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## Miscellaneous Theorems

A mind without choice is only truth, and only uncertain. As life presents choice, and we fain understanding, that uncertainty gives way to belief in what is possible, and what is not. At first that belief can be very specific, since it isn't very complicated. But after a while, the whole thing gets too big, and something must be done. For this, we develop theories. These theories are a few simple parts that can be assembled to build everything in our experience. Sometimes they're mnemonics, sometimes they're spiritual, and sometimes they're predictive. Regardless of the type, theories take one idea, and define it in terms of other, hopefully more simple, ideas.

### Domains, Parses, and Power Sets

$$\big|\{\bar{\alpha}^{2^n}\}\big|=\big|2^{\{\textbf{A}^n\}}\big|\ni(\textbf{A}^n)_\alpha$$

### Universal Quantifier

$$\forall x(\mathscr{Y})\Leftrightarrow\Big<\big((\textbf{A}^n)_\alpha,\mathscr{Y}\big)_\mathtt{0xD}\Big>^{(\textbf{A}^n)}_\cap\ni x=(\textbf{A}^n)_\text{CV}$$

### Existential Quantifier

$$\exists x(\mathscr{Y})\Leftrightarrow\Big<\big((\textbf{A}^n)_\alpha,\mathscr{Y}\big)_\mathtt{0x8}\Big>^{(\textbf{A}^n)}_\cup\ni x=(\textbf{A}^n)_\text{CV}$$