Axiom of choice
If Axiom of Choice is not assumed, non-Lebesgue measurable subsets of
the reals may cease to exist.
Why Axiom of Choice is deeply embedded in standard mathematics?
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Relevant Pages
- Re: Skolems Paradox and why is math the way it is?
... > of the real number can't even be formulated in the language of ZF. ... that real is in the set of reals in ZF. ... do NOT assume the existance of first order language, ... ANOTHER axiom system that is NOT the ZF system. ...
(sci.math) - Re: Skolems Paradox and why is math the way it is?
... ZF proves the LACK of a bijection IN the ZF system, ... > set of reals is uncountable is perfectly precisely rendered into ... If you look at the axiom of equality two sets are the same IFF they ... since all you need to do is add the existance of a bijection B from N ...
(sci.math) - Re: Finitely additive becoming countably additive
... Howewver, if the axiom of determinacy holds, ... all sets of reals are Lebesgue measurable. ... problem with the countable additivity? ... cardinality of the reals has to be what is called a measurable ...
(sci.math) - Re: Special Functions
... then elements of that field may be used to form linear ... The next fact is that any vector space has a basis. ... the reals in the sense desired. ... And this requires the Axiom of Choice? ...
(sci.math) - Re: Skolems Paradox and why is math the way it is?
... there are an uncountable number of reals ... I eschew axioms, but if you call infinity an axiom then it generates ... > You, for instance, cited a theorem about ONE set (the cantor set). ... > there is at least one set in the universe. ...
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