♣ Rings . A ring is a non-empty set R with two binary operations ( + , · ) , called addition and multiplication, respectively satisfying : Axiom 1. Closure ( + ) : ∀x, y ∈ R , x + y ∈ R . Axiom 2. Commutative ( + ) : For every x, y ∈ R , x + y = y + x . Axiom 3. Associative ( + ) : ∀x, y, z ∈ R , x + (y + z) = (x + y) + z . Axiom 4. Neutral ( + ) : ∃ θ ∈ R , such that ∀x ∈ R, x + θ = θ + x = x ...

The following notion of forcing was introduced by Grigorieff [2]: Let I ⊂ ω be an ideal, then P is the set of all functions p : ω → 2 such that dom(p) ∈ I. The usual Cohen forcing corresponds to the case when I is the ideal of finite subsets of ω. In [2] Grigorieff proves that if I is the dual of a p-point ultrafilter, then ω1 is preserved in the generic extension. Later, when Shelah introduced...

We define a finitary model of first-order Peano Arithmetic in which quantification is interpreted constructively in terms of Turing-computability, and show that it is inconsistent with the standard interpretation of PA. 1 Hilbert’s interpretation of quantification Hilbert interpreted quantification in terms of his ε-function as follows [Hi27]: IV. The logical ε-axiom 13. A(a)→ A(ε(A)) Here ε(A)...

First, Padoa’s principle is used to prove the non-definability of the fundamental qualitative concepts of comparative probability, independence and comparative uncertainty in terms of each other. Second, the qualitative axioms of uncertainty leading to an entropy representation are new. Third, a qualitative random-variable axiomatization of these concepts is given, but the random variables are ...

A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of zfc has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion v=hod that...