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quantum123
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Which one do you prefer? Which do you think is better?
ZFC and NBG are two different mathematical axiom systems used to build the foundation of mathematics. ZFC stands for Zermelo-Fraenkel set theory with the Axiom of Choice, while NBG stands for Von Neumann-Bernays-Gödel set theory.
One of the main differences is that ZFC allows for the existence of a universal set, while NBG does not. ZFC also has a stronger version of the Axiom of Infinity, which states the existence of an infinite set, while NBG has a weaker version. Additionally, ZFC has a principle called the Axiom of Choice, which allows for the selection of an element from each non-empty set, while NBG does not have this principle.
ZFC is more widely accepted and used in the mathematics community. It has been the standard system of mathematics since the early 20th century and is used as the foundation for most of modern mathematics.
One advantage of NBG is that it allows for the existence of proper classes, which are collections that are too large to be considered sets. This can be useful in certain areas of mathematics, such as category theory. NBG also has a more rigorous treatment of classes than ZFC.
Neither system is considered to be more "correct" than the other. They both have their own strengths and weaknesses, and are used in different areas of mathematics depending on their applicability. It is important for mathematicians to understand both systems and to choose the appropriate one for their specific needs.