles increase learning power. In [7] it was shown that if A is 1-generic and A T K then access to oracle A does not increase learning power. Here we show that these are the only such oracles, i.e., we show that the following two statements are equivalent conditions on the non-recursive set A : (1) access to A does not increase learning power (i.e., EX = EX A ), (2) A is 1-generic and A ≥ T K. The proof uses a (rather complex) finite injury priority argument.

On Partitions into Stationary Sets

We shall apply some of the results of Jensen [4] to deduce new combinatorial consequences of the axiom of constructibility, V = L . We shall show, among other things, that if V = L then for each cardinal λ there is a set A ⊆ λ such that neither A nor λ – A contain a closed set of type ω 1 . This is an extension of a result of Silver who proved it for λ = ω 2 , providing a partial answer to Problem 68 of Friedman [2]. The main results of this paper were obtained independently by both authors. If λ is an ordinal, E is said to be Mahlo (or stationary) in λ, if λ – E does not contain a closed cofinal subset of λ. Consider the statements: (J 1 ) There is a class E of limit ordinals and a sequence C λ defined on singular limit ordinals λ such that (i) E ⋂ μ is Mahlo in μ for all regular > ω; (ii) C λ is closed and unbounded in λ; (iii) if γ C λ , then γ is singular, γ ∉ E and C γ = γ ⋂ C λ . For each infinite cardinal κ: (J 2,κ ) There is a set E ⊂ κ + and a sequence C λ (Lim(λ), λ + ) such that (i) E is Mahlo in κ + ; (ii) C λ is closed and unbounded in λ; (iii) if cf(λ) C λ (iv) if γ C λ then γ ∉ E and C γ = ϣ ⋂ C λ .

Definability of measures and ultrafilters

Nonprincipal ultrafilters are harder to define in ZFC, and harder to obtain in ZF + DC, than nonprincipal measures. The function μ from P(X) to the closed interval [0, 1] is a measure on X if μ. is finitely additive on disjoint sets and μ( X ) = 1. ( P is the power set.) μ is nonprincipal if μ ({ x }) = 0 for each x Є X . μ is an ultrafilter if Range μ= {0, 1}. The existence of nonprincipal measures and ultrafilters on any infinite X follows from the axiom of choice. Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo–Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on ω is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF + DC (dependent choice) in which no nonprincipal measure on ω exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF + DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a different method. Our construction will be sketched in 4.1.

Some new results on decidability for elementary algebra and geometry

Abstract We carry out a systematic study of decidability for theories (a) of real vector spaces, inner product spaces, and Hilbert spaces and (b) of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable.

Explicit Henkin sentences

On donne une formalisation precise de la notion de Hofstadter et on montre que des phrases de Henkin explicites existent

Strong measure zero and infinite games

We show that strong measure zero sets (in a

STRONG AXIOMS OF INFINITY AND ELEMENTARY EMBEDDINGS

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