A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the collection of introreducible sets is $Π^1_1$-complete, so that there is no simple characterization of the introreducible sets; and that every introenumerable set has an introreducible subset.
There is an infinite subset of a Martin-Löf random set of integers that does not compute any Martin-Löf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a Martin-Löf random set of integers, and apply a result of Miller.
A theorem of Erdos asserts that every infinite subset of Euclidean n-space R^n has a subset of the same cardinality having no repeated distances. This theorem is generalized here as follows: If (R^n,E) is an algebraic hypergraph that does not have an infinite, complete subset, then every infinite subset of it has an independent subset of the same cardinality.
A MAD (maximal almost disjoint) family is an infinite subset A of the infinite subsets of {0,1,2,..} such that any two elements of A intersect in a finite set and every infinite subset of {0.1.2...} meets some element of $å$ in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G-delta set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology in inherits a subset of the Power set of {0,1,2,..}, ie the Cantor set.
Erdös proved that for every infinite $X \subseteq \mathbb{R}^d$ there is $Y \subseteq X$ with $|Y|=|X|$, such that all pairs of points from $Y$ have distinct distances, and he gave partial results for general $a$-ary volume. In this paper, we search for the strongest possible canonization results for $a$-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose $T$ is a stable theory, $Δ$ is a finite set of formulas of $T$, $M \models T$, and $X$ is an infinite subset of $M$. Then there is $Y …
The theory of finitely supported algebraic structures represents a reformulation of Zermelo-Fraenkel set theory in which every construction is finitely supported according to the action of a group of permutations of some basic elements named atoms. In this paper we study the properties of finitely supported sets that contain infinite uniformly supported subsets, as well as the properties of finitely supported sets that do not contain infinite uniformly supported subsets. For classical atomic sets, we study whether they contain or not infinite uniformly supported subsets.
In the present paper we are interested in properties of forcing notions which measure in a sense the distance between the ground model reals and the reals in the extension. We look at the ways the ``new'' reals can be aproximated by ``old'' reals. We consider localizations for infinite subsets of omega. Though each member of [omega]^omega can be identified with its increasing enumeration, the (standard) localizations of the enumeration does not provide satisfactory information on successive points of the set. They give us ``candidates'' for the n-th point of the set but the same candidates can…
The tree forcing method given by (Liu 2015) enables the cone avoiding of strong enumeration of a given tree, within a subset or co-subset of an arbitrary given set, provided the given tree does not admit computable strong enumeration. Using this result, we settled and reproduced a series of problems in reverse mathematics. In this paper, we demonstrate cone avoiding results within an infinite subset of a given 1-random set. We show that for any given 1-random set $X$, there exists an infinite subset $Y$ of $X$ such that $Y$ does not compute any real with positive effective Hausdorff dimension,…
We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of 1-types and the length of the sequences. Specifically, if $κ\leq λ$, then $$\sup_{|A| = λ} |S^κ(A)| = (\sup_{|A| = λ} |S^1(A)|)^κ$$ We show that this holds for any abstract elementary class with $λ$ amalgamation, but it is new for first order theories when $κ$ is infinite. No such calculation is possible for nonalgebraic types. We introduce a generalization of nonalgebraic types for which the same upper bound holds.
We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if $X\in\mathcal A$ then $X$ has an infinite subset $Y$ such that no element of $\mathcal A$ is Turing computable from $Y$.
The infinite pigeonhole principle for 2-partitions asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we develop a new notion of forcing enabling a fine analysis of the computability-theoretic features of the pigeonhole principle. We deduce various consequences, such as the existence, for every set $A$, of an infinite subset of it or its complement of non-high degree. We also prove that every $Δ^0_3$ set has an infinite low${}_3$ solution and give a simpler proof of Liu's theorem that every set has an infinite subset in it or its complem…
We call a subset S of a topological vector space V linearly Borel, if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It will be shown that a Hamel base of an infinite dimensional Banach space can never be linearly Borel. This answers a question of Anatolij Plichko.
Matthew Baker investigated, in previous work, an elegant, infinite-length game that may be used to study subsets of real numbers. We present two accessible examples of how an important technique from set theory, or a different technique from infinite game theory, may be used to answer Baker's question on whether this game provides a precise characterization for countable subsets of real numbers, and we connect this game to the well-studied Banach-Mazur game from topology.
We give another proof that for every lambda >= beth_omega for every large enough regular kappa < beth_omega we have lambda^{[kappa]}= lambda, dealing with sufficient conditions for replacing beth_omega by aleph_omega. In section 2 we show that large pcf(a) implies existence of free sets. An example is that if pp(aleph_omega)> aleph_{omega_1} then for every algebra M of cardinality aleph_omega with countably many functions, for some a_n in M (for n< omega) we have a_n notin cl_M({a_l: l not= n, l<omega}). Then we present results complementary to those of section 2 (but not close enough): if IND…
We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary positive transcendence degree over $F$ and of infinite algebraic degree such that each $A_i$ is first-order definable over $K$. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable.
Hamkins and Kikuchi (2016 and 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. If $\mathcal{M}$ is a model of set theory, then $\langle M, \subseteq^\mathcal{M} \rangle$ is an atomic unbounded relatively complemented distributive lattice. If $\mathcal{M}$ is model of class theory, then $\langle M, \subseteq^\mathcal{M} \rangle$ is an infinite atomic boolean algebra. We identify the minimal subsystem of $\mathrm{ZF}$, $\mathrm{BAS}$, that ensures that the definable subset relation is an atomic unbounded…
Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show that, for almost every graph on $\mathbb N$, $(1)$ every set of positive upper density is universal, and $(2)$ every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on $\mathbb N$ will almost surely contain non-universal sets with divergent reciprocal sums) and, more generally, that neithe…
A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of $\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no uncountable equilateral sets is also considered.
The approximation of natural numbers subsets has always been one of the fundamental issues in computability theory. Computable approximation, $Δ_2$-approximation, as well as introducing the generically computable sets have been some efforts for this purpose. In this paper, a type of approximation for natural numbers subsets by computably enumerable sets will be examined. For an infinite and non-c.e set, $W_i$ will be an $A$.maximal (maximal inside $A$) if $W_i \subseteq A$, is infinite and $\forall j (W_i \subseteq W_j \subseteq A) \to Δ(W_i, W_j )< \infty$, where $Δ$ is the symmetric differen…
We prove that for each universal algebra $(A,\mathcal A)$ of cardinality $|A|\ge 2$ and an infinite set $X$ of cardinality $|X|\ge|\mathcal A|$, the $X$-th power $(A^X,\mathcal A^X)$ of the algebra $(A,\mathcal A)$ contains a free subset $\mathcal F\subset A^X$ of cardinality $|\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\mathcal I\subset\mathcal P(X)$ of cardinality $|\mathcal I|=|\mathcal P(X)|$ in the Boolean algebra $\mathcal P(X)$ of subsets of an infinite set $X$.
