6 problems

Let $R$ be the set of Kolmogorov-random binary strings defined using the fixed universal Turing machine $U$, and let $k$ be the threshold supplied by Chaitin's incompleteness theor…

Let $\mathcal{S}$ and $\mathcal{T}$ be consistent theories, with $\mathcal{T}$ strictly stronger than $\mathcal{S}$, and let $\mathfrak{I}$ be a collection of sentences unprovable…

Let $Q$ be a propositional proof system. A theory $\mathcal{T}$ is a conservative extension of a base theory when it extends that theory without proving any new sentences in the ba…

Let $\mathcal{T}$ be the theory used to define $H'$, and let $H'$ be the collection of dense sets of true $\mathcal{T}$-unprovable sentences described in the source. Given a nondet…

Let $H'$ be the set comprising $H$ and other dense sets of true sentences unprovable in $\mathcal{T}$, including the specified sets associated with universal Turing machines and Tu…

Let $\mathcal T$ be the class of theories under consideration, and let $Con_S$ mean that $S$ is consistent. Consistency-provability strengthening conjecture. For every…