1. First order logic and the axioms of set theory 2 1.1. Syntax 2 1.2. Semantics 2 1.3. Completeness, compactness and consistency 3 1.4. Foundations of mathematics and the incompleteness theorems 3 1.5. The axioms 4 2. Review of basic set theory 5 2.1. Classes 5 2.2. Well-founded relations and recursion 5 2.3. Ordinals, cardinals and arithmetic 6 3. The consistency of the Axiom of Foundation 8 ...

An inexact Newton method is a generalization ofNewton's method for solving F(x) 0, F n _ _ in, in which, at the kth iteration, the step sk from the current approximate solution xk is required to satisfy a condition F (x) + F'(xk) sk _< F(x)II for a "forcing term" r/ [0, 1). In typical applications, the choice of the forcing terms is critical to the efficiency of the method and can affect robust...

Assume ZF(j) and there is a Reinhardt cardinal, as witnessed by the elementary embedding j:V→V. We investigate linear iterates (Nα,jα) of (V,j), their relationship to forcing definability, including that for each infinite α, every set set-generic over Nα, but Nα not set-ground. Morse-Kelley theory without Axiom Choice. prove existence super cardinals total affected small forcing. And if V[G] ha...

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...

In this paper we study the question assuming MA+¬CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalant to question of what is the additivity of Marczewski’s ideal s. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals. Introduction Let S be Sacks perfe...

Uncountable superperfect forcing is tree forcing on regular uncountable cardinals κ with κ<κ = κ , using trees in which the heights of nodes that split along any branch in the tree form a club set, and such that any node in the tree with more than one immediate extension has measure-one-many extensions, where the measure is relative to some κ-complete, nonprincipal normal filter (or p-filter) F...

Given a set F of graphs, a graph G is F-free if G does not contain any member of F as an induced subgraph. We say that F is a degree-sequence-forcing set if, for each graph G in the class C of F-free graphs, every realization of the degree sequence of G is also in C. We give a complete characterization of the degree-sequence-forcing sets F when F has cardinality at most two. © 2007 Wiley Period...