The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brelot

Peter Loeb; Bertram Walsh

Displaying similar documents to “The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brelot”

On the Leibniz-Mycielski axiom in set theory

Ali Enayat (2004)

Fundamenta Mathematicae

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Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that $(V_{α}, \in)$ satisfies φ(x) ∧¬ φ(y). We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows: 1. In the presence of ZF, the following are equivalent: (a)...

Hyperplanes in matroids and the axiom of choice

Marianne Morillon (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show that in set theory without the axiom of choice ZF, the statement sH: “Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it” implies AC $^{fin}$ , the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC $^{fin}$ in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?

Axioms weaker then $R_{0}$

J. Guia (1984)

Matematički Vesnik

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Complexity of the axioms of the alternative set theory

Antonín Sochor (1993)

Commentationes Mathematicae Universitatis Carolinae

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If is a complete theory stronger than $_{Fin}$ such that axiom of extensionality for classes + + $(\exists X) Φ_{i}$ is consistent for 1 $\leq i \leq k$ (each alone), where $Φ_{i}$ are normal formulae then we show + $(\exists X) Φ_{1} + \dots + (\exists X) Φ_{k}$ + scheme of choice is consistent. As a consequence we get: there is no proper $Δ_{1}$ -formula in + scheme of choice. Moreover the complexity of the axioms of is studied, e.gẇe show axiom of extensionality is $Π_{1}$ -formula, but not $Σ_{1}$ -formula and furthermore prolongation axiom, axioms of choice and cardinalities...

The gap between I₃ and the wholeness axiom

Paul Corazza (2003)

Fundamenta Mathematicae

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∃κI₃(κ) is the assertion that there is an elementary embedding $i : V_{λ} \to V_{λ}$ 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...

The interdependence of certain consequences of the axiom of choice

A. Lévy (1964)

Fundamenta Mathematicae

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[unknown]

M. Jelić (1990)

Matematički Vesnik

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When is $𝐍$ Lindelöf?

Horst Herrlich, George E. Strecker (1997)

Commentationes Mathematicae Universitatis Carolinae

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Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $ℕ$ is a Lindelöf space, (2) $ℚ$ is a Lindelöf space, (3) $ℝ$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $ℝ$ is separable, (6) in $ℝ$ , a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$ , (7) a function $f : ℝ \to ℝ$ is continuous at a point $x$ iff $f$ is sequentially continuous...