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In article <Z7Odne1QP94KvETZnZ2dnUVZ_u6dnZ2d@xxxxxxxxxxx>,
"Russell Easterly" <logiclab@xxxxxxxxxxx> wrote:
> "Virgil" <virgil@xxxxxxxxxxx> wrote in message
> news:virgil-951F59.14035408082006@xxxxxxxxxxxxxxxxxxxxxxxxx
> > In article <vcCdnZAhPKIufkXZnZ2dnUVZ_tqdnZ2d@xxxxxxxxxxx>,
> > "Russell Easterly" <logiclab@xxxxxxxxxxx> wrote:
> >
> >> "Virgil" <virgil@xxxxxxxxxxx> wrote in message
> >> news:virgil-315A96.12471008082006@xxxxxxxxxxxxxxxxxxxxxxxxx
> >> > In article <fLednYgKWsB5U0XZnZ2dnUVZ_s6dnZ2d@xxxxxxxxxxx>,
> >> > "Russell Easterly" <logiclab@xxxxxxxxxxx> wrote:
> >> >
> >> >> This should be a simple question for all you experts.
> >> >> Let X be a non-empty ordinal. Consider X-Union(X).
> >> >> Some examples:
> >> >>
> >> >> { 0 } - 0 = 0
> >> >> { 0, {0} } - { 0 } = { 0 }
> >> >> { 0, {0}, {0, {0}} } - { 0,{0} } = { 0, {0} }
> >> >> etc.
> >> >>
> >> >> Does N - Union(N) = N?
> >> >> If not, what is N- Union(N) equal to?
> >> >
> >> > If by A - B, Russell means {x in A: x not in B}, the usual notation is
> >> > A \ B or A\B.
> >>
> >> OK
> >>
> >> > Note that for any successor ordinal, X, Union(X) is a member of and a
> >> > PROPER
> >> > subset of X, and is, in fact, its predecessor.
> >>
> >> Is Union(N) a member of N?
> >
> > No!
>
> So, Union(N) has elements that aren't members of any element of N.
What short ciruit it whatever you are using in place of a brain misleads
you to that stupid and false conclusion?
It has clearly been stated several times that the union of any limit
ordinal it itself, and that no ordinal can ever be a member of itself,
so why do you think N can have any members that N does not have?.
Since N is a limit ordinal, Union(N) = N.
>
> >> Is Union(N) a proper subset of N?
> >
> > No!
>
> N is an element of N.
No ordinal can be a member of itself, and that includes N.
>
> >> Is N its own predecessor?
> >
> > No!
> >
> >>
> >> For a non-empty successor ordinal X, Union( Union(X), {Union(X)} ) = X.
> >
> > Successor ordinals are necessarily non-empty. Think about it!
>
> OK
>
> >> What is Union( Union(N), {Union(N)} )?
> >
> > Why is Russell unable to figure out these things for himself?
>
> Union( Union(N), {Union(N)} } = Union(N,{N}) = w+1.
>
> So, not only is N its own predecessor, N is also its own successor.
Wrong twice in one sentence. N has no (immediate) predecessor, which is
what makes it a limit ordinal instead of a successor ordinal.
And the successor to N is the union of N and {N} which contains a member
which is not a member of N.
> Good thing you have the axiom of regularity so you can "prove"
> N is not a member of itself.
It is the being an ordinal which allows proof that N is not a member of
itself. Any ordinal contains as members those ordinals and only those
ordinals which are proper subsets of it.
Since by definition no set can be a PROPER subset of itself, no ordinal
can be a member of itself. The axiom of regularity was not needed at all.
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