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

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

posted 2022-03-11 | tags: set theory