Re: ZFC in another shape.

On Feb 6, 2:59 pm, "MoeBlee" <jazzm...@xxxxxxxxxxx> wrote:
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.

well, my original theory didn't include Power.
The idea was clear and simple. It was to use replacement as the
centeral player in this theory.
Axiom of ordinal succession was meant to provide the existance a
mantle upon which the replacement of the members of each ordinal we
can form any other set of the same or lower cardinality, using the
appropriate replacement function.
So a set like {0,{0}} can be converted to any other binary
set( pairing) , or to any singlton set , or to the emtpy set, by
replacing its members using the approporiate replcement functions.

This was my primary idea. Power shouldn't be axiomatized. It should be
a theorum in this theory. In a similar maner I thought union,pairing
are also theorums in this theory.

Anyhow? obviouselly I failed.

I think I should modefy the axiom of replacement to acheive such a
goal.
anyhow I don't have the time for that.

Anyhow , I don't regard anything I wrote a pointless waste of time.

MoeBlee


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Relevant Pages

  • Re: ZFC in another shape.
    ... we can simply prove that using power and replacement alone. ... Since every ordinal subsets its power set, and is a member of its ... from the other axioms (if I'm not mistaken, ... regularity) without union. ...
    (sci.math)
  • Re: ZFC in another shape.
    ... we can simply prove that using power and replacement alone. ... Since every ordinal subsets its power set, and is a member of its ... from the other axioms (if I'm not mistaken, ... regularity) without union. ...
    (sci.math)
  • Re: ZFC in another shape.
    ... we can simply prove that using power and replacement alone. ... Since every ordinal subsets its power set, and is a member of its ... I think you can find proofs that union is not derivable ... from the other axioms (if I'm not mistaken, ...
    (sci.math)
  • Re: ZFC in another shape.
    ... we can simply prove that using power and replacement alone. ... Since every ordinal subsets its power set, and is a member of its ... from the other axioms (if I'm not mistaken, ...
    (sci.math)
  • Re: ZFC+
    ... Extensionality: As in ZFC ... since the first form would be a theorum ... (small replacement). ... since we have extensionality,regularity,Infinity,pairing as axioms. ...
    (sci.math)