Standard axioms of set theory

Overview

The standard axioms of set theory are the Zermelo–Fraenkel axioms together with the Axiom of Choice. They are expressed in a first-order language with one non-logical binary predicate $\in$, meaning “is a member of.” Quantifiers range over sets alone, including a so-called “empty set.”

Set theory is significant because the axioms of all other branches of mathematics can be interpreted in it, meaning those axioms can be understood as claims about sets (@2020clarke-doaneMorality, p. 39). ¹ This implies that if the axioms of set theory are consistent, then so are the other axioms of mathematics.

Related notes: Naive conception of set and Russell’s paradox

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Standard axioms of set theory

• Extensionality: Sets are identical if they have the same members. $(x)(y)(z)(z\in x⟺z\in y)⟹x=y))$
• Pairing: For any sets $z,w$, there exists a set containing exactly $z$ and $w$. $(z)(w)(\exists y)(x)(x\in y)⟺(x=z\vee x=w))$
• Union: For any set $z$, there is a set $\bigcup z$ containing exactly the members of $z$. $(z)(\exists y)(x)(x\in y⟺(\exists w)(w\in z\wedge x\in w))$
• Powerset: For any set $x$, there is a set $P(z)$ containing exactly the subsets of $z$. $(z)(\exists y)(x)(x\in y⟺(w)(w\in x⟹w\in z))$
• Subsets (Restricted Comprehension) Schema: For any set $z$ and any condition $\Phi$, there is a set that contains exactly those members of $z$ which satisfy $\Phi$. $(z)(\exists y)(x)(x\in y⟺(x\in z\wedge \Phi )),$ where $y$ is not free in $\Phi$.
• Infinity: There is a set containing $\varnothing$ and containing the successor of $x$ (i.e., $x\cup \{x\}$) whenever it contains $x$. $(\exists y)(((\exists x)(x\in y\wedge (z)(z\notin x)\wedge (x)(x\in y⟹(\exists z)(z\in y\wedge (w)(w\in z⟺(w\in x\vee w=x)))))$
• Foundation (Regularity) Schema: For any condition $\Phi$, if there is something that satisfies $\Phi$, then there is a minimal $x$ that does (i.e., an $x$ such that $\Phi$ and no $y\in x$ such that $\Phi$. $(\exists x)\Phi ⟹(\exists x)[\Phi \vee (y)(y\in x⟹\neg {\Phi }^{\ast })],$ where $\Phi$ does not contain $y$, and ${\Phi }^{\ast }$ is just $\Phi$ but contains $y$ whenever $\Phi$ contains free occurrences of $x$.

Remarks on the axioms:

• Pairing allows for the construction of singletons, which are unique by extensionality.
• The power set is philosophically interesting, since the axiom itself doesn’t tell you what its size should be.
• The Subsets Schema implies the existence of an empty set $\varnothing$ via the condition $x\ne x$.
• Without infinity, one can prove the existence of infinitely many things, but not the existence of a set of them.
• The Regularity Schema gives content to the conception of set: sets are not created ad-hoc, but every set occurs at some level of the hierarchy. Contra-posing gives a Principle of Set-Theoretic Induction.

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Footnotes

1. A theory $T$ interprets another $T^{\ast }$ if there exists a translation of $T^{\ast }$ into the language of $T$ such that, if $S^{\ast }$ is a theorem of $T^{\ast }$, then its translation $S$ is also a theorem of $T$, and the translation is reasonable. ↩