The end compactification |Γ| of a locally finite graph Γis the union of the graph and its ends, endowed with a suitable topology. We show that π₁(|Γ|) embeds into a nonstandard free group with hyperfinitely many generators, i.e. an ultraproduct of finitely generated free groups, and that the embedding we construct factors through an embedding into an inverse limit of free groups. We also show how to recover the standard description of π₁(|Γ|) given by Diestel and Sprüssel (2011). Finally, we give...

This paper presents an approach to the problem of quantifying the inequality of a finite population with respect to a (social, economical, etc.) fuzzy-valued attribute. For this purpose, the fuzzy hyperbolic inequality index is introduced, and some properties extending the basic ones for real-valued attributes are examined.

∃κI₃(κ) is the assertion that there is an elementary embedding $i:V_{\lambda }\to V_{\lambda }$ with critical point below λ, and with λ a limit. The Wholeness Axiom, or WA, asserts that there is a nontrivial elementary embedding j: V → V; WA is formulated in the language ∈,j and has as axioms an Elementarity schema, which asserts that j is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by j; and includes every instance of the Separation schema for j-formulas. Because no instance of Replacement...

It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of (ℚ, +) by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring 𝔄 of finite adèles. ...

This article is the second in a series of two Mizar articles constituting a formal proof of the Gödel Completeness theorem [15] for uncountably large languages. We follow the proof given in [16]. The present article contains the techniques required to expand a theory such that the expanded theory contains witnesses and is negation faithful. Then the completeness theorem follows immediately.

Schöning [14] introduced a notion of helping and suggested the study of the class ${\mathrm{P}}_{\mathrm{help}}\left(𝒞\right)$ of the languages that can be helped by oracles in a given class $𝒞$. Later, Ko [12], in order to study the connections between helping and “witness searching”, introduced the notion of self-helping for languages. We extend this notion to classes of languages and show that there exists a self-helping class that we call $\mathrm{SH}$ which contains all the self-helping classes. We introduce the Helping hierarchy whose levels are...

Schöning [14] introduced a notion of helping and suggested the study of the class ${\mathrm{P}}_{\mathrm{help}}\left(𝒞\right)$ of the languages that can be helped by oracles in a given class $𝒞$. Later, Ko [12], in order to study the connections between helping and "witness searching" , introduced the notion of self-helping for languages. We extend this notion to classes of languages and show that there exists a self-helping class that we call SH which contains all the self-helping classes. We introduce the Helping hierarchy whose levels...

According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X has the Hurewicz property if, and only if, each large open cover of X contains a groupable subcover. This...

We prove that the ideal (a) defined by the density topology is not $G_{\delta }$ generated. This answers a question of Z. Grande and E. Strońska.

By a ternary structure we mean an ordered pair $(X_0,T_0)$, where $X_0$ is a finite nonempty set and $T_0$ is a ternary relation on $X_0$. By the underlying graph of a ternary structure $(X_0,T_0)$ we mean the (undirected) graph $G$ with the properties that $X_0$ is its vertex set and distinct vertices $u$ and $v$ of $G$ are adjacent if and only if $$\{x\in X_0{T}_0(u,x,v)\}\cup \{x\in X_0{T}_0(v,x,u)\}=\{u,v\}.$$ A ternary structure $(X_0,T_0)$ is said to be the B-structure of a connected graph $G$ if $X_0$ is the vertex set of $G$ and the following statement holds for all $u,x,y\in X_0$: $T_0(x,u,y)$ if and only if $u$ belongs to an induced $x-y$...