Let L be a computable vocabulary, let X_L be the space of L-structures with universe ω and let f:2^\omega \rightarrow X_L be a hyperarithmetic function such that for all x,y \in 2^\omega, if x \equiv _h y then f(x) \cong f(y). One of the following two properties must hold. (1) The Scott rank of f(0) is \omega _1^{CK} + 1. (2) For all x \in 2^\omega, f(x) \cong f(0).

We generalize two concepts from special cases of Polish group actions to the general case. The two concepts are elementary embeddability, from model theory, and analytic sets, from the usual descriptive set theory.

Let L be a countable language, let I be an isomorphism-type of countable L-structures, and let a∈2ω. We say that I is a-strange if it contains a computable-from-a structure and its Scott rank is exactly ω1a. For all a, a-strange structures exist. Theorem : If C is a collection of ℵ1 isomorphism-types of countable structures, then for a Turing cone of a’s, no member of C is a-strange.

Science is one faith and has one great commandment. The faith is simply that man can control his future in his own interest. In the face of the appalling discrepancy between the powers of the natural and the social sciences, a discrepancy that leads to visions of man's liquidation by man, the assertion is nevertheless made that the remedy for the consequences of science is more science. The long record of scientific victories in the past gives some warrant, let us (...) suppose, for faith in continued victory, but at bottom there is no compelling proof. There never has been, and, I make bold to say, there never will be. The role of scientist in the human drama always contains lines reminiscent of Prometheus; an utterly nonrational belief in ultimate triumph lends him strength to confront what to others may seem a hopeless future. The frequency with which it will appear in the references is not because the writer regards it as of major importance, but because it dovetails more neatly with this paper than do many other more significant books. Moreover, it has fairly full footnote and chapter bibliographies. The statements in the non-parenthetical part of this paragraph are good examples of the near-elliptical character of much of what is said here. Expansion is badly needed, but the reader must be referred to ch. vi of. The position represented is not new; many writers, among them William James, have held it in various ways.). (shrink)

We show that all the projective ordinals $\delta^1_n$ are supercompact through their supremum $\aleph_{\varepsilon 0}$, and a ways beyond.

We consider sparseness, smoothness and the Glimm-Effros Dichotomy for the restriction of a Borel equivalence relation on a Polish space to definable subsets of that space.