Counting models of set theory

Set Theory: Counting the Uncountable

If f : X → Y is a bijection, then for every y ∈ Y there is a unique x ∈ X such that f(x) = y. That is, f matches up all the elements of X and Y in a 1–1 fashion. We can thus consider X and Y to have the same number of elements. If f : X → Y is an injection, then f matches up all of the elements of X with only some of the elements of Y , so we can consider X to be smaller than Y . This motivates...

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 ...