User:Binary198/MST set theory

MST set theory is a set theory I created more as a joke to be the strongest axiomatic set theory, and it certainly achieved its goal! Since it isn't a true set theory, and was made more of a joke, I probably won't turn this into an actual Wikipedia article :)

Axioms

This theory has a LOT of (27) axioms! Here is a complete list:

1. Axiom of extensionality: $\forall x\forall y(\forall z(z\in x\leftrightarrow z\in y)\rightarrow x=y)$
2. Axiom of regularity: $\forall x(\exists a(a\in x)\rightarrow \exists y(y\in x\land \nexists z(z\in y\land z\in x)))$
3. Axiom schema of specification: $\phi$ is a formula in MST with all free variables among $x$, $w_{1}$, ..., $w_{n}$ ($y$ is not free in $\phi$). Then: $\forall z\forall w_{1}\forall w_{2}...\forall w_{n}\exists y\forall x(x\in y\leftrightarrow ((x\in z)\land \phi ))$
4. Axiom of pairing: $\forall x\forall y\exists z((x\in z)\land (y\in z))$
5. Axiom of union: $\forall x\forall y\exists z(\forall a\in x(a\in z)\land \forall b\in y(b\in z))$.
6. Axiom schema of replacement: $\phi$ is a formula in MST with all free variables among $x$, $y$, $A$, $w_{1}$, ..., $w_{n}$ ($B$ is not free in $\phi$). Then: $\forall A\forall w_{1}\forall w_{2}...\forall w_{n}(\forall x(x\in A\rightarrow \exists !y\phi )\rightarrow \exists B\forall x(x\in A\rightarrow \exists y(y\in B\land \phi )))$, where $\exists !$ is uniqueness quantification.
7. Axiom of infinity: $\exists X(\varnothing \in X\land \forall y(y\in X\rightarrow S(y)\in X))$
8. Axiom of powerset: $\forall x\exists y\forall z(z\subseteq x\rightarrow z\in y)$
9. Well-ordering theorem: $\forall X\exists R(R\;{\textrm {well-orders}}\;X)$
10. Axiom of induction: $\forall X(0\in X\land \forall x\in \mathbb {N} (x\in X))\rightarrow \mathbb {N} \subseteq X$
11. Axiom of empty set: $\exists \varnothing \forall x(x\notin \varnothing )$
12. Axiom schema of Σ₀-separation: For a set $X$ and Σ₀-formula $\varphi (x)$, $\exists Y\subseteq X(\forall x(\varphi (x(x\in Y))))$.
13. Axiom schema of Σ₁-separation: For a set $X$ and Σ₁-formula $\varphi (x)$, $\exists Y\subseteq X(\forall x(\varphi (x(x\in Y))))$.
14. Axiom schema of Σ₀-collection: For a set $X$ and Σ₀-formula $\varphi (x,y)$, $\forall x\exists y(\varphi (x,y))$ and $\forall u\exists v(\forall x\in u\exists y\in v(\varphi (x,y)))$.
15. Axiom schema of Σ₁-collection: For a set $X$ and Σ₁-formula $\varphi (x,y)$, $\forall x\exists y(\varphi (x,y))$ and $\forall u\exists v(\forall x\in u\exists y\in v(\varphi (x,y)))$.
16. $\forall x\in \mathbb {N} (S(x)\neq 0)$
17. $S(x)=S(y)\rightarrow x=y$
18. $\forall x(x=0\lor \exists y(S(y)=x))$
19. $\forall x(x+0=x)$
20. $\forall x,y(x+S(y)=S(x+y))$
21. $\forall x(x\cdot 0=0)$
22. $\forall x,y(x\cdot S(y)=(x\cdot y)+y)$
23. $\forall x\in \mathbb {N} (x<0=\bot )$
24. $\forall m\forall n(m<S(n)\leftrightarrow m<n\lor m=n)$
25. $\forall m\forall n((S(m)<n\lor S(m)=n)\leftrightarrow m<n)$
26. Full second-order induction schema: For all second-order arithmetic and $\Sigma _{1}$ formulas φ(n) with a free variable n and possible other free number or set variables (written m_• and X_•), $\forall m\forall X((\varphi (0)\land \forall n(\varphi (n)\rightarrow \varphi (Sn)))\rightarrow \forall n\varphi (n))$.
27. Comprehension axiom schema: For all $\Delta _{0}^{0}$, $\Sigma _{1}^{0}$, $\Delta _{1}^{0}$ and arithmetical formulas φ(n) with a free variable n and possibly other free variables, but not the variable Z, $\exists Z\forall n(n\in Z\leftrightarrow \varphi (n))$