Related papers: Definable Hamel bases and $AC_\omega(R)$
A proof of G\"odel's incompleteness theorem is given. With this new proof a transfinite extension of G\"odel's theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a…
In this note we present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hern\'andez-Guti\'errez and Hru\v{s}\'ak. The method of the proof also allows us to obtain a…
We make use of a finite support product of $\omega_1$ clones of the Jensen minimal $\varPi^1_2$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface…
A realization of the affine Lie algebra ${A^{(1)}_1}$ and the relevant $Z$-algebra at negative level $-k$ is given in terms of parafermions. This generalizes the recent work on realization of the affine Lie algebra at the critical level.
This article investigates pathological behavior at the first limit stage in the sequence of inner mantles, obtained by iterating the definition of the mantle to get smaller and smaller inner models. I show: (A) it is possible that the…
We study the definability of ultrafilter bases on $\omega$ in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in $L$ we can construct $\Pi^1_1$ P-point and…
In this note we are concerned with the validity of an uncountable analogue of a combinatorial lemma due to Vlastimil Pt\'ak. We show that the validity of the result for $\omega_1$ can not be decided in ZFC alone. We also provide a…
Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_\delta$ be a non-trivial $\in$-cofinal $\Sigma_1$-elementary embedding, where $\varepsilon,\delta$ are limit ordinals. We prove some restrictions on the constructibility of…
Suppose that there's no transitive model of ZFC + there's a strong cardinal, and let K denote the core model. It is shown that if \delta has the tree property then \delta^{+K} = \delta^+ and \delta is weakly compact in K.
We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length $\omega_2$. This implies that $\omega_1$ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument…
Assuming PFA, every uncountable subset E of the plane meets some C^1 arc in an uncountable set. This is not provable from MA(aleph_1), although in the case that E is analytic, this is a ZFC result. The result is false in ZFC for C^2 arcs,…
Let $\mathbf{c}_0$ be the lowest generalized two-sided cell of an extended affine Weyl group W. We determine the structure of the based ring of $\mathbf{c}_0$. For this we show that certain conjectures of Lusztig on generalized cells…
False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular…
We classify flat strict nearly K\"ahler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K\"ahler factor of maximal dimension and a strict flat nearly K\"ahler manifold of split…
If we apply an extension of the Deduction meta-Theorem to Goedel's meta-reasoning of "undecidability", we can conclude that Goedel's formal system of Arithmetic is not omega-consistent. If we then take the standard interpretation…
We observe how a classical model-theoretic fact proves the existence of many strong amalgamation bases for the class of R^\omega-embeddable von Neumann algebras, where R is the hyperfinite II_1 factor. In particular, we shows that R itself…
We sow that there exists a generic extension of the G\"{o}del's constructible universe in which diamond holds and there exists a subset $Y \subseteq \omega_1$ such that for stationary many $\delta < \omega_1,$ the set $Y \cap \delta$ is not…
Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge3$, there exists a lightface $\varPi^1_n$ set of reals, which is a ${\mathsf E}_0$ equivalence class, hence a countable set, and…
Assuming that there is no inner model with a strong cardinal, the following is shown: any subset of \omega_1 can be made \Delta^1_3 (in the codes) by a reasonable set-forcing; there is a reasonable set-generic extension with a \Delta^1_3…
It is suggested a topological hierarchical classification of the infinite many Localized phases figuring in the phase diagram of the Harper equation for anisotropy parameter $\epsilon$ versus Energy $E$ with irrational magnetic flux…