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zuhair wrote:
> Actually that is not the case. People are only giving me statements but
> no proofs.
No, we gave you proofs.
> No body proved to me that the standard definition of Ordinals doesn't
> depend on the axiom of regularity. They only said that without
> mentioning any proove including you.
We've been over this before. I gave you the proof. It is utterly
trivial. The standard definition stipulates that an ordinal is well
ordered, so we don't require the axiom of regularity to show that every
ordinal is well ordered.
> If a set is well ordered then there should exist a relation between its
> member that is asymmetric, transitive and connective. Now how come this
> definition doesn't depend on the axiom of regularity, while we need
> this axiom to show that any non empty set that is a member of an
> ordinal is not the same as the set which contain it.
No, we do NOT need the axiom of regularity to show that any member of
an ordinal is not equal to the singleton of that set. We need the axiom
of regularity to show, in the arbitrary case, that no set equals the
singleton of that set, but we don't need the axiom of regularity to
show this in the case of ORDINALS, where 'is an ordinal' is defined
with a standard defintion such as I gave.
You just jump to make so many assumptions. Don't assume that because we
need an axiom to prove something in the arbitrary case that we need
that axiom in special cases. The general, or arbitrary, case requires
the axiom of regularity, but the special case of ordinals (such as I
defined 'is an ordinal') does not require the axiom of regularity.
MoeBlee
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