All of Jaime Sevilla Molina's Comments + Replies

Generalized fixed point theorem:

Suppose that $A_i(p_1,...,p_n)$ are $n$ modal sentences such that $A_i$ is modalized in $p_n$ (possibly containing sentence letters other than $p_{j}s$).

Then there exists $H_1,...,H_n$ in which no $p_j$ appears such that $GL⊢{\wedge }_{i\le n}\{⊡(p_i\leftrightarrow A_i(p_1,...,p_n)\}\leftrightarrow {\wedge }_{i\le n}\{⊡(p_i\leftrightarrow H_i)\}$.

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We will prove it by induction.

For the base step, we know by the fixed point theorem that there is $H$ such that $GL⊢⊡(p_1\leftrightarrow A_i(p_1,...,p_n\left)\right)\leftrightarrow ⊡(p_1\leftrightarrow H(p_2,...,p_n\left)\right)$

Now suppose that for $j$ we have $H_1,...,H_j$ such that $GL⊢{\wedge }_{i\le j}\{⊡(p_i\leftrightarrow A_i(p_1,...,p_n)\}\leftrightarrow {\wedge }_{i\le j}\{⊡(p_i\leftrightarrow H_i(p_{j+1},...,p_n\left)\right)\}$.

By the second substitution theo

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Uniqueness of arithmetic fixed points:

Notation: $⊡A=\square A\wedge A$

Let $H$ be a fixed point on $p$ of $\varphi \left(p\right)$; that is, $GL⊢⊡(p\leftrightarrow \varphi (p\left)\right)\leftrightarrow (p\leftrightarrow H)$.

Suppose $I$ is such that $GL⊢H\leftrightarrow I$. Then by the first substitution theorem, $GL⊢F\left(I\right)\leftrightarrow F\left(H\right)$ for every formula $F\left(q\right)$. If $F\left(q\right)=⊡(p\leftrightarrow q)$, then $GL⊢⊡(p\leftrightarrow H)\leftrightarrow ⊡(p\leftrightarrow I)$, from which it follows that $GL⊢⊡(p\leftrightarrow \varphi (p\left)\right)\leftrightarrow (p\leftrightarrow I)$.

Conversely, if $H$ and $I$ are fixed points, then $GL⊢⊡(p\leftrightarrow H)\leftrightarrow ⊡(p\leftrightarrow I)$, so since $GL$ is closed under substitution, $GL⊢⊡(H\leftrightarrow H)\leftrightarrow ⊡(H\leftrightarrow I)$. Since $GL⊢⊡(H\leftrightarrow H)$, it follows that $GL⊢(H\leftrightarrow I)$.

(Taken from The Logic of Provability, by G. Boolos.)