User:Binary198/Binary198's notepad

This is my notepad, where I just write about maths stuff. Kinda like a talk page, except just for me. I could've put this on my website, but it doesn't support MathJax / LaTeX.

ID1

 * $$0 \in \N$$: 0 is a natural number.
 * $$\forall x(x = x)$$: Equality is reflexive.
 * $$\forall x \forall y = x \rightarrow x = y$$: Equality is symmetric.
 * $$\forall x \forall y \forall z(x = y \land y = z \rightarrow x = z)$$: Equality is transitive.
 * $$\forall x \forall y(x \in \N \land x = y \rightarrow y \in \N)$$: The natural numbers are closed under equality.
 * $$\forall x \in \N(S(x) \in \N)$$: The natural numbers are closed under S (the successor operation).
 * $$\forall x, y \in \N(x = y \leftrightarrow S(x) = S(y))$$: S is an injection.
 * $$\forall x \in N(S(x) \neq 0)$$: There is no natural number who's successor is zero.
 * $$\forall X(0 \in X \land \forall x \in \N(x \in X)) \rightarrow \N \subseteq X$$: If K is a set such that 0 is in K, and for every natural number n, n being in K implies that S(n) is in K, then K contains every natural number.
 * The induction scheme for $$L_{ID_1}$$ formulas.
 * $$A(I_A) \subseteq I_A$$
 * $$A(F) \subseteq F \rightarrow I_A \subseteq F$$

Note that the second-to-last axiom expresses that $$I_A$$ is closed under the arithmetically definable set operator $$\Gamma_A(S) = \{x \in \N | \N \models A(S, x)\}$$, while the last axiom expresses that $$I_A$$ is the least such (at least among sets definable in $$L_{ID_1}$$).

Thus, $$I_A$$ is meant to be the least pre-fixed-point, and hence the least fixed point of the operator $$\Gamma_A$$. Also, $$F(x)$$ represents the set $$\{x \in N | F(x)\}$$, $$s \in F$$ means $$F(s)$$, and for two formulas $$F$$ and $$G$$, $$F \subseteq G$$ means $$\forall x F(x) \rightarrow G(x)$$.

IDν

 * $$0 \in \N$$: 0 is a natural number.
 * $$\forall x(x = x)$$: Equality is reflexive.
 * $$\forall x \forall y = x \rightarrow x = y$$: Equality is symmetric.
 * $$\forall x \forall y \forall z(x = y \land y = z \rightarrow x = z)$$: Equality is transitive.
 * $$\forall x \forall y(x \in \N \land x = y \rightarrow y \in \N)$$: The natural numbers are closed under equality.
 * $$\forall x \in \N(S(x) \in \N)$$: The natural numbers are closed under S (the successor operation).
 * $$\forall x, y \in \N(x = y \leftrightarrow S(x) = S(y))$$: S is an injection.
 * $$\forall x \in N(S(x) \neq 0)$$: There is no natural number who's successor is zero.
 * $$\forall X(0 \in X \land \forall x \in \N(x \in X)) \rightarrow \N \subseteq X$$: If K is a set such that 0 is in K, and for every natural number n, n being in K implies that S(n) is in K, then K contains every natural number.
 * The induction scheme for $$L_{ID_\nu}$$ formulas.
 * $$TI(\prec, F)$$ expresses transfinite induction along $$\prec$$ for an arbitrary $$L_{ID_\nu}$$ formula $$F$$.
 * $$\forall \mu \prec \nu; A^\mu(J^\mu_A) \subseteq J^\mu_A$$ for an arbitrary $$L_{ID_\nu}$$ formula $$F$$.
 * $$\forall \mu \prec \nu; A^\mu(F) \subseteq F \rightarrow J^\mu_A \subseteq F$$ for an arbitrary $$L_{ID_\nu}$$ formula $$F$$.

In these last two axioms we used the abbreviation $$A^\mu(F)$$ for the formula $$A(F, (\lambda\gamma y; \gamma \prec \mu \land y \in J^\gamma_A), \mu, x)$$, where $$x$$ is the distinguished variable. We see that these express that each $$J^\mu_A$$, for $$\mu \prec \nu$$, is the least fixed point (among definable sets) for the operator $$\Gamma^\mu_A(S) = \{n \in \N | (\N, (J^\gamma_A)_{\gamma \prec \mu}\}$$. Note how all the previous sets $$J^\gamma_A$$, for $$\gamma \prec \mu$$, are used as parameters.

We then define $$ID_{\prec \nu} = \bigcup \limits_{\xi \prec \nu}ID_\xi$$.

$$\widehat{\mathsf{ID}}_\nu$$

 * $$0 \in \N$$: 0 is a natural number.
 * $$\forall x(x = x)$$: Equality is reflexive.
 * $$\forall x \forall y = x \rightarrow x = y$$: Equality is symmetric.
 * $$\forall x \forall y \forall z(x = y \land y = z \rightarrow x = z)$$: Equality is transitive.
 * $$\forall x \forall y(x \in \N \land x = y \rightarrow y \in \N)$$: The natural numbers are closed under equality.
 * $$\forall x \in \N(S(x) \in \N)$$: The natural numbers are closed under S (the successor operation).
 * $$\forall x, y \in \N(x = y \leftrightarrow S(x) = S(y))$$: S is an injection.
 * $$\forall x \in N(S(x) \neq 0)$$: There is no natural number who's successor is zero.
 * $$\forall X(0 \in X \land \forall x \in \N(x \in X)) \rightarrow \N \subseteq X$$: If K is a set such that 0 is in K, and for every natural number n, n being in K implies that S(n) is in K, then K contains every natural number.
 * A weakened induction scheme for $$L_{ID_\nu}$$ formulas.
 * $$TI(\prec, F)$$ expresses transfinite induction along $$\prec$$ for an arbitrary $$L_{ID_\nu}$$ formula $$F$$.
 * $$\forall \mu \prec \nu; A^\mu(J^\mu_A) \subseteq J^\mu_A$$ for an arbitrary $$L_{ID_\nu}$$ formula $$F$$.
 * $$\forall \mu \prec \nu; A^\mu(F) \subseteq F \rightarrow J^\mu_A \subseteq F$$ for an arbitrary $$L_{ID_\nu}$$ formula $$F$$.

In this weakened version, $$I_A$$ is just a fixed point, not necessarily the least fixed point, of the operator $$\Gamma_A$$.