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Virgil wrote:
In article <LdadnbRzxMF6ZUrZnZ2dnUVZ_vadnZ2d@xxxxxxxxxxx>,
"Russell Easterly" <logiclab@xxxxxxxxxxx> wrote:
...
R1: For any non-empty set, X, there is some Y, element of X,
such that Y intersect X is the empty set.
In plain English, R1 says every set has a smallest element.
False interpretation. In ZF, including that axiom of regularity, Russell
Easterly's own set, A, has no smallest member.
Recall that Russell defined A_i = {x inN: x >= i} and
Russell defined A = {A_i: i in N}.
Russell's A, as define above, is a set without any smallest member, but
the R1 form of the axiom of regularity is valid for A, since the
intersection of A with ANY of its members is the empty set.
So Russell provided the evidence to prove himself wrong. How fitting!
That shows a set that meets the quoted condition R1 but does not have a
smallest element.
Just to complete the picture, consider a set X whose only element is X
itself. R1 rejects it, because the intersection of X with its only
element contains X. However, any set with only one element contains a
smallest element, so it passes Russell's restatement.
Patricia
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