Related papers: A club guessing toolbox I
Uncovering the hidden regularities and organizational principles of networks arising in physical systems ranging from the molecular level to the scale of large communication infrastructures is the key issue for the understanding of their…
We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings…
We show that the decidability of the first-order theory of the language that combines Boolean algebras of sets of uninterpreted elements with Presburger arithmetic operations. We thereby disprove a recent conjecture that this theory is…
Starting from suitable large cardinals, we force the failure of (weak) diamond at the least inaccessible cardinal. The result improves an unpublished theorem of Woodin and a recent result of Ben-Neria, Garti and Hayut.
In [Sh893], Shelah proves that (on a stationary set of cardinals) an AEC has not too many models or every model has extensions of arbitrary cardinality. We show that, if we assume limited amalgamation, then the second condition holds for a…
We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_\kappa=\kappa^+$. Based on this statement, we separate tiltan from superclub at a successor…
We provide two types of guessing principles for ultrafilter ($\diamondsuit^{-}_{\lambda}(U), \ \diamondsuit^p_\lambda(U)$) on $\omega$ which form subclasses of Tukey-top ultrafilters, and construct such ultrafilters in $ZFC$. These…
We use forcing over admissible sets to show that, for every ordinal $\alpha$ in a club $C\subset\omega_1$, there are copies of $\alpha$ such that the isomorphism between them is not computable in the join of the complete $\Pi^1_1$ set…
Part of the title of this article is taken from writings of Einstein, which argue that we need to exercise our ability to analyse familiar concepts, to demonstrate the conditions on which their justification and usefulness depend, and the…
In order to address complex systems, apply pattern recongnition on their evolution could play an key role to understand their dynamics. Global patterns are required to detect emergent concepts and trends, some of them with qualitative…
New sets (typically found by computer search) with Sidon constant equal to the square root of their cardinalities are given. For each integer $N$ there are only a finite number of groups of prime order containing $N$-element extreme sets.…
It was suggested recently that the study of 1-dimensional QCD with fermions in the adjoint representation could lead to an interesting toy model for strange metals and their holographic formulation. In the high density regime, the infrared…
We study the approachability ideal I[\kappa^+] in the context of large cardinals properties of the regular cardinals below a singular \kappa. As a guiding example consider the approachability ideal I[\aleph_{\omega+1}] assuming that…
The notion of $\theta$-FAM-linkedness, introduced in the second author's master thesis, is a formalization of the notion of strong FAM limits for intervals, whose initial form and applications have appeared in the work of Saharon Shelah,…
The work presents the first part of second edition of the previous edition of 2000 under the same title containing the proof (in ZF) of the nonexistence of inaccessible cardinals, now enriched and improved. This part contains the apparatus…
A structural analysis of construction schemes is developed. That analysis is used to give simple and new constructions of combinatorial objects which have been of interest to set theorists and topologists. We then continue the study of…
Fairly deep results of Zermelo-Frenkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is K*K = K,…
An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree,…
H. Furstenberg introduced the notion of central sets in terms of topological dynamics and established the famous Central Sets Theorem. Later in [A new and stronger Central Sets Theorem, Fund. Math. 199 (2008), 155-175], D. De, N. Hindman,…
We investigate an extension of ZFC set theory (in an extended language) that stipulates the existence of a proper class of indiscernibles over the universe. One of the main results of the paper shows that the purely set-theoretical…