Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised 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 AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3

On the Relative Consistency Strength of Determinacy Hypothesis

For any collection of sets of reals C, let C-DET be the statement that all sets of reals in C are determined. In this paper we study questions of the form: For given C ⊆ C', when is C'-DET equivalent, equiconsistent or strictly stronger in consistency strength than C-DET (modulo ZFC)? We focus especially on classes C contained in the projective sets

Introduction to Q-Theory

Working in the context of Projective Determinacy (PD), we introduce and study in this paper a countable ∏^1_(2n+1) set of reals Q_(2n+l) and an associated real y^0_(2n+l) for each n ≥ 0 (real means element of ω^ω in this paper). Our theory has analytical (descriptive set theoretic) as well as set theoretic aspects, strongly interrelated with each other

The axiom of determinacy and the prewellordering property

Let ω = (0,1,2,...) be the set of natural numbers and R = ω^ω the set of all functions from ω into ω, or for simplicity reals. A product space is of the form X = X_1 x X_2 x ... x X_k, where X_1 = ω or R. Subsets of these product spaces are called pointsets. A boldface pointclass is a class of pointsets closed under continuous preimages and containing all clopen pointsets (in all product spaces)

Introduction to Q-theory

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