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 , ) 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)...

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 + + ( X ) Φ i is consistent for 1 i k (each alone), where Φ i are normal formulae then we show + ( X ) Φ 1 + + ( 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 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...

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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) is a Lindelöf space, (2) is a Lindelöf space, (3) is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of is separable, (6) in , 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 : is continuous at a point x iff f is sequentially continuous...