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forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
4
votes
What are the known implications of "There exists a Reinhardt cardinal" in the theory "ZF + j"?
First, let's show that I-1 implies I0:
Suppose $I-1(\kappa,\delta, j)$; we may assume, by forcing if necessary, that $V_\kappa$ (and hence $V_\delta$) satisfies the Axiom of Choice. Now if j is a ran …
2
votes
Is theory with domain of interpretation in second order objects a First Order Theory?
To answer your first question: No, a theory can be well-defined formally without having any specified domain of interpretation. But if it is a first-order theory, a technique developed by Godel (in hi …