The concept of definability of physical fields in a set-theoretical foundation is introduced and an axiomatic set theory is proposed which provides precisely the tools necessay for a nonlinear sigma model. In this theory quantization of the model derives from a null postulate and becomes equivalent to definability. We also obtain scale invariance and compactification of the spatial dimensions e...

The concept of definability of physics fields is introduced and a set-theoretical foundation is proposed. In this foundation, we obtain a scale invariant nonlinear sigma model and then show that quantization of the model is necessary and sufficient for definability. We also obtain com-pactification of the spatial dimensions effectively and show its equivalence to quantization. We look to provid...

zw Let G be a finite abelian group and 1 ≤ r < ∞. We prove that every locally definable CG manifold admits a unique locally definable C∞G manifold structure up to locally definable C∞G diffeomorphism. zh 2000 Mathematics Subject Classification. 14P10, 14P20, 57S05, 57S15, 58A05, 58A07, 03C64.

Let N be an o-minimal structure. In this paper we develop group extension theory over N and use it to describe N -definable solvable groups. We prove an o-minimal analogue of the Lie-Kolchin-Mal’cev theorem and we describe N -definable G-modules and N -definable

We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field L is definably isomorphic to the group of L-rational points of an algebraic group defined over L.

In this note we show: Let R = 〈R, <, +, 0, . . . 〉 be a semi-bounded (respectively, linear) o-minimal expansion of an ordered group, and G a group definable in R of linear dimension m ([Ed1]). Then G is a definable extension of a bounded (respectively, definably compact) definable group B by 〈Rm, +〉.

We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G00, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, Mext, of a model M . In the light of these results we continue the study of the “definable topological dynamics” of groups in NIP theories. In particular we prove the Ellis gro...

We study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We...

The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the Σk relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k ≥ 7, the Σk relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that Low1 is parameter definable, and we provide a new example of a ∅–...

In Pawlak rough sets, the structure of the definable set families is simple and clear, but in generalizing rough sets, the structure of the definable set families is a bit more complex. There has been much research work focusing on this topic. However, as a fundamental issue in relation based rough sets, under what condition two relations induce the same definable set family has not been discus...