Answer by Emil Jeřábek for A "Completion" of $ZFC^-$
Without a definition of “large cardinal axiom”, I’m going to ignore Q1 and Q3.The answers to Q2 and Q4 are negative by the following general principle. (For Q4, we take $T_0$ to be the set of $\mathcal...
View ArticleAnswer by Danielle Ulrich for A "Completion" of $ZFC^-$
Overnight the following occurred to me...The answer to Question 2 is negative (with an asterisk), and so the same is true of Question 1. Namely, let $T$ be a set of $\Pi_2$ sentence with $ZFC^- \cup T$...
View ArticleA "Completion" of $ZFC^-$
Let $T_0$ be the set theory axiomatized by $ZFC^-$ (that is $ZFC$ without powerset) + every set is countable + $\mathbb{V}=\mathbb{L}$. Question 1: Suppose $\phi$ is a sentence of set theory. Must...
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