Re: Axiom of Pairing, Scheme of Replacement from others

MoeBlee wrote:

Stephen J. Herschkorn 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.



Also, from http://en.wikipedia.org/wiki/Axiom_of_empty_set,





any axiom of set theory or logic that implies the existence of any set
will imply the existence of the empty set, if one has the axiom schema
of separation.



The axiom schema of separation alone is sufficient.



However, if separation is derived as a theorem schema
from the axiom schema of replacement (as is sometimes done), then that
derivation requires the axiom of empty set.



That is incorrect. An empty set axiom is not needed to derive a theorem
schema of separation from the axiom schema of replacement.



So it could not be used to
eliminate the axiom of empty set.



That is incorrect.


The first Wikipedia page cited gives a brief, explicit proof of separation from replacement, given we are assured the existence of an empty set. The page also desribes in detail why the existence of an empty set is required.

MB's repetitions demonstrate how easy it is to claim something is not true. However, without explicit presentation of a proof which does not rely on the existence of the empty sets, such claims are not credible in this case. MB, can you present such a proof?


--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan
.

Relevant Pages

  • 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: Axiom of Pairing, Scheme of Replacement from others
    ... separation follows from the axiom of replacement together with the axiom ... An empty set axiom is not needed for that. ...
    (sci.math)
  • 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)
  • Re: ZFC in another shape.
    ... countable language (and, I'm supposing that you have only a countable ... Anyhow I think it is the strong replacement, ... existance of the empty set, ... your axiom might not say what you want it to say. ...
    (sci.math)
  • Re: Axiom of Pairing, Scheme of Replacement from others
    ... need existence in the empty-set part of the axiom infinity. ... to prove the existence of the empty set, then the axiom of infinity, as ... replacement entails separation. ...
    (sci.math)
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