Re: Axiom of Pairing, Scheme of Replacement from others

MoeBlee wrote:
G. Frege wrote:
On 19 Jan 2007 12:17:45 -0800, "MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:


According to Wikipedia
(http://en.wikipedia.org/wiki/Axiom_of_separation), "the axiom of
separation follows from the axiom of replacement together with the axiom
of empty set." See the discussion before at the cited page before the
quoted text. Also, from http://en.wikipedia.org/wiki/Axiom_of_empty_set,

The axiom schema of separation follows from the axiom schema of
replacement. An empty set axiom is not needed for that.


Hey Moe!

How about a p r o o f to support your claim?! :-)


"Unproven statements carry little weight in the world of
mathematics." - Amir D. Aczel


By all means! See my reply to Stephen J. Herschkorn.

P.S., as to proving things, the poster of that Wikipedia article does
not PROVE that separation is independent of replacement or that
existence of an empty set is indepedent of separation. Instead, he
"explains" (my scare quotes) why (he thinks) they are. And, as far as I
can tell, the fault in his "explanation" is that he demurs from
applying the full force of first order logic as he seems to prefer to
suppose that set theory must also prove its theorems in (I guess) some
kind of free logic or other system that does not have a proof system to
match the non-empty domain principle. But the fact is that standard
classical first order logic does have a proof system that matches the
non-empty domain principle, so standard first order logic (which is the
presumed logic for a formal Z set theory) does prove separation from
replacement and empty set from separation.

MoeBlee

.

Relevant Pages

  • Re: Axiom of Pairing, Scheme of Replacement from others
    ... separation follows from the axiom of replacement together with the axiom ... Your hypothesis has uniqueness but not existence. ... Proof of separation from replacement: ... empty set is required. ...
    (sci.math)
  • Re: Axiom of Pairing, Scheme of Replacement from others
    ... To prove separation from the formulation of replacement I gave, ... alone entails the existence of an empty set, ...
    (sci.math)
  • Re: Axiom of Pairing, Scheme of Replacement from others
    ... separation follows from the axiom of replacement together with the axiom ... empty set is required. ...
    (sci.math)
  • Re: Restricting Separation
    ... restriction on separation would eventually mutilate the mathematical ... It seems to me, your new axiom of separation, with side parameters x1, ... ... and replacement, and consider these axioms replacing usual separation and replacement to ... accessed the Xi dynamically inside the members where a copy of the ...
    (sci.logic)
  • Re: Some basic set theory questions
    ... Let y =, which exists by separation. ... There's no need for x to be a set satisfying the axiom of infinity. ... If there is something that has the empty set ...
    (sci.math)