ZFC Axioms

notes
The Zermelo-Frankel axioms of set theory: extension, foundation, pairing, union, power, infinity, separation, replacemeht, choice.
Author

Stephen J. Mildenhall

Published

2022-03-11

ZFC Axioms of Set Theory

Main Act 3

  • Extension \(\forall x\ \forall y\ \forall z\ (z \in x \leftrightarrow z \in y)\rightarrow x=y\)
  • Foundation \(\forall x\ [\exists y\ (y\in x)\rightarrow \exists y\ (y\in x \wedge \neg\exists z(z\in x \wedge z\in y))]\)
  • Pairing \(\forall a\ \forall b\ \exists x\ [a\in x \wedge b\in x]\)
  • Union \(\forall X\ \exists U\ [\forall Y\ \forall x\ (x\in Y \wedge Y \in X)\rightarrow x\in U]\)
  • Power \(\forall X\ \exists P\ \forall z\ [z\subset X\rightarrow z\in P]\)
  • Infinity \(\exists x\ [\forall z\ (z=\emptyset)\rightarrow z\in x \wedge \forall x\in x\forall z\ (z=S(x)\rightarrow z\in x)]\)
  • Separation \(\forall x\ \forall p\ \exists y[\forall u(u\in y\leftrightarrow(u\in x\wedge \phi(u,p)))]\)
  • Replacement \(\forall A\ \forall p\ [\forall x\in A\ \exists !y\ \phi(x, y, p)\rightarrow\exists Y\ \forall x\in A\ \exists y\in Y\phi(x, y,p)]\)
  • Choice \(\forall X[\forall x\in X(x\not=\emptyset) \wedge \forall x\in X\forall y\in X(x=y\vee x\cap y=\emptyset)]\rightarrow\exists S\forall x\in X\exists !z(z\in S\wedge z\in x)\)

Warm Up

  • \(x\subset X\leftrightarrow \forall z(z\in x\rightarrow z\in X)\)
  • \(S(x) = x\cup \{x\}\)
  • \(\emptyset = \forall x(x!=x)\)
  • \(\exists !x\phi(x)\leftrightarrow \exists x\phi(x)\wedge \forall x\forall y(\phi(x)\wedge \phi(y)\rightarrow x=y)\)

Main Act

  • \(\forall x\forall y[\forall z(z \in x \leftrightarrow z \in y)\rightarrow x=y]\)
  • \(\forall x[\exists y(y\in x)\rightarrow \exists y(y\in x \wedge \neg\exists z(z\in x \wedge z\in y))]\)
  • \(\forall a\forall b\exists x[a\in x \wedge b\in x]\)
  • \(\forall X\exists U[\forall Y\forall x(x\in Y \wedge Y \in X)\rightarrow x\in U]\)
  • \(\forall X\exists P\forall z[z\subset X\rightarrow z\in P]\)
  • \(\exists x[\forall z(z=\emptyset)\rightarrow z\in x \wedge \forall x\in x\forall z(z=S(x)\rightarrow z\in x)]\)
  • \(\forall x\forall p\exists y[\forall u(u\in y\leftrightarrow(u\in x\wedge \phi(u,p)))]\)
  • \(\forall A\forall p[\forall x\in A\exists !y\phi(x, y, p)\rightarrow\exists Y\forall x\in A\exists y\in Y\phi(x, y,p)]\)
  • \(\forall X[\forall x\in X(x\not=\emptyset) \wedge \forall x\in X\forall y\in X(x=y\vee x\cap y=\emptyset)]\rightarrow\exists S\forall x\in X\exists !z(z\in S\wedge z\in x)\)