Re: Cardinality: a definition

zuhair wrote:
how the usual definition of ordinal do not require the
axiom of regularity?

Because the usual definition stipulates that an ordinal is well ordered
by the membership relation on the ordinal, so, a fortiori, the oridnal
has a membership-least element.

from what I understand of the usual definition of ordinal, is that
there is a successor relation defined on x , were x subset Px, and this
successor relation that I know is simply
xU{x}. since the first x={ } .

That is not the usual defintion of 'ordinal'. I already posted the
definition I'm referring to about a dozen times for you. Morevover,
what you just said doesn't even make sense, let alone that it is not
even a definition, let alone that it is not even a definition of
'ordinal'.

so xU{x} is the immediate successor of x and, xU{x} U {x U {x} } is the
successor of
xU {x} and so on........

Yeah, that's true. So what?

It is obvious that this definition is dependant on the axiom of
regularity.

It's not a definition of 'ordinal'.

I posted the definition about a dozen times already. Go READ it
already.

MoeBlee

.