Axiomatic set theory

Material set theory

A material set theory is a set theory based on a global membership relation ∈ where sets are characterized by ∈ 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 W of objects with the primitive notions¹

• 𝑥 =𝑦 iff 𝑥 is the same object as 𝑦;
• 𝔐⁡(𝑥) iff 𝑥 is a set;
• 𝑥 ∈𝑦 iff 𝔐⁡(𝑦) and 𝑥 is a member of 𝑦;

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

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 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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tidy | en | SemBr

Footnotes

1. 2006. Notes on set theory, pp. 23ff ↩