23 research outputs found
Chain models, trees of singular cardinality and dynamic EF games
Let κ be a singular cardinal. Karp's notion of a chain model of size ? is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf(κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf(κ) = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf(κ)κ.The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models
On properties of theories which preclude the existence of universal models
We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality ? when certain cardinal arithmetic assumptions about ? implying the failure of GCH (and close to the failure of SCH) hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to non-universality, as was raised by the earlier work of Shelah. One of our examples is the theory View the MathML source for which non-universality results similar to the ones we obtain are already known; hence we may view our results as an abstraction of the known results from a concrete theory to a class of theories. We show that no theory with the oak property is simple
Extension and reconstruction theorems for the Urysohn universal metric space
We prove some extension theorems involving uniformly continuous maps of the
universal Urysohn space. We also prove reconstruction theorems for certain
groups of autohomeomorphisms of this space and of its open subsets.Comment: Final and shortened version, 25 pages, to appear in Czechoslovak
Math.
Tree indiscernibilities, revisited
We give definitions that distinguish between two notions of indiscernibility
for a set \{a_\eta \mid \eta \in \W\} that saw original use in \cite{sh90},
which we name \textit{\s-} and \textit{\n-indiscernibility}. Using these
definitions and detailed proofs, we prove \s- and \n-modeling theorems and
give applications of these theorems. In particular, we verify a step in the
argument that TP is equivalent to TP or TP that has not seen
explication in the literature. In the Appendix, we exposit the proofs of
\citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.Comment: submitte
Club does not imply the existence of a Suslin tree
We prove that club does not imply the existence of a Suslin tree, so answering a question of I. Juhasz
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.Comment: Final versio
Universality of uniform Eberlein compacta
We prove that if , then there is no family of less than c-algebras of size which are jointly universal for c-algebras of size . On the other hand, it is consistent to have a cardinal as large as desired and satisfying \lambda^{\lambda^{++}, while there are c-algebras of size that are jointly universal for c-algebras of size . Consequently, by the known results of M. Bell, it is consistent that there is as in the last statement and uniform Eberlein compacta of weight such that at least one among them maps onto any Eberlein compact of weight (we call such a family universal). The only positive universality results for Eberlein compacta known previously required the relevant instance of to hold. These results complete the answer to a question of Y. Benyamini, M. E. Rudin and M. Wage from 1977 who asked if there always was a universal uniform Eberlein compact of a given weight
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Formalizing Ordinal Partition Relations Using Isabelle/HOL
This is an overview of a formalisation project in the proof assistant
Isabelle/HOL of a number of research results in infinitary combinatorics and
set theory (more specifically in ordinal partition relations) by
Erd\H{o}s--Milner, Specker, Larson and Nash-Williams, leading to Larson's proof
of the unpublished result by E.C. Milner asserting that for all , \omega^\omega\arrows(\omega^\omega, m). This material has been
recently formalised by Paulson and is available on the Archive of Formal
Proofs; here we discuss some of the most challenging aspects of the
formalisation process. This project is also a demonstration of working with
Zermelo-Fraenkel set theory in higher-order logic