153 research outputs found
Reverse mathematics and well-ordering principles
The paper is concerned with generally Pi^1_2 sentences of the form 'if X is well ordered then f(X) is well ordered', where f is a standard proof theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded omega-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we shall focus on the well-known psi-function which figures prominently in so-called predicative proof theory. However, the approach taken here lends itself to generalization in that the techniques we employ can be applied to many other proof-theoretic functions associated with cut elimination theorems. In this paper we show that the statement 'if X is well ordered then 'X0 is well ordered' is equivalent to ATR0. This was first proved by Friedman, Montalban and Weiermann [7] using recursion-theoretic and combinatorial methods. The proof given here is proof-theoretic, the main techniques being Schuette's method of proof search (deduction chains) [13], generalized to omega logic, and cut elimination for infinitary ramified analysis
A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice
We present a Kleene realizability semantics for the intensional level of the
Minimalist Foundation, for short mtt, extended with inductively generated
formal topologies, Church's thesis and axiom of choice. This semantics is an
extension of the one used to show consistency of the intensional level of the
Minimalist Foundation with the axiom of choice and formal Church's thesis in
previous work. A main novelty here is that such a semantics is formalized in a
constructive theory represented by Aczel's constructive set theory CZF extended
with the regular extension axiom
Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman
The paper proves a conjecture of Solomon Feferman concerning the indefiniteness of the continuum hypothesis relative to a semi-intuitionistic set theory
Constructing the Constructible Universe Constructively
We study the properties of the constructible universe, L, over intuitionistic
theories. We give an extended set of fundamental operations which is sufficient
to generate the universe over Intuitionistic Kripke-Platek set theory without
Infinity. Following this, we investigate when L can fail to be an inner model
in the traditional sense. Namely, we show that over Constructive
Zermelo-Fraenkel (even with the Power Set axiom) one cannot prove that the
Axiom of Exponentiation holds in L.Comment: 26 pages. Revised following referee's recommendation
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