Condensation and Large Cardinals- a simplified version of my dissertation

We give a corrected and simplified, self-contained account of the proof of the main theorem of the author’s dissertation ([4]): We show that over any model of set theory we may perform a cofinality-preserving forcing to obtain a model of set theory which satisfies Local Club Condensation while preserving an ω-superstrong cardinal. To simplify reference, chapter numbers in this note correspond with chapter numbers in [4]. 1 Canonical Functions Lemma 1.1 Assume β has regular cardinality κ and for every γ ≤ β, fγ is a bijection from card γ to γ. Then there is a club of δ < κ such that fα[δ] = fβ [δ] ∩ α for all α ∈ fβ [δ] \ κ. Proof: See [2] or [4]. 2 Large Cardinal Basics Definition 2.1 κ is ω-superstrong if there is an elementary embedding j: V→ M with critical point κ such that Vjω(κ) ⊆M.1 3 Forcing Basics Definition 3.1 If P is a notion of forcing and η is a cardinal, we say that P is η+-strategically closed iff Player I has a winning strategy in the following two player game of perfect information: Player I and Player II alternately make moves where in each move, each player plays a condition of P. Player I has to start and play 1P in the first move. Player II is allowed to play any condition stronger than the condition just played by Player I in each of his moves. Player I has to play a condition stronger than all previously played conditions in each move, Player I has to make a move at every limit step of the game. We say that Player I wins if he can find conditions to play in any such game of length η+ (arriving at η+, the game ends, no condition has to be played at stage η+). 1Such an embedding with Vjω(κ)+1 ⊆M is known to be inconsistent by Kunen’s Theorem.