Re: Axiom of Pairing, Scheme of Replacement from others

On 22 Jan 2007 11:42:27 -0800, "MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:

How about the following (shorter) argument? (Just a sketch.)

Given (a certain formulation of the Replacement schema):

Au e z Av Aw(R[u,v] & R[u,v]) -> v = w) -> EbAy(y e b <-> Eu e z R[u,y]) (*)


Let R[u,x] be "x =/= x".

Then the antecedence of (*) holds (vacuously). And we get:

EbAy(y e b <-> Eu(u e z & y =/= y)).

Since Eu(u e z & y =/= y) is false for any y, we get:

EbAy(y !e b).

With other words, there is an empty set.

This shows/proves: If Separation follows from (this form of)
Replacement _and_ the empty set axiom, then it follows from
(this form of) Replacement alone.


F.

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