[[Axiomatic set theory]]
# Material set theory

A **material set theory** is a set theory based on a global membership relation $\in$
where sets are characterized by $\in$ and propositional equality.
These theories introduce a [[Cumulative hierarchy]].
[[F. William Lawvere]] describes such set theories as prioritizing [[Substance]] over [[Form]].


## Setup

Unless otherwise specified, we deploy [[1st-order logic]] on a universe $\mathcal{W}$ of objects with the primitive notions[^2006]

- $x = y$ iff $x$ is the same object as $y$;
- $\shood(x)$ iff $x$ is a set;
- $x \in y$ iff $\shood(y)$ and $x$ is a member of $y$;

where if there exists an object in $\mathcal{W}$ that is not a set it is called an [[Urelement]].
While most treatments do without urelements by considering only [[Pure set|pure sets]], these notes allow for their existence unless otherwise stated,
which occasionally complicates the statements of axioms somewhat.

  [^2006]: 2006\. [[Sources/@moschovakisNotesSetTheory2006|Notes on set theory]], pp. 23ff

## Possible systems

- Sets alone
  - [[ZF]]
  - [[ZFC]]
  - [[NFU]]
- Sets and classes
  - [[NBG]]
  - [[ML (set theory)]]
- Small and large sets
  - [[TG]]

## Possible axioms and axiom schemata

- [[Axiom of Extensionality]]
- [[Axiom of Purity]]
- [[Emptyset Axiom]]
- [[Axiom of Pairing]]
- [[Axiom of Union]]
- [[Specification Axiom Schema]]
- [[Powerset Axiom]]
- [[Axiom Schema of Replacement]]

### Infinity and large cardinals

- [[Axiom of Infinity]]

### Foundation

- [[Axiom of Foundation]]
- [[Aczel's Antifoundation Axiom]]

### Choices

- [[Axiom of Choice]]
- [[Axiom of Dependent Choice]]
- [[Countable Axiom of Choice]]

### Classes

- [[Elementhood Relation Class Axiom]]
- [[Axiom of Intersection#Axiom of Intersection for Classes]]
- [[Complement Axiom for classes]]
- [[Universal Relation Axiom]]
- [[Axioms of Permutation for classes]]
- [[Axiom of Subsets]]
- [[Axiom of Replacement for classes]]

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