General models of set theory.

Models of Set Theory

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 ...

Leibnizian models of set theory

A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the LeibnizMycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF , T has a Leibnizian model iff T proves LM. Here we prove: Theorem A. Every complete theory T extending ZF + LM has 2 א 0 nonisomorphic countable Leibnizian models. Theorem B. ...

MODELS OF SET THEORY 1 Models

Slim Models of Zermelo Set Theory

Working in Z+KP , we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence 〈Aλ | λ a limit ordinal 〉 where for each λ, Aλ ⊆ 2, there is a supertransitive inner model of Zermelo containing all ordinals in which for every ...

Models of set theory from topological groups

The Boolean prime ideal theorem is a very slightly weaker version of the axiom of choice. In 2011, Andreas Blass showed that models of set theory (with atoms) in which the Boolean prime ideal theorem holds but the full axiom of choice fails correspond with a certain type of topological group. The path taken to prove this surprising result draws not just from set theory but also from combinatori...