An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa.$ We prove that if there is a model of $\ZFC$ with two supercompact cardinals, then there is a model of \ZFC where simultaneously $(\aleph_2, \mu)$-ITP and $(\aleph_3, \mu')$-ITP hold, for all $\mu\geq \aleph_2$ and $\mu'\geq \aleph_3.$
An inaccessible cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every natural number n greater than 1 and for every ordinal mu greater than or equal to aleph_n, we have (aleph_n, mu)-ITP.
We discuss classical realizability, a branch of mathematical logic that investigates the computational content of mathematical proofs by establishing a correspondence between proofs and programs. Research in this field has led to the development of highly technical constructions generalizing the method of forcing in set theory. In particular, models of realizability are models of ZF, and forcing models are special cases of realizability models.
Oscillations are a powerful tool for building examples of colorings witnessing negative partition relations. We survey several results illustrating the general technique and present a number of applications.
Starting from infinitely many supercompact cardinals, we force a model of ZFC where $\aleph_{\omega^2+1}$ satisfies simultaneously a strong principle of reflection, called $\Delta$-reflection, and a version of the square principle, denoted $\square(\aleph_{\omega^2+1}).$ Thus we show that $\aleph_{\omega^2+1}$ can satisfy simultaneously a strong reflection principle and an anti-reflection principle.
We prove that successors of singular limits of strongly compact cardinals have the strong tree property. We also prove that aleph_{omega+1} can consistently satisfy the strong tree property.
The metabolism of zetidoline, a new neuroleptic, in the rat and the dog has been studied. From the urine of rats and dogs given 5 mg/kg of [2-14C] zetidoline orally, unchanged drug and five metabolites were isolated and the structures of four of them assigned by physicochemical analysis. They are: metabolite B, 4'-hydroxy-3'-chlorophenyl zetidoline; metabolite D, zetidoline without the aryl group; metabolite E, the 6'-hydroxy-4'-beta-D-glucuronide of metabolite B, and metabolite F, the 4'-beta-D-glucuronide of metabolite B. The plasma levels of zetidoline and its metabolites after iv administration show that the drug is rapidly excreted and/or metabolized in both animal species. The plasma radioactivity in the dog consists mainly of the pharmacologically active (neuroleptic) metabolite B, whereas in the rat it consists of the more polar metabolites. After oral administration, elimination in both species occurs mostly via the kidneys. In the dog, within a 24-hr period, 6.2 +/- 0.4% of the dose is accounted for as unchanged zetidoline, 7.6 +/- 0.5% as metabolite B, 10.1 +/- 0.7% as the unidentified metabolite C, and 21.4 +/- 1.1% as metabolite F. In the rat, over the same period, zetidoline is present in traces, metabolite B accounts for 6.9 +/- 0.3% of the dose, metabolite D for 6.6 +/- 0.9%, metabolite E for 15.2 +/- 1.4%, and metabolite F for 31.7 +/- 2.2%.
Abstract We develop a technique for representing and preserving cardinals in realizability models, and we apply this technique to define a realizability model of Zorn’s lemma restricted to an ordinal.
