Ladder mice

Farmer Schlutzenberg (TU Vienna)

Jun 28. 2024, 11:15 — 12:00

We will discuss a new analysis of ladder mice, first introduced and studied by Rudominer, and then Woodin and Steel. Our analysis establishes a (lightface) mouse set theorem, which appears to be more general than what was known earlier: OD_{alpha n} is a mouse set for every ordinal alpha of countable L(R)-cofinality such that [alpha,alpha] is a projective-like gap and alpha is not the successor of a strong gap, and for every integer n>=1. The analysis also gives an alternate proof of this in the case "just past projective", avoiding the stationary tower. It also establishes an associated on-a-cone "anti-correctness" result. Anti-correctness is the generalization of, for example, the facts that (Pi^1_3)^V truth about reals in M_1 is (Sigma^1_3)^{M_1}, and that (Pi^1_3)^{M_1} truth (about reals in M_1) is (Sigma^1_3)^V. Time permitting, we may also mention a version of ladder mice at the end of a weak gap / successor of a strong gap. This work appears to be a useful component toward a positive resolution of the Rudominer-Steel conjecture on optimal wellorders of the reals, a related question on which the author and Steel are working.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Determinacy, Inner Models and Forcing Axioms (Workshop)
Organizer(s):
Sandra Müller (TU Vienna)
Grigor Sargsyan (Polish Academy of Science, Warsaw)
Ralf Schindler (WWU Münster)
John Steel (UC, Berkeley)