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Cantor's Attic is a really great website for the various descriptions of large finite numbers, large countable ordinals, and large cardinal axioms. However, after looking through the archives of the …

Given an ordinal $\alpha$, I define $F_{n}(\alpha)$ as follows: $F_0(\alpha)=\alpha$ $F_{n+1}(\alpha)$ is the smallest $\beta$ such that no first-order $\phi$ in the language of $\{\in\}$ has $(\ma …

A common trend in large cardinal axioms dealing with critical points of elementary embeddings from $V$ into a transitive class $M$ is to make some large cardinal axiom with an ordinal parameter, simil …

I was reading Kanamori's The Higher Infinite, when I came across the fact that for any extendible cardinal $\kappa$ and any $\mathcal{L}_{\kappa,\kappa}^n$-theory $T$, $Sat(T)\Leftrightarrow \forall t …

You can guarantee $F_n(\alpha)$ is countable. Assume the contrary. There is a first-order formula for every countable ordinal $\phi$ such that $(V\models\phi(S,F_1(\alpha)...))\Leftrightarrow S=\alp …

Let a cardinal $\kappa$ be $n$-shadow iff $\kappa$ does not have cofinality $\omega$ and for any $n$-th order sentence $\varphi$ in the language $\mathcal{L}_\in$, $\varphi\Leftrightarrow V_\kappa\mod …

It is quite well-known that $\Pi_2^0$-Indescribability is the same as Strong Inaccessibility and $\Pi_n^0$-Indescribability for every $n>2$. It is also quite simple to show that $\Pi_0^0$-Indescribab …