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Mirna Džamonja

Publications and source records attributed to Mirna Džamonja.

At least 37 records · Page 2Linked to original sources
arXiv2019

A new foundational crisis in mathematics, is it really happening?

The article reconsiders the position of the foundations of mathematics after the discovery of HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics. The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.

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arXiv2016

A Framework for Forcing Constructions at Successors of Singular Cardinals

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size is less than $2^κ$ and such that any graph on $κ^+$ embeds into one of the graphs in the collection.

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arXiv2015

Some Banach spaces added by a Cohen real

We study certain Banach spaces that are added in the extension by one Cohen real. Specifically, we show that adding just one Cohen real to any model adds a Banach space of density $\aleph_1$ which does not embed into any such space in the ground model such a Banach space can be chosen to be UG This has consequences on the the isomorphic universality number for Banach spaces of density $\aleph_1$, which is hence equal to $\aleph_2$ in the standard Cohen model and the same is true for UG spaces. Analogous universality results for Banach spaces are true for other cardinals, by a different proof.

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arXiv2014

The singular world of singular cardinals

The article uses two examples to explore the statement that, contrary to the common wisdom, the properties of singular cardinals are actually more intuitive than those of the regular ones.

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arXiv2014

Small universal families of graphs on $\aleph_{ω+1}$

We prove that it is consistent that $\aleph_ω$ is strong limit, $2^{\aleph_ω}$ is large and the universality number for graphs on $\aleph_{ω+1}$ is small. The proof uses Prikry forcing with interleaved collapsing.

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arXiv2013

Isomorphic universality and the number of pairwise non-isomorphic models in the class of Banach spaces

We study isomorphic universality of Banach spaces of a given density and a number of pairwise non-isomorphic models in the same class. We show that in the Cohen model the isomorphic universality number for Banach spaces of density $\aleph_1$ is $\aleph_2$, and analogous results are true for other cardinals (Theorem 1.2(1)) and that adding just one Cohen real to any model destroys the old universality of Banach spaces of density $\aleph_1$ (Theorem 1.5). Moreover, adding one Cohen real adds a weakly compactly generated Banach space of density $\aleph_1$ which does not embed into any Banach space of density $\aleph_1$ in the ground model. We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces and use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embeddings (very positive embeddings), is high (Theorem 4.8(1)), and similarly for the number of pairwise non-isomorphic models (Theorem 4.8(2)).

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arXiv2011

On the isomorphism problem for measures on Boolean algebras

The paper investigates possible generalisations of Maharam's theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure. We give some partial analogues to be used for a classification of algebras that support a finitely additive non-atomic measure with a higher uniform regularity number. We show that some naturally induced equivalence relations connected to metric isomorphism are quite complex even on the Cantor algebra and therefore probably we cannot hope for a nice general classification theorem for finitely--additive measures. We present an example of a Boolean algebra which supports only separable measures but no uniformly regular one.

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arXiv2011

On triangleleft^*-maximality

This paper investigates a connection between the ordering triangleleft^ast among theories in model theory and the (N)SOP_n hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP_2 and SOP_1, and gives a strong connection between SOP_1 and the maximality in Keisler ordering. Together with the known results about the connection between the (N)SOP_n hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property.

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arXiv2010

Introducing a nontrivial square_omega

We define a nontrivial version of the square principle $\Box_ω$, which we then show consistent by means of forcing with finite conditions. This paper has been withdrawn by the author due to the fact that the presented $\Box_ω$ can be constructed in ZFC, as shown by Lajos Soukup and Philip Welch. Note, that the forcing itself has not been found flawed.

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arXiv2006

On $D$-spaces and Discrete Families of Sets

We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using the concept of discrete families. We note the connection between non-reflection arguments involving discrete families and the well known question of the existence of families allowing partial transversals without having a transversal themselves, and use it to give non-trivial instances of the incompactness phenomenon in the context of discretisations.

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arXiv2006

On club-like principles on regular cardinals above $\beth_ω$

We prove that for regular $λ$ above a strong limit singular $μ$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $λ$ and $μ$ there are coboundedly many regular $κ<μ$ such that $\clubsuit^-(S^λ_κ)$ holds whenever $λ=λ^{<κ}$.

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arXiv2006

Parametrized $\diamondsuit$ principles

We will present a collection of guessing principles which have a similar relationship to $\diamond$ as cardinal invariants of the continuum have to $\CH$. The purpose is to provide a means for systematically analyzing $\diamond$ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of $\CH$ and $\diamond$ in models such as those of Laver, Miller, and Sacks.

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arXiv2006

On the existence of universal models

Suppose that $λ=λ^{<λ} \ge\aleph_0$, and we are considering a theory $T$. We give a criterion on $T$ which is sufficient for the consistent existence of $λ^{++}$ universal models of $T$ of size $λ^+$ for models of $T$ of size $\leλ^+$, and is meaningful when $2^{λ^+}>λ^{++}$. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind possible applications in analysis, we further observe that for such $λ$, for any fixed $μ>λ^+$ regular with $μ=μ^{λ^+}$, it is consistent that $2^λ=μ$ and there is no normed vector space over ${\Bbf Q}$ of size $<μ$ which is universal for normed vector spaces over ${\Bbf Q}$ of dimension $λ^+$ under the notion of embedding $h$ which specifies $(a,b)$ such that $\norm{h(x)}/\norm{x}\in (a,b)$ for all $x$.

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arXiv2006

An application of CAT

We comment on a question of Justin Moore on colourings of pairs of nodes in an Aronszajn tree and solve an instance of it.

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arXiv2006

Measure Recognition Problem

This is an article in mathematics, specifically in set theory. On the example of the Measure Recognition Problem (MRP) the article highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra $\algB$ and a property $Φ$ of measures one can recognize by purely combinatorial means if $\algB$ supports a strictly positive measure with property $Φ$. The most famous instance of this problem is MRP(countable additivity), and in the first part of the article we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP(nonatomic) and MRP(separable). Then we show how our recent work D\v zamonja and Plebanek (2006) gives an easy solution to the former of these problems, and gives some partial information about the latter. The long term goal of this line of research is to obtain a structure theory of Boolean algebras that support a finitely additive strictly positive measure, along the lines of Maharam theorem which gives such a structure theorem for measure algebras.

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