The chapter represents a comparison of ultrapower fields (loosely surreals, hyper-reals, long line) and number fields generated by infinite primes having a physical interpretation in Topological Geometrodynamics. Ultrapower fields are discussed in very physicist friendly manner in the articles of Elemer Rosinger and these articles are taken as a convenient starting point. The physical interpret...

Our motivating question was whether all traces on a U-ultrapower of a C*-algebra A, where U is a non-principal ultrafilter on N, are necessarily U-limits of traces on A. We show that this is false so long as A has infinitely many extremal traces, and even exhibit a 22 א0 size family of such traces on the ultrapower. For this to fail even when A has finitely many traces implies that A contains o...

We construct a model without precipitous ideals but so that for each τ < א3 there is a normal ideal over א1 with generic ultrapower wellfounded up to the image of τ .

A syntax and semantics of types, terms and formulas for coalgebras of polynomial functors is developed, extending earlier work [4] on monomial coalgebras to include functors constructed using coproducts. A modified ultrapower construction for polynomial coalgebras is introduced, adapting the conventional ultrapower to retain only those states that evaluate observable terms in a standard way. A ...

• At stage 0, M0 = M , U0 = U and κ0 = κ. • At stage α+1, if it is well-founded, we build Ult(Mα, Uα) and let Mα+1 be its transitive collapse, where the ultrapower is computed from the point of view of Mα. Let jα,α+1 : Mα → Mα+1 be the ultrapower embedding. Then set κα+1 = j(κα) and Uα+1 = j(Uα). • At limit stages λ, if it is well-founded, we define Mλ as the direct limit of the directed system...

The Connes Embedding Problem (CEP) asks whether every separable II1 factor embeds into an ultrapower of the hyperfinite II1 factor. We show that the CEP is equivalent to the computability of the universal theory of every type II1 von Neumann algebra. We also derive some further computability-theoretic consequences of the CEP.

We prove that a nonstandard extension of arithmetic is eeectively conservative over Peano arithmetic by using an internal version of a deenable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same proof-theoretic strength as second order arithmetic, where comprehension is restricted to arithmetical formulas.