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poutsos.A
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since a lot of talking is going on with sets, will somebody write down the axioms in ZFC theory as a point of reference , when a discussion is opened up.
thanx
thanx
loop quantum gravity said:You misunderstood him Dragonfall, I think he meant that someone should post a sticky in this subforum underlying all the axioms of ZFC, or something like this.
Cause checking in mathworld or wiki is really a triviality, nowadays.
Dragonfall said:I don't think we should favor any formalism over another, lest someone thinks that ZFC is gods-given or something.
Dragonfall said:If anyone here discusses set theory without explicitly mentioning the formalism then it's assumed to be ZFC. .
Dragonfall said:I don't think we should favor any formalism over another, lest someone thinks that ZFC is gods-given or something.
evagelos said:What that suppose to mean??
evagelos said:mention couple of formalisms,if you like,please
Dragonfall said:It means "If anyone here discusses set theory without explicitly mentioning the formalism then it's assumed to be ZFC."
Morse-Kelley, type theory, category theory, von-Neumann-Godel. You can google the rest yourself.
Dragonfall said:Because no person in the right mind would prove things straight from the axioms.
I think he has a fair objection. Dragonfall has insulted several demographics of mathematicians, computer scientists, students (and probably people in other fields too). While one might assume Dragonfall really just meant something to the effect of "leave set theory to the set theorists", I believe it is quite reasonable to call Dragonfall out on his comment.morphism said:evagelos, are you being deliberately dense?
ZFC set theory is a foundational theory in mathematics that aims to provide a rigorous and consistent framework for understanding and reasoning about sets. It stands for Zermelo-Fraenkel set theory with the Axiom of Choice, and is based on a set of axioms and rules for constructing and manipulating sets.
The axioms in ZFC set theory are the fundamental assumptions that form the basis of the theory. They include the Axiom of Extension, Axiom of Pairing, Axiom of Union, Axiom of Infinity, Axiom of Power Set, and the Axiom of Choice. These axioms provide the rules for defining and constructing sets and ensure the consistency and completeness of the theory.
The Axiom of Choice is controversial because it allows for the creation of infinite sets that are non-measurable, meaning they cannot be assigned a precise size or measure. This has implications for certain areas of mathematics, such as analysis and probability, and has been the subject of much debate among mathematicians.
Axioms are used in ZFC set theory as the starting points for constructing and manipulating sets. They provide the rules for defining sets and determining their properties, and allow for the development of more complex mathematical structures, such as groups and topological spaces.
The axioms in ZFC set theory have far-reaching implications for mathematics, as they provide the foundation for all mathematical reasoning involving sets. They allow for the development of rigorous and consistent mathematical theories and play a crucial role in many areas of mathematics, including analysis, algebra, and topology.