Answer by Yair Hayut for Do these ordinals exist?
Let $V$ be a transitive model of $ZFC$. Without special assumptions about the model $V$, it is possible that $F_0(0)$ does not exist (in other words - it is possible that every ordinal is definable...
View ArticleAnswer by Andreas Blass for Do these ordinals exist?
For the definition of $F_{n+1}$ to make sense, we need, in addition to the usual axiomatic apparatus of ZFC, a notion of "satisfaction of formulas in $V$." If we have this additional notion and if we...
View ArticleAnswer by Zetapology for Do these ordinals exist?
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...
View ArticleDo these ordinals exist?
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...
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