The impossibility of an effective theory of policy in a complex economy

theory of policy,dynamical system,computation universality,recursive rule,complex economy

Sraffa's Mathematical Economics - A Constructive Interpretation

The claim in this paper is that Sraffa employed a rigorous logic of mathematical reasoning in his book, Production of Commodities by Means of Commodities (PCC), in such a way that the existence proofs were constructive. This is the kind of mathematics that was prevalent at the beginning of the 19th century, which was dominated by the concrete, the constructive and the algorithmic. It is, therefore, completely consistent with the economics of the 19th century, which was the fulcrum around which the economics of PCC was conceived.Existence Proofs, Constructive Mathematics, Algorithmic Mathematics, Mathematical Economics, Standard System.

Resource Bounded Immunity and Simplicity

Revisiting the thirty years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we propose the k-immune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in the Proceedings of the 3rd IFIP International Conference on Theoretical Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France, August 23-26, 200

An Embedding into a Substructure of the r.e. Turing Degrees

Let $(I_m,\le)$ be the partial ordering of the $m$-introimmune r.e. Turing degrees. We wonder if such structure is an upper semi-lattice. We give a partial answer, by embedding some Boolean algebras into $(I_m,\le)$

On the proof-theoretic strength of monotone induction in explicit mathematics

AbstractWe characterize the proof-theoretic strength of systems of explicit mathematics with a general principle (MID) asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the Σ12-axiom of choice and Π12-comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a stable ordinal. In all cases, the exact strength depends on what forms of induction are admitted in the respective systems

a minimal pair of turing degrees

Let $P(A)$ be the following property, where $A$ is any infinite set of natural numbers: \begin{displaymath} (\forall X)[X\subseteq A\wedge |A-X|=\infty\Rightarrow A\not\le_m X]. \end{displaymath} Let $({\bf R}, \le)$ be the partial ordering of all the r.e. Turing degrees. We propose the study of the order theoretic properties of the substructure $({\bf S}_{m},\le_{{\bf S}_m})$, where ${\bf S}_{m}=_{\rm dfn}\{{\bf a}\in {\bf R}$: ${\bf a}$ contains an infinite set $A$ such that P(A) is true$\}$, and $\le_{{\bf S}_m}$ is the restriction of $\le$ to ${\bf S}_m$. In this paper we start by studying the existence of minimal pairs in ${\bf S}_{m}$

Osservazioni su autoriferimento e verità

The present essay deals with the fundamental role of self-referential notions in contemporary logic. As a special case study, we survey recent ideas and results in formal semantics