Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas.

Math connections, Divisions, Automata, Yates index set theorem, Adjunction

Understand how the arithmetical hierarchy relates to the foursome and sevensome.

Study the classification of sets in the jump hierarchy as in Soare's book.

Readings

Three minds

Answering mind

Knows that for some input {$x$} and some time {$t$} the machine {$z$} will halt, there will be an answer, an output {$y$}.
{$Nonempty(z)\Leftrightarrow W_z\neq \varnothing \Leftrightarrow \exists x \; \exists t \;\exists y\;\neg \neg U(z,t,y,x)$}

Questioning mind

Knows that once the input {$x$} is large enough, the machine {$z$} will never halt, there will never be an answer.
In other words, answers are exceptional.
{$Finite(z)\Leftrightarrow W_z \textrm{ is finite }\Leftrightarrow \exists X \; \forall x \; \exists q \;\neg (x>X\Rightarrow U(z,t,y,x))$} where {$q$} is superfluous

Investigating mind

Aligns what is known (what will halt) and what is not known (what will never halt).
Complements a machine that knows (halts) and a machine that does not know (does not halt).

{$Recursive(z)\Leftrightarrow W_z\textrm{ is recursive }\Leftrightarrow \exists m \forall x (x\in W_z \Leftrightarrow x\notin W_m) \Leftrightarrow $}

{$\exists m\; \forall x\; \exists t_{m,g}\; \forall t_{z,g}\; \exists t_{z,c}\; \forall t_{m,c}(((U(z,t_{z,g},y_{z,x},x)\Rightarrow \neg (U(m,t_{m,c},y_{m,x},x))∧(\neg U(m,t_{m,g},y_{m,x},x)\Rightarrow (U(z,t_{z,c},y_{z,x},x)))$}

Ideas

The ways of figuring things out in mathematics include four patterns from analysis which exhibit stages in the arithmetical hierarchy:

{$\Pi_1$} Induction {$\forall x P(x)$}
{$\Sigma_2$} Maximum or minimum {$\forall x \exists y (P(x)\Rightarrow P(y))$}
{$\Pi_3$} Least upper bound or greatest lower bound {$\forall x \exists y \forall z ((x \leq y) \wedge (x \leq z \Rightarrow y \leq z))$}
{$\Sigma_4$} Limit {$\forall \delta \exists \epsilon \forall x \exists x_0 (|x-x_0|<\delta \Rightarrow |F(x)-F(x_0)|<\epsilon )$}

In general, a logical form has three parts:

Given...
Quantifiers...
Statement...

Hartley Rogers. Theory of recursive functions and effective computability.

Notes

Consider what delta-epsilon statements mean in terms of the arithmetical hierarchy. There exists and x_0 such that for every epsilon there exists a delta such that for every x... It starts out most fixed (x_0) and ends most free (x).
How do notions of variables (fixed, varying) relate to the arithmetical hierarchy?
{$\exists ! P(x) = \exists x \forall y (P(y)\leftrightarrow y=x)$}
{$\forall x \exists y$} defines a function {$f(x)=y$}.
Nested quantifiers, arithmetic hierarchy. God's perspective {$\forall$}, human's perspective {$\exists$}.