24 pages. To appear in: 'Information and Computation' Special Issue WoLLIC19 post-proceedings. Same "Preliminaries" section as e.g. arXiv:1905.04058, arXiv:1908.05676
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These counterexamples are often crucial in establishing reversals in the Reverse Mathematics program. All the previous is essentially limited to a language that can only express countable mathematics directly. The aim of this paper is to show that reversals and recursive counterexamples, countable in nature as they might be, directly yield new and interesting results about uncountable mathematics with little-to-no modification. We shall treat the following topics/theorems: the monotone convergence theorem/Specker sequences, compact and closed sets in metric spaces, the Rado selection lemma, the ordering and algebraic closures of fields, and ideals of rings. The higher-order generalisation of sequence is of course provided by nets (aka Moore-Smith sequences ).
Solecki proved that the group of automorphisms of a countable structure cannot be an uncountable free abelian group. See more in Just, Shelah and Thomas math.LO/0003120 where as a by product we can say something on on uncountable structures. We prove here the following Theorem: If A is a countable model, then Aut(M) cannot be a free uncountable group.
We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.
We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a proof-theoretic bound on definable countable ordinals whose existence is provable from the existence of second order indescribable cardinals over ZF.
D'Aquino, Knight and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of $ω_1$-like models of PA.
Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality κ. We denote by μ(respectively \hatμ) the number of cardinals (respectively infinite cardinals) less than or equal to κ. We prove that I(T,κ), as a function of κ> \aleph_0, is the minimum of 2^κ and one of the following functions: 1. 2^κ; 2. the constant function 1; 3. |\hatμ^n/{\sim_G}|-|(\hatμ- 1)^n/{\sim_G}| if \hatμ<ωfor some 1<n<ωand \hatμif \hatμ>= ωsome group G <= Sym(n); 4. the constant function \beth_2; 5. \be…
We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, by analogy with the way in which Ramsey's Theorem can be generalized to weakly compact cardinals. But unlike Ramsey's Theorem, the outcome of this paper is that the natural generalizations of Hindman's Theorem proposed here tend to fail at all uncountable cardinals.
Every mathematical structure has an elementary extension to a pseudo-countable structure, one that is seen as countable inside a suitable class model of set theory, even though it may actually be uncountable. This observation, proved easily with the Boolean ultrapower theorem, enables a sweeping generalization of results concerning countable models to a rich realm of uncountable models. The Barwise extension theorem, for example, holds amongst the pseudo-countable models -- every pseudo-countable model of ZF admits an end extension to a model of ZFC+V=L. Indeed, the class of pseudo-countable m…
We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $κ$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both $T$ and $A$ have cardinality $κ$, (2) $|\bar{a}\cap T|=κ$ for each $a\in A$, (3) for each uncountable subset of $T$ contains some elements of $A$, and so (i) there is an almost disjoint family of subsets of the reals with size and chromatic number $κ$, (ii) there is a locally compact, locally countable $T_2$ space with cardinality spectrum $\{ω,κ\}$.
We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\geq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erdös--Rényi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measure…
Combinatorial covering properties in an uncountable setting: canonical examples
Bar-Ilan University（巴伊兰大学）；Ariel University（阿里埃勒大学）；Lodz University of Technology（罗兹理工大学）；University of Warsaw（华沙大学）；Technische Universität Wien（维也纳工业大学）
We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. $[0,1]$ as 'almost finite', while the latter allows one to treat uncountable sets like e.g. $\mathbb{R}$ as 'almost countable'. This reduction of the uncountable to the finite/coun…
This paper shows that, even at the most basic level, the parallel, countable branching and uncountable branching recurrences of Computability Logic (see http://www.cis.upenn.edu/~giorgi/cl.html) validate different principles.
In [2] Su Gao proves that the following are equivalent for a countable $M$ (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of $M$. (II) There exists some $j\in \overline{Aut(M)}\setminus Aut(M)$, where $\overline{Aut(M)}$ is the closure of $Aut(M)$ under the product topology in $ω^ω$. (III) There is an $L_{ω_1,ω}$- elementary embedding $j$ from $M$ to itself such that $range(j)\subset M$. We generalize his theorem to all cardinals $κ$ of of cofinality $ω$ (cf. theorem 4.2). The following are equivalent: (I$^*$) There is a model of the Scott sentence of $M$ of size…
Shelah shows that certain revised countable support (RCS) iterations do not add reals. His motivation is to establish the independence (relative to large cardinals) of Avraham's problem on the existence of uncountable non-constuctible sequences all of whose proper initial segments are constructible, Friedman's problem on whether every 2-coloring of $S^2_0=\{α<ω_2\colon\cf(α)=ω\}$ has an uncountable sequentially closed homogeneous subset, and existence of a precipitous normal filter on $ω_2$ with $S^2_0\in{\cal F}$. The posets which Shelah uses in these constructions are Prikry forcing, Namba f…
Forcing in bounded arithmetic and set theory: analogies and differences
Charles University, Faculty of Arts（查理大学文学院）；University of Warwick, Department of Computer Science（华威大学计算机科学系）；Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）
In this paper we analyse some questions concerning trees on $κ$, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in \cite[Question 5.2]{FKK16} about the diagram for regularity properties.
We extend the concept of "almost indiscernible theory" introduced by Pillay and Sklinos in [Bull. Symb. Log., 2015] (which was itself a modernization and expansion of Baldwin and Shelah [Algebra Universalis, 1983]), to uncountable languages and uncountable parameter sequences. Roughly speaking a theory $T$ is almost indiscernible if some saturated model is in the algebraic closure of an indiscernible set of sequences. We show that such a theory $T$ is nonmultidimensional, superstable, and stable in all cardinals $\ge |T |$ . We prove a structure theorem for sufficiently large $a$-models $M$: T…
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Physics and Mechanics（数学、力学与物理研究所）；Iowa State University（爱荷华州立大学）
The Heine-Borel theorem for uncountable coverings has recently emerged as an interesting and central principle in higher-order Reverse Mathematics and computability theory, formulated as follows: HBU is the Heine-Borel theorem for uncountable coverings given as $\cup_{x\in [0,1]}(x-Ψ(x), x+Ψ(x))$ for arbitrary $Ψ:[0,1]\rightarrow \mathbb{R}^{+}$, i.e. the original formulation going back to Cousin (1895) and Lindelöf (1903). In this paper, we show that HBU is equivalent to its restriction to functions continuous almost everywhere, an elegant robustness result. We also obtain a nice splitting HB…
