We prove the following surprising result: there exist a 1-counter Büchi automaton and a 2-tape Büchi automaton such that the ω-language of the first and the infinitary rational relation of the second in one model of ZFC are Π2-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an ω-language accepted by a 1-counter Büchi au...

Non-equilibrium aging dynamics in 3D Ising spin glass Cu0.5Co0.5Cl2-FeCl3 GBIC has been studied by zero-field cooled (ZFC) magnetization and low frequency AC magnetic susceptibility (f = 0.05 Hz), where Tg = 3.92± 0.11 K. The time dependence of the relaxation rate S(t) = (1/H)dMZF C/dln t for the ZFC magnetization after the ZFC aging protocol, shows a peak at a characteristic time tcr near a wa...

In this paper we prove Jensen’s Coding Theorem, assuming ∼ 0#, via a proof that makes no use of the fine structure theory. We do need to quote Jensen’s Covering Theorem, whose proof uses fine-structural ideas, but make no direct use of these ideas. The key to our proof is the use of “coding delays.” Coding Theorem (Jensen) Suppose 〈M,A〉 is a model of ZFC + O# does not exist. Then there is an 〈M...

On unfoldable cardinals, !-closed cardinals, and the beginning of the inner model hierarchy Abstract. Let be a cardinal, and let H be the class of sets of hereditary cardinality less than ; let ( ) > be the height of the smallest transitive admissible set containing every element of H . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in t...

Applied mathematicians increasingly use computers to answer mathematical questions. We want to provide them domain-specific languages. The languages should have exact meanings and computational meanings. Some proof assistants can encode exact mathematics and extract programs, but formalizing the required theorems can take years. As an alternative, we develop λZFC, a lambda calculus that contain...

Since the definition of the Busy Beaver function by Radó in 1962, an interesting open question has been the smallest value of n for which BB(n) is independent of ZFC set theory. Is this n approximately 10, or closer to 1,000,000, or is it even larger? In this paper, we show that it is at most 7,910 by presenting an explicit description of a 7,910-state Turing machine Z with 1 tape and a 2-symbo...

dc magnetization measurements, for zero-field cooled (M ZFC) and field-cooled (M FC) cases, have been carried out for flash-evaporated Pd-doped NiMn thin films. These included reentrant phases (Ni762xPdx)Mn24 , for 0<x<5, and Ni75Mn23Pd2 , a pure spinglass phase. The studies were performed over the temperature range 3–300 K. Low-field magnetization measurements show the irreversibility effect (...

large cardinal hypothesis used to prove I1~+1 determinacy in [MS] cannot be substantially improved. Our theorem is an application of the theory of core models or canonical inner models for large cardinal hypotheses. Our work develops further some of the basic ideas of this theory. The theory itself has many other applications, particularly in the realm of relative consistency results. (As a quo...