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Boban Velickovic

Publications and source records attributed to Boban Velickovic.

17 recordsLinked to original sources
arXiv2025

Adding cofinal countable sequences through multiple regular cardinals by ssp forcing

We present a direct construction of stationary set preserving forcings that make $\omega$-cofinal all the members of some arbitrary set $\mathcal{K}$ of regular cardinals $\kappa > \omega_1$. In addition, it is made possible to ensure that no other uncountable regular cardinals from the ground model acquire countable cofinality in the forcing extension. Our method is elementary, being based on a combinatorial argument by Foreman and Magidor together with generalizations of typical side-condition arguments and needs no assumptions beyond $\mathsf{ZFC}$.

math.LO↗
arXiv2024

On Some Infinitary Logics

We define a new class of infinitary logics $\mathscr L^1_{\kappa,\alpha}$ generalizing Shelah's logic $\mathbb L^1_\kappa$ defined in \cite{MR2869022}. If $\kappa=\beth_\kappa$ and $\alpha <\kappa$ is infinite then our logic coincides with $\mathbb L^1_\kappa$. We study the relation between these logics for different parameters $\kappa$ and $\alpha$. We give many examples of classes of structures that can or cannot be defined in these logics. Finally, we give a different version of Lindstr\"{o}m's Theorem for $\mathbb L^1_\kappa$ in terms of the $\phi$-submodel relation.

math.LO↗
arXiv2023

On Indestructible Strongly Guessing Models

In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(\omega_3,\omega_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\omega_2$ and $\omega_3$. In this paper we formulate a strengthening of ${\rm GM}^+(\omega_3,\omega_1)$ that we call ${\rm SGM}^+(\omega_3,\omega_1)$. We also prove, modulo the consistency of two supercompact cardinals, that ${\rm SGM}^+(\omega_3,\omega_1)$ is consistent with ZFC. In addition to all the consequences of ${\rm GM}^+(\omega_3,\omega_1)$, the principle ${\rm SGM}^+(\omega_3,\omega_1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $\omega_2$ either adds a real or collapses some cardinal. This gives a partial answer to a question of Abraham \cite{AvrahamPhD} and extends a previous result of Todor\v{c}evi\'{c} \cite{Todorcevic82} in this direction.

math.LO↗
arXiv2023

Characterization of $\mathcal L^1_κ$

The logic $\mathcal L^1_κ$ was introduced by Shelah in [3]. In [4], he proved that for a strongly compact cardinal $κ$, it admits the following algebraic characterization: two structures are $\mathcal L^1_κ$-equivalent if and only if they have isomorphic iterated ultrapowers via $κ$-complete ultrafilters. We give a presentation of the logic $\mathcal L^1_κ$ and a simplified and slightly modified proof of this result.

math.LO↗
arXiv2022

Increasing the second uniform indiscernible by strongly ssp forcing

We introduce a new and natural stationary set preserving forcing $\mathbb P^{c-c}({\lambda},{\mu})$ that (under $\mathsf{NS}_{\omega_1}$ precipitous + existence of $H_{\theta}^#$ for a sufficiently large regular ${\theta}$) increases the second uniform indiscernible $\mathbf{u}_2$ beyond some given ordinal ${\lambda}$. The forcing $\mathbb P^{c-c}$ shares this property with forcings defined in [2] and [9]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games $\mathbf{G}_M^{cap}(X)$ and the catching-capturing games $\mathbf{G}_M^{c-c}(X)$. In particular, these games are used to isolate a special family of countable elementary submodels $M \prec H_{\theta}$ that occur as side conditions in $\mathbb P^{c-c}$ and thus allow to control the forcing in a strong way.

math.LO↗
arXiv2022

Games on AF-algebras

We analyze $\mathrm{C}^\ast$-algebras, particularly AF-algebras, and their $K_0$-groups in the context of the infinitary logic $\mathcal{L}_{ω_1 ω}$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_0$-groups as ordered unital groups, we prove that $K_0(A) \equiv_{ω\cdot α} K_0(B)$ implies $A \equiv_αB$, where $M \equiv_βN$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $β$. This implication is proved using techniques from Elliott's classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the metric setting. We use moreover this result to build a family $\{ A_α\}_{α< ω_1}$ of pairwise non-isomorphic separable simple unital AF-algebras which satisfy $A_α\equiv_αA_β$ for every $α< β$. In particular, we obtain a set of separable simple unital AF-algebras of arbitrarily high Scott rank. Next, we give a partial converse to the aforementioned implication, showing that $A \otimes \mathcal{K} \equiv_{ω+ 2 \cdot α+2} B \otimes \mathcal{K}$ implies $K_0(A) \equiv_αK_0(B)$, for every unital $\mathrm{C}^\ast$-algebras $A$ and $B$.

math.LO↗
arXiv2021

Non-vanishing higher derived limits

In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups $\mathbf A$ indexed by elements of $ω^ω$. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits $\lim^n \mathbf A$ must vanish, for $n>0$. They also proved that under the Continuum Hypothesis $\lim^1 \mathbf A \neq 0$. The question whether $\lim^n \mathbf A$ vanishes, for $n>0$, has attracted considerable interest from set theorists. Dow, Simon and Vaughan showed that under PFA $\lim^1 \mathbf A =0$. Bergfalk show that it is consistent that $\lim^2\mathbf A$ does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that $\lim^n \mathbf A =0$, for all $n$. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Henson. We complete the picture by showing that, for any $n>0$, it is relatively consistent with ZFC that $\lim^n \mathbf A \neq 0$.

math.LO↗
arXiv2019

Guessing models and the approachability ideal

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call ${\rm GM}^+(ω_3,ω_1)$ holds. This principle implies ${\rm ISP}(ω_2)$ and ${\rm ISP}(ω_3)$, and hence the tree property at $ω_2$ and $ω_3$, the Singular Cardinal Hypothesis, and the failure of the weak square principle $\square(ω_2,λ)$, for all regular $λ\geq ω_2$. In addition, it implies that the restriction of the approachability ideal $I[ω_2]$ to the set of ordinals of cofinality $ω_1$ is the non stationary ideal on this set. The consistency of this last statement was previously shown by Mitchell.

math.LO↗
arXiv2016

Ranks of Maharam algebras

Solving a well-known problem of Maharam, Talagrand [17] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This gives rise to uncountably many non isomorphic separable atomless Maharam algebras.

math.FA↗
arXiv2016

A direct proof of the five element basis theorem

We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders. Our argument is based on the approach of notion of saturation of Aronszajn trees considered by Koenig, Larson, Moore and Velickovic and simplifies the original proof of Moore.

math.LO↗
arXiv2013

Positional strategies in long Ehrenfeucht-Fraissé games

We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fraïssé-game of length ω_1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.

math.LO↗
arXiv2013

Proper forcing remastered

In these notes we present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: the Friedman-Mitchell poset for adding a club in ω_2 with finite conditions, Koszmider's forcing construction of a strong chain of length ω_2 of functions from ω_1 to ω_1, and the Baumgartner-Shelah forcing construction of a thin very tall superatomic Boolean algebra.

math.LO↗
arXiv2013

Stationary reflection principles and two cardinal tree properties

We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi stationary set reflection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two reflection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set reflection is analogous to strong compactness.

math.LO↗
arXiv1995

The complexity of the reals in inner models of set theory

The usual definition of the set of constructible reals is $Σ^1_2$. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be analytic or even Borel in a nontrivial way. A related problem was posed by K. Prikry: can there exist a nonconstructible perfect set of constructible reals? The main result of this paper is a negative answer to Friedman's question. In fact we prove that if $M$ is an inner model of set theory and the set of reals in $M$ is analytic then either all reals are in $M$ or else $\aleph _1^M$ is countable. We also extend this result to higher levels of the projective hierarchy under appropriate large cardinal assumptions. Concerning Prikry's problem we show that the answer is negative if "perfect" is replaced by "superperfect" but that it can be positive if "constructible" is replaced by "belonging to some inner model $M$".

math.LO↗