Constructible set

Abstract: Publisher Summary This chapter presents Church's thesis, which is the reducibility axiom for constructive mathematics. The chapter focusses on the development of the theory of abstract constructions by use of the notion of graspable domain and a connection between Scott's model and the theory of lawless sequences. The chapter considers the abstract notion of arbitrary subset and the power set operation (collecting all subsets of a given set); correspondingly, the notion of constructible set, obtained by iterating the process of collecting the subsets that are definable by means of formulae in the language of set theory, using names for elements of the given set. The chapter also considers the (abstract) notion of constructive function (with integral arguments and values) as understood in intuitionistic mathematics; correspondingly, the notion of recursive function, defined by means of recursion equations. The notions and laws of mechanics (e.g. the equations for incompressible fluids) were derived from general qualitative experience, not from delicate measurements, which can be stated only in terms of the theoretical notions.

Abstract: Let $\mathbb K$ be an algebraically closed field, let $X$ be a $\mathbb K$-variety, and let $X(\mathbb K)$ be the set of closed points in $X$. A constructible set $C$ in $X(\mathbb K)$ is a finite union of subsets $Y(\mathbb K)$ for subvarieties $Y$ in $X$. A constructible function $f:X(\mathbb K)\rightarrow\mathbb Q$ has $f(X(\mathbb K))$ finite and $f^{-1}(c)$ constructible for all $c e 0$. Write CF$(X)$ for the vector space of such $f$. Let $\phi:X\rightarrow Y$ and $\psi: Y\rightarrow Z$ be morphisms of ${\mathbb C}$-varieties. MacPherson defined a linear pushforward CF$(\phi):{\rm CF}(X)\rightarrow{\rm CF}(Y)$ by ?integration? with respect to the topological Euler characteristic. It is functorial, that is, CF$(\psi\circ\phi)={\rm CF}(\psi)\circ{\rm CF}(\phi)$. This was extended to $\mathbb K$ of characteristic zero by Kennedy. This paper generalizes these results to $\mathbb K$-schemes and Artin $\mathbb K$-stacks with affine stabilizer groups. We define the notions of Euler characteristic for constructible sets in $\mathbb K$-schemes and $\mathbb K$-stacks, and pushforwards and pullbacks of constructible functions, with functorial behaviour. Pushforwards and pullbacks commute in Cartesian squares. We also define pseudomorphisms, a generalization of morphisms well suited to constructible functions problems

Abstract: In the proof of the consistency of the Continuum Hypothesis and the Axiom of Choice with the other axioms of set theory, Godel [ l ] introduced the notion of a constructible set and showed that the constructible sets form a model for set theory. These sets are intuitively those which can be reached by means of a transfinite sequence of several simple operations. He then showed that the Axiom of Choice and Continuum Hypothesis held in the collection of constructible sets. If the original universe of sets is sufficiently rich in ordinal numbers, it will follow that every set is constructible, in which case we say that the Axiom of Constructibility is satisfied. This axiom implies the two axioms previously mentioned. However, from one point of view it may seem that this notion of constructibility does not intuitively correspond to what is meant by constructive since it may happen that all sets in the universe are constructive. In this paper we show that a more restricted notion of "construction" will yield a class of sets which form a minimal model for set theory. In this manner we prove the consistency of a stronger form of the Axiom of Constructibility. We observe that the idea of a minimal collection of objects satisfying certain axioms is well known in mathematics, for example, in group theory one often considers the subgroup generated by a collection of elements, and in measure theory we define the Borel sets as the smallest y) holds, then there exists a set B consisting of precisely those y. Since much of the proofs of the theorems we state follow quite closely the arguments of [ l ] , we shall be rather brief. Our main result is

Abstract: Let K be an algebraically closed field, X a K-scheme, and X(K) the set of closed points in X. A constructible set C in X(K) is a finite union of subsets Y(K) for finite type subschemes Y in X. A constructible function f : X(K) --> Q has f(X(K)) finite and f^{-1}(c) constructible for all nonzero c. Write CF(X) for the Q-vector space of constructible functions on X. Let phi : X --> Y and psi : Y --> Z be morphisms of C-varieties. MacPherson defined a Q-linear "pushforward" CF(phi) : CF(X) --> CF(Y) by "integration" w.r.t. the topological Euler characteristic. It is functorial, that is, CF(psi o phi)=CF(psi) o CF(phi). This was extended to K of characteristic zero by Kennedy. This paper generalizes these results to K-schemes and Artin K-stacks with affine stabilizers. We define notions of Euler characteristic for constructible sets in K-schemes and K-stacks, and pushforwards and pullbacks of constructible functions, with functorial behaviour. Pushforwards and pullbacks commute in Cartesian squares. We also define "pseudomorphisms", a generalization of morphisms well suited to constructible functions problems.