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: $$\forall X(0 \in X \land \forall x \in \N(x \in X)) \rightarrow \N \subseteq X$$
 * 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) $$\forall x \in \N(S(x) \neq 0)$$
 * 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) $$\forall x \in \N(x < 0 = \bot)$$
 * 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 $$\Delta^0_0$$, $$\Sigma^0_1$$, $$\Delta^0_1$$ and arithmetical formulas φ(n) with a free variable n and possibly other free variables, but not the variable Z, $$\exists Z \forall n (n\in Z \leftrightarrow \varphi(n))$$