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In article <Q4OdnQfrxp4DJUrZnZ2dnUVZ_vudnZ2d@xxxxxxxxxxx>,
"Russell Easterly" <logiclab@xxxxxxxxxxx> wrote:
> "Aatu Koskensilta" <aatu.koskensilta@xxxxxxxxx> wrote in message
> news:TcNBg.4380$7q4.2663@xxxxxxxxxxxxxxxxxxxxxxxxxx
> > MoeBlee wrote:
> >> Russell Easterly wrote:
> >>> I have been reading up on the Axiom of Regularity,
> >>> also called the Axiom of Foundation.
> >>> I admit it, I do read this stuff sometimes.
> >>>
> >>> There are several ways the Axiom of Regularity
> >>> can be stated, all of which are supposedly "equivalent".
> >>> http://planetmath.org/encyclopedia/AxiomOfRegularity.html
> >>
> >> The rest of your post is rife with confusions. Why not just start with
> >> the axioms and work systematically so that you understand each step
> >> clearly and firmly?
> >
> > That would be so much less fun. One often finds that when one understands
> > a subject thoroughly things that once appeared to be pregnant with all
> > sorts of intriguing and interesting epiphanies are, in the end, just
> > somewhat boring technicalities.
>
> I guess I don't understand set theory "thoroughly".
Understatement of the year!
> I will admit, given "every set has a smallest member",
> proving there are no infinite descending posets
> is a "somewhat boring technicality".
> I gave such a proof, but you snipped it.
>
> What isn't so boring is that the same proof
> shows there are no infinite ascending posets, either.
WRONG! Russell relies on properties that have been shown to be false in
his proof.
>
> I think it is deliberately misleading to say
> "every set has a smallest element" is equivalent to
> "there are no infinite descending posets", when
> "every set has a smallest element" is really equivalent
> to "there are no infinite posets."
But in ZF, the empty set does not have a smallest element so your
equivalent statement is also false.
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