Re: Axiom of Pairing, Scheme of Replacement from others

MoeBlee wrote:

Stephen J. Herschkorn wrote:


MoeBlee wrote:



Stephen J. Herschkorn wrote:




Does

(Extensionality, Pairing, Union, Power, Infinity, Comprehension) |-
Replacement?




Which comprehension schema? I take it you don't mean the separation
schema.



Hmm, looking at Halmos, Kunen, and Suppes, it seems to me that
"Comprehension," "Separation," and "Specifcation" (not respectively) all refer to the same thing:

For any logical formula f wherein the variable "x" is not free, we
have the universal closure of

There exists x such that for all y, y in x iff (y in z and f).

From looking at http://en.wikipedia.org/wiki/Axiom_of_separation , I
suspect maybe MB thought I was referring to unrestricted comprehension.
Does anyone do that any more?



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. I had a vague recollection that Extensionality, Union, Power, Infinity, and Comprehension were sufficient. I guess the correct collection is Extensionality, Union, Power, Infinity, and Replacement, where Infinity implies the existence of the empty set.

Thanks for your input, Moe (and others).

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

.

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