Re: Axiom of Pairing, Scheme of Replacement from others

Stephen J. Herschkorn wrote:

MoeBlee wrote:

I just wanted to make sure, especially since as taken in the sense of
the axiom scyhema of separation, your question seems to be whether Z |-
ZF, while it is famous that it is not the case that Z |- ZF. Am I
missing something here?

Actually, I wasn't familiar with the last non-implication. 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.

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. ...
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  • 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 ...
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  • 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 ...
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