arXiv subjects

Radek Honzik

Publications and source records attributed to Radek Honzik.

8 recordsLinked to original sources

arXiv2026

Rado's Conjecture and the random algebra

Rado's Conjecture (RC) is a compactness principle for a certain class of partial orders, namely trees $T$ of height $\omega_1$ without cofinal branches, postulating that a partial order $P$ from this class can be decomposed into at most countably many antichains if and only if all its suborders of size $\omega_1$ can be decomposed into at most countably many antichains. Rado's Conjecture is thus an uncountable version of Mirsky's theorem asserting that for every natural number $n$, every infinite partial order $P$ can be decomposed into at most $n$ many antichains if and only if all its finite suborders can be decomposed into at most $n$ many antichains. Todorcevic showed that RC is consistent modulo a strongly compact cardinal. RC implies $2^\omega \le \omega_2$, and has powerful consequences such as the Singular Cardinal Hypothesis, the failure of $\square(\kappa)$ for every regular $\kappa \ge \omega_2$ (and hence in particular the Projective Determinacy), and the Strong Chang Conjecture. It is also known that it is incompatible with Martin Axiom. We show that RC is consistent with $2^\omega = \omega_2$ and the cardinal invariants in Cichon diagram corresponding to forcing with the random algebra, i.e., $\mathfrak{d} = \omega_1$, $\mathrm{cov}(\mathcal{N}) = \omega_2$, $\mathrm{non}(\mathcal{N}) = \omega_1$. This provides a new pattern of cardinal invariants known to be consistent with RC. To prove the theorem, we first observe that random algebras do not specialize non-special trees of height $\omega_1$ without cofinal branches. Then we use the random algebra $\mathcal{B}_\kappa$ for a strongly compact $\kappa$ to define a new version of Mitchell forcing which yields the required result.

math.LO↗

arXiv2026

Forcing with random variables in bounded arithmetics and set theory

We analyse the Boolean-valued random forcing $B_{M,\Omega}$ in bounded arithmetics developed in Krajicek (Forcing with random variables and proof complexity, vol. 382, Cambridge University Press, 2011) from the perspective of the forcing in set theory. We observe that under the assumption that $M$ is a non-standard $\omega_1$-saturated model of true arithmetics of size $\omega_1$, and $\Omega \in M$ is a non-standard number, $B_{M,\Omega}$ is isomorphic to the probability (random) algebra corresponding to the product measure space on $2^{\omega_1}$ (and hence does not depend on $M$ and $\Omega$). Thus, in a well-defined sense, the forcing $B_{M,\Omega}$ adds a "random integer" to the model $M$, using a non-separable algebra corresponding to $2^{\omega_1}$. If $G$ is a generic filter for $B_{M,\Omega}$ over a transitive model of set theory $V$, we naturally define in $V[G]$ two-valued generic extensions $M[G]^{R}$ of $M$ which correspond to Boolean-valued models in Krajicek's book (where $R$ ranges over collections of random variables which function as names for new integers). We study the relationship between the linear order $(M,<)$ and its extensions $(M[G]^R,<)$, proving several results on the extent of the mutual density of new integers in $M[G]^{R}$ and the "ground-model" integers in $M$. At the end, we discuss some advantages and limitations of interpreting forcing in bounded arithmetics (and other weak theories) in the framework of set-theoretic forcing, providing an alternative to an axiomatic approach to forcing in bounded arithmetics formulated by Atserias and M\"uller in Partially definable forcing and bounded arithmetic, Archive for Mathematical Logic 54 (2015), 1-33.

math.LO↗

arXiv2025

Compactness for small cardinals in mathematics: principles, consequences, and limitations

We discuss some well-known compactness principles for uncountable structures of small regular sizes ($\omega_n$ for $2 \le n<\omega$, $\aleph_{\omega+1}$, $\aleph_{\omega^2+1}$, etc.), consistent from weakly compact (the size-restricted versions) or strongly compact or supercompact cardinals (the unrestricted versions). We divide the principles into logical principles (various tree properties) and mathematical principles, which directly postulate compactness for structures like groups, graphs, or topological spaces (for instance, countable chromatic and color compactness of graphs, compactness of abelian groups, $\Delta$-reflection, Fodor-type reflection principle, and Rado's Conjecture). We focus on indestructibility, or preservation, of these principles in forcing extensions. Using the existing preservation results we observe that many traditional problems such as Suslin Hypothesis, Whitehead's Conjecture, Kaplansky's Conjecture, and Baumagartner's Axiom, are independent from some of the strongest forms of compactness at $\omega_2$. Additionally, we observe that Rado's Conjecture plus $2^\omega = \omega_2$ is consistent with the negative solutions of some of these conjectures (as they hold in $V = L$), verifying that they hold in suitable Mitchell models. Finally, we comment on whether the compactness principles under discussion are good candidates for axioms. We consider their consequences and the existence or non-existence of convincing unifications (such as Martin's Maximum or Rado's Conjecture). This part is a modest follow-up to the articles by Foreman ``Generic large cardinals: new axioms for mathematics?'' (1998) and Feferman et al. ``Does mathematics need new axioms?'' (2000).

math.LO↗

arXiv2023

Generalized cardinal invariants for an inaccessible $\kappa$ with compactness at $\kappa^{++}$

We show that if the existence of a supercompact cardinal $\kappa$ with a weakly compact cardinal $\lambda$ above $\kappa$ is consistent, then the following are consistent as well (where $\mathfrak{t}(\kappa)$ and $\mathfrak{u}(\kappa)$ are the tower number and the ultrafilter number, respectively): (i) There is an inaccessible cardinal $\kappa$ such that $\kappa^+ < \mathfrak{t}(\kappa)= \mathfrak{u}(\kappa)< 2^\kappa$ and $SR(\kappa^{++})$ hold, and (ii) There is an inaccessible cardinal $\kappa$ such that $\kappa^+ = \mathfrak{t}(\kappa) < \mathfrak{u}(\kappa)< 2^\kappa$ and $SR(\kappa^{++}), TP(\kappa^{++})$ and $\neg wKH(\kappa^+)$ hold. The cardinals $\mathfrak{u}(\kappa)$ and $2^\kappa$ can have any reasonable values in these models. We obtain these results by combining the forcing construction from Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results for compactness principles. Apart from $\mathfrak{u}(\kappa)$ and $\mathfrak{t}(\kappa)$ we also compute the values of $\mathfrak{b}(\kappa)$, $\mathfrak{d}(\kappa)$, $\mathfrak{s}(\kappa)$, $\mathfrak{r}(\kappa)$, $\mathfrak{a}(\kappa)$, $\mathrm{cov}(M_\kappa)$, $\mathrm{add}(M_\kappa)$, $\mathrm{non}(M_\kappa)$, $\mathrm{cof}(M_\kappa)$ which will all be equal to $\mathfrak{u}(\kappa)$. In (ii), we compute $\mathfrak{p}(\kappa) = \mathfrak{t}(\kappa) = \kappa^+$ by observing that the $\kappa^+$-distributive quotient of the Mitchell forcing adds a tower of size $\kappa^+$. Finally, we observe that (i) and (ii) hold also for the traditional invariants on $\kappa = \omega$, using Mitchell forcing up to a weakly compact cardinal; in this case we also obtain the disjoint stationary sequence property $DSS(\omega_2)$, which implies the negation of the approachability property $\neg AP(\omega_2)$.

math.LO↗

arXiv2022

Indestructibility of some compactness principles over models of PFA

We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $ω$ will not add an $ω_2$-Aronszajn tree or a weak $ω_1$-Kurepa tree, and moreover no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree (a tree of height and size $ω_1$ with at least $ω_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $ω_2$-Aronszajn or $ω_1$-Kurepa trees. We actually prove more: We show that a consequence of $\mathsf{PFA}$, namely the guessing model principle, $\mathsf{GMP}$, which is equivalent to the ineffable slender tree property, $\mathsf{ISP}$, is preserved by adding any number of Cohen subsets of $ω$. And moreover, $\mathsf{GMP}$ implies that no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree. For more generality, we study the principle $\mathsf{GMP}$ at an arbitrary regular cardinal $κ= κ^{<κ}$ (we denote this principle $\mathsf{GMP}_{κ^{++}}$), and as an application we show that there is a model in which there are no weak $\aleph_{ω+1}$-Kurepa trees and no $\aleph_{ω+2}$-Aronszajn trees.

math.LO↗

arXiv2019

Small u(kappa) at singular kappa with compactness at kappa++

We show that the tree property, stationary reflection and the failure of approachability at $κ^{++}$ are consistent with $\mathfrak{u}(κ) = κ^+ < 2^κ$, where $κ$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $λ$ is a regular cardinal, then stationary reflection at $λ^+$ is indestructible under all $λ$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).

math.LO↗

arXiv2019

Indestructibility of the tree property

In the first part of the paper, we show that if $ω\le κ< λ$ are cardinals, $κ^{<κ} = κ$, and $λ$ is weakly compact, then in $V[\M(κ,λ)]$ the tree property at $λ= κ^{++V[\M(κ,λ)]}$ is indestructible under all $κ^+$-cc forcing notions which live in $V[\Add(κ,λ)]$, where $\Add(κ,λ)$ is the Cohen forcing for adding $λ$-many subsets of $κ$ and $\M(κ,λ)$ is the standard Mitchell forcing for obtaining the tree property at $λ= (κ^{++})^{V[\M(κ,λ)]}$. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that $λ$ is supercompact and generalize the construction and obtain a model $V^*$, a generic extension of $V$, in which the tree property at $(κ^{++})^{V^*}$ is indestructible under all $κ^+$-cc forcing notions living in $V[\Add(κ,λ)]$, and in addition by all forcing notions living in $V^*$ which are $κ^+$-closed and ``liftable'' in a prescribed sense (such as $κ^{++}$-directed closed forcings or well-met forcings which are $κ^{++}$-closed with the greatest lower bounds).

math.LO↗

arXiv2013

Easton functions and supercompactness

Suppose $κ$ is $λ$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ with critical point $κ$, and further suppose that $F$ is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton's theorem: (1) $\forallα$ $α<\textrm{cf}(F(α))$ and (2) $α<β$ $\Longrightarrow$ $F(α)\leq F(β)$. In this article we address the question: assuming GCH, what additional assumptions are necessary on $j$ and $F$ if one wants to be able to force the continuum function to agree with $F$ globally, while preserving the $λ$-supercompactness of $κ$? We show that, assuming GCH, if $F$ is any function as above, and in addition for some regular cardinal $λ>κ$ there is an elementary embedding $j:V\rightarrow M$ with critical point $κ$ such that $κ$ is closed under $F$, the model $M$ is closed under $λ$-sequences, $H(F(λ))\subseteq M$, and for each regular cardinal $γ\leq λ$ one has $(|j(F)(γ)|=F(γ))^V$, then there is a cardinal-preserving forcing extension in which $2^δ=F(δ)$ for every regular cardinal $δ$ and $κ$ remains $λ$-supercompact. This answers a question of B. Cody, M. Magidor, On supercompactness and the continuum function, Ann. Pure Appl. Logic, (2013).

math.LO↗