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What is the complexity (e.g. is it $\Sigma^0_1$, arithmetic, fully $\Pi^1_1$) of the relation $|a| < |b|$ given two notations $a, b \in \mathscr{O}$ (Kleene's O)? What about the case where only one of the notations must be in $\mathscr{O}$ (where $b \not\in \mathscr{O} \implies |b| = \infty$)? Fo...

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A Heyting algebra is a bounded distributive lattice $(L,\vee,\wedge,0,1)$ together with a binary operation $\rightarrow$ called implication or relative pseudocomplementation with the property that, for all $x,y,z\in L$, $$x\le y\rightarrow z\quad\text{ if and only if } x\wedge y\le z.$$ Boolean...

In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that (c ∧ a) ≤ b is equivalent to c ≤ (a → b). From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by Arend Heyting (1930) to...