Two consistency results on set mappings

2000 ◽  
Vol 65 (1) ◽  
pp. 333-338 ◽  
Author(s):  
Péter Komjáth ◽  
Saharon Shelah
Keyword(s):  
Natural Number ◽  
Consistency Results ◽  
Set Mappings ◽  
Free Sets

AbstractIt is consistent that there is a set mapping from the four-tuples of ωn into the finite subsets with no free subsets of size tn for some natural number tn. For any n < ω it is consistent that there is a set mapping from the pairs of ωn into the finite subsets with no infinite free sets. For any n < ω it is consistent that there is a set mapping from the pairs of ωn into ωn with no uncountable free sets.

scholarly journals A Note on the Number of $(k,l)$-Sum-Free Sets

10.37236/1508 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Tomasz Schoen
Keyword(s):  
Natural Number ◽  
Positive Integers ◽  
Free Sets

A set $A\subseteq {\bf N}$ is $(k,\ell)$-sum-free, for $k,\ell\in {\bf N}$, $k>\ell$, if it contains no solutions to the equation $x_1+\dots+x_k=y_1+\dots+y_{\ell}$. Let $\rho=\rho (k-\ell)$ be the smallest natural number not dividing $k-\ell$, and let $r=r_n$, $0\le r < \rho$, be such that $r\equiv n \pmod {\rho }$. The main result of this note says that if $(k-\ell)/\ell$ is small in terms of $\rho$, then the number of $(k,\ell)$-sum-free subsets of $[1,n]$ is equal to $(\varphi(\rho)+\varphi_r(\rho)+o(1)) 2^{\lfloor n/\rho \rfloor}$, where $\varphi_r(x)$ denotes the number of positive integers $m\le r$ relatively prime to $x$ and $\varphi(x)=\varphi_x(x)$.

Free sets for nowhere-dense set mappings

1981 ◽  
Vol 39 (1-2) ◽  
pp. 167-176 ◽  
Author(s):  
Uri Avraham
Keyword(s):  
Set Mappings ◽  
Free Sets ◽  
Nowhere Dense Set ◽  
Dense Set

On a combinatorial principle of Hajnal and Komjáth

1986 ◽  
Vol 51 (4) ◽  
pp. 1056-1060 ◽  
Author(s):  
Dan Velleman
Keyword(s):  
General Principle ◽  
Cardinal Number ◽  
Order Type ◽  
Infinite Cardinal ◽  
Combinatorial Principle ◽  
Mahlo Cardinal ◽  
One Step ◽  
Set Mappings ◽  
Free Sets

In their paper [3], Hajnal and Komjáth define the following combinatorial principle:Definition 1.1. Suppose κ is an infinite cardinal and n < ω. Then Hn(κ) is the statement: There is a function F: [κ]n → [[κ]ω]≤ω such that(a) ∀A ∈[κ]n ∀Y ∈ F(A)(Y ⊆ min (A)), and(b) .Hn(κ) is related to a more general principle introduced by Hajnal and Nagy in [4]. For applications of these principles to free sets for set mappings and Ramsey games we refer the reader to [3] and [4].In [3] Hajnal and Komjáth prove the consistency of ZFC + GCH + ∀n ∈ ω(Hn + 1(ωn + 1)), relative to an ω-Mahlo cardinal. They conjecture that L is a model of this theory, and suggest that the proof might require higher gap morasses. The first few cases of this conjecture are known to be true; it is easy to see that if CH holds then H1 (ω1) is true, and Laver proved that V = L implies H2(ω2). In this paper we go one step further and prove V = L → H3(ω3). Unfortunately our methods do not appear to give Hn (ωn) for n ≥ 4.Most of our notation is standard. If X is any set and κ is a cardinal number then [X]κ is the set of subsets of X with cardinality κ, and [X]≤κ is the set of subsets of X with cardinality ≤ κ. If X is a set of ordinals then tp(X) is the order type of X.

Free sets for set-mappings relative to a family of sets

2017 ◽  
Vol 63 (6) ◽  
pp. 605-613
Author(s):  
Antonio Avilés ◽  
Claribet Piña
Keyword(s):  
Set Mappings ◽  
Free Sets

Free sets for some special set mappings

1987 ◽  
Vol 58 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Ludomir Newelski
Keyword(s):  
Set Mappings ◽  
Free Sets

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

scholarly journals The Limits of Computation

Axiomathes ◽  
2021 ◽  
Author(s):  
Andrew Powell
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Lambda Calculus ◽  
Second Order ◽  
Functional Interpretation ◽  
Arbitrary Type ◽  
Computable Functions

AbstractThis article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions.

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