disciple: Your proof was quite beautiful. speaker: Thank you! disciple: Actually, sometimes, I had a bit a problem following. There was quite a jump from the Fundamental Lemma to Theorem A. speaker: Well, yes, because the time was short. But I can give the details. disciple: Rigorous? speaker: Yes, of course. disciple: And Theorem B? speaker: The Fundamental Lemma also implies Theorem B; you ca...

Hauser, K., The indescribability of the order of the indescribable cardinals, Annals of Pure and Applied Logic 57 (1992) 45-91. We prove the following consistency results about indescribable cardinals which answer a question of A. Kanamori and M. Magidor (cf. [3]). Theorem 1.1 (m 22, n 22). CON(ZFC+ 3~, K’ (K is Ii: indescribable, K’ is XF indescribable, and K < K’)) j CON(ZFC + fl> n: + GCH). ...

We consider the role of the foundation axiom and various anti-foundation axioms in connection with the nature and existence of elementary self-embeddings of the set-theoretic universe. We shall investigate the role of the foundation axiom and the various anti-foundation axioms in connection with the nature and existence of elementary self-embeddings of the set-theoretic universe. All the standa...

Roitman's combinatorial principle $\Delta$ is equivalent to monotone normality of the nabla product, $\nabla (\omega +1)^\omega$. If $\{ X_n : n\in \omega\}$ a family metrizable spaces and $\nabla_n X_n$ monotonically normal, then hereditarily paracompact. Hence, if holds box product $\square +1)^\omega$ Large fragments hold in $\mathsf{ZFC}$, yielding large subspaces (\omega+1)^\omega$ that ar...

We prove that the determinacy of Gale-Stewart games whose winning sets are accepted by realtime 1-counter Büchi automata is equivalent to the determinacy of (effective) analytic GaleStewart games which is known to be a large cardinal assumption. We show also that the determinacy of Wadge games between two players in charge of ω-languages accepted by 1-counter Büchi automata is equivalent to the...

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there ...

Magnetic measurements on a high-T, superconductor YBa2Cu30q are reported. Diamagnetism sets in, at low fields, below a temperature T=90 K. At low T the field-cooled magnetization becomes positive again. We show that the low-T phase is still superconducting and the apparent reentry behavior is due to strong paramagnetic contributions. Glassy features appear below a temperature whose field depend...