Re: Axiom of Pairing, Scheme of Replacement from others

MoeBlee wrote:

Stephen J. Herschkorn wrote:



I was trying to remember what I had read sometime before, viz., that the axioms of ZF - Foundation are not independent.



Right, if you include schema of separation and the pairing axiom in
that axiom set, then it's not an indedependent axiom set.



I had a vague recollection that Extensionality, Union, Power, Infinity, and Comprehension were sufficient.



They're not sufficient for proving the schema of replacement.



I guess the correct collection is Extensionality, Union, Power, Infinity, and Replacement,



Right, that is sufficient for ZF (without regularity).



where Infinity implies the existence of the empty set.



We don't need infinity for existence of the empty set.

Existence of an empty set follows from the schema of replacement (or
schema of separation, if we want to take it back to Z). And uniqueness
of such an empty set follows from extensionality.


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,

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


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

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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
    ... separation follows from the axiom of replacement together with the axiom ... empty set is required. ...
    (sci.math)
  • 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: Revising Separation:
    ... |6) Relation schema: if F is a formula in which x is not free then any ... |from this axiom we can define Cartesian product of any two sets ... can have free variables, since for any P we can always ... |With the above revised axiom schema of separation we cannot have this ...
    (sci.logic)
  • Re: Revising Separation:
    ... |6) Relation schema: if F is a formula in which x is not free then any ... |from this axiom we can define Cartesian product of any two sets ... This is basically selection for Cartesian products. ... |With the above revised axiom schema of separation we cannot have this ...
    (sci.logic)
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