ZFC

ZFC

ZFC is an abbreviation for Zermelo-Fraenkel set theory, a commonly used axiom system for treating sets in mathematics. It was constructed by mathematicians Ernst Zermelo and Abraham Fraenkel and includes the axiom of choice (hence the "C" in the abbreviation). The commonly applied version is known as the ZFC system, which was initially created by Zermelo in 1908 and then further polished by Frankel. In short, ZFC refers to the axiom set used for mathematics, which includes the Zermelo-Fraenkel axioms and the axiom of choice.From: Introduction to Mathematics of Satisfiability [2019], Foundations of mathematics under neuroscience conditions of lateral inhibition and lateral activation [2018], A Systemic Perspective on Cognition and Mathematics [2019], Student Handbook for Discrete Mathematics with Ducks [2019]

Preliminaries

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Published in Aliakbar Montazer Haghighi, Indika Wickramasinghe, Probability, Statistics, and Stochastic Processes for Engineers and Scientists, 2020

Aliakbar Montazer Haghighi, Indika Wickramasinghe

These days, the set of axioms used are of Zermelo and Fraenkel combination referred to as ZF axioms, and if the axiom of choice (AC) is included, they are referred to as ZFC. Indeed, the most widely accepted form and strength of axiomatic set theory is ZFC (Zermelo–Fraenkel choice). In addition to many text references in English and other languages, Montazer-Haghighi (1973, in Farsi Language) has a complete discussion of the ZFC. We will state the axioms of sets, including the AC, later.

Foundations of mathematics under neuroscience conditions of lateral inhibition and lateral activation

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Journal Information

Published in International Journal of Parallel, Emergent and Distributed Systems, 2018

Andrew Schumann, Alexander V. Kuznetsov

Now there are some basic formal theories which are regarded as start points in the foundations of mathematics. This means that these theories, in the way how it seems to mathematicians, can cover big fragments of mathematics by their extensions. For instance, it is assumed that in the foundations for number theory we should start from the five Peano’s axioms, introduced by Giuseppe Peano in 1889 and now called the Peano arithmeticPA. Also, it is supposed that any set-theoretic reasoning in mathematics (like reasoning in topology) can be reduced to statements formalized in the Zermelo–Fraenkel set theory, constructed by mathematicians Ernst Zermelo and Abraham Fraenkel and denoted by the abbreviation ZFC, where C means axiom of choice.

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Preliminaries

Foundations of mathematics under neuroscience conditions of lateral inhibition and lateral activation