Related papers: Set theory without choice: not everything on cofin…

Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $\omega^{\omega}$-bounding sufficiently absolute forcing notions $\mathbb Q$.

This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a…

Individual choices often depend on the order in which the decisions are made. In this paper, we expose a general theory of measurable systems (an example of which is an individual's preferences) allowing for incompatible (non-commuting)…

We show that the common theory of all modules over a tubular algebra (over a recursive algebraically closed field) is decidable. This result supports a long standing conjecture of Mike Prest which says that a finite-dimensional algebra…

We consider natural cardinal invariants hm_n and prove several duality theorems, saying roughly: if I is a suitably definable ideal and provably cov(I)>=hm_n, then non(I) is provably small. The proofs integrate the determinacy theory,…

Commutative totally ordered monoids abound, number systems for example. When the monoid is not assumed commutative, one may be hard pressed to find an example. One suggested by Professor Orr Shalit are the countable ordinals with addition.…

Suppose that lambda = mu^+. We consider two aspects of the square property on subsets of lambda. First, we have results which show e.g. that for aleph_0 <= kappa =cf (kappa)< mu, the equality cf([mu]^{<= kappa}, subseteq)= mu is a…

Consider a commutative monoid $(M,+,0)$ and a biadditive binary operation $\mu \colon M \times M \to M$. We will show that under some additional general assumptions, the operation $\mu$ is automatically both associative and commutative. The…

We prove the undecidability of MSO on $\omega$-words extended with the second-order predicate $U_1(X)$ which says that the distance between consecutive positions in a set $X \subseteq \mathbb{N}$ is unbounded. This is achieved by showing…

We solve two long-standing open problems regarding the combinatorics of $\aleph_{\omega+1}$. We answer a question of Shelah by showing that it is consistent for any $n\geq 1$ that $\mathsf{GCH}$ holds and there is a stationary set of points…

We investigate the provability of classical combinatorial theorems in ZF. Using combinatorial arguments, we establish the following results for each infinite cardinal ${\kappa}\in On$, (1) ${\kappa}^+\to ({\kappa},{\omega}+1)$, (2) any…

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $\mathbf{ZF}$, some are shown to be independent of…

A Steinhaus set $S \subseteq \RR^d$ for a set $A \subseteq \RR^d$ is a set such that $S$ has exactly one point in common with $\tau A$, for every rigid motion $\tau$ of $\RR^d$. We show here that if $A$ is a finite set of at least two…

Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $\delta$ of countable cofinality and every regular uncountable $\mu<\delta$ there are stationarily many non-approachable…

For each Turing machine T, we construct an algebra A'(T) such that the variety generated by A'(T) has definable principal subcongruences if and only if T halts, thus proving that the property of having definable principal subcongruences is…

In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C <= D or D <= C. However, in ZF this is no longer so. For a given infinite…

We show using a proof of the Global Square property in Core Models below a measurable of Mitchell order o(kappa)=kappa^++ (a result originally due to Jensen & Zeman) that Foreman and Magidor's Mutual Stationarity property MS(Aleph_n…

It is shown that if $G$ is an uncountable Polish group and $A\subseteq G$ is a universally measurable set such that $A^{-1}A$ is meager, then the set $T_l(A)=\{\mu\in P(G): \mu(gA)=0 \text{for all} g\in G\}$ is co-meager. In particular, if…

We consider the fragment F of first order arithmetic in which quantification is restricted to ''for all but finitely many.'' We show that the integers form an F-elementary substructure of the real numbers. Consequently, the F-theory of…

We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary…