Re: ZFC in another shape.

On Feb 6, 11:20 am, "zuhair" <zaljo...@xxxxxxxxx> wrote:
Not only that. I discovered that The axiom of ordina succession I've
made , is in reality a theorum in a theory consisting of the other six
axioms. we can simply prove that using power and replacement alone.
Since every ordinal subsets its power set, and is a member of its
power set, then by replacement the the sucessor ordinal is a subset of
the power set.
what I want to say is that S(x) is a subset of P(x) for all x: x is an
ordinal.

Then the axiom of ordinal succession should be converted into the
theorum of ordinal succession.

You don't even need replacement. Power set with separation will do the
job.

Meanwhile, I think you can find proofs that union is not derivable
from the other axioms (if I'm not mistaken, in the usual treatments of
the independence proofs). So your new theory is just ZFCR (ZFC with
regularity) without union. So you've set up all this rigmarole just to
state ZFCR without union.

MoeBlee

.

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