Question

Show how a contradiction arises from the following taken as an assumption (∃y)(x)(x∈y≡∼(x∈x))

Show how a contradiction arises from the following taken as an assumption

(∃y)(x)(x∈y≡∼(x∈x))

Math   Advanced-Math   
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Answer #1

The assumption tells us that

The statement that there exists such that forall , belongs to is equivalent to the statement is not contained in .

The contradiction arises because the statement " is not contained in " is not true even if the statement "there exists such that forall " is true.

Consider the set of rationals .

Obviously there exists the set of reals such that i.e. is contained in but is also contained in i.e. also holds.

Hence the assumption is wrong.

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