Mutual dependence of replacement and power set axioms I read in Gittman, Hamkins, et al, that ZFC without the power set causes the axiom of replacement to fail. Yet I also read (generally, throughout the literature, but mostly in connection with Cantor's theorem) that the power set is generated by replacing one set with another. Is the term replacement in the context of the power set informally used? Or does this apparent circularity simply explain why power set and replacement are axiomatic?
 A: First of all, Victoria Gitman's last name has only one "t", and Thomas A. Johnstone is a co-author. Our paper is: 


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*V. Gitman, J. D. Hamkins, and T. A. Johnstone, What is the theory ZFC without power set?, under review.


But you haven't described the result correctly. The main result is that many things go completely wrong, if one axiomatizes ZFC without power set using the replacement axiom, instead of the collection axiom plus separation. Although replacement is equivalent to collection plus separation over the usual version of ZFC without those axioms, when one omits the power set axiom this equivalence is no longer true, and there are a number of surprising issues that arise, as we explain in our paper. For example, one cannot prove that $\omega_1$ is regular in the version of ZFC-powerset where one has only replacement and not collection, even though the well-order principle is there, and similarly the Los theorem fails and there are many other problems. 
The final conclusion is that the right way to axiomatize ZFC without power set is to use the collection and separation axioms, rather than merely replacement.
A: Just to answer the title of the question: The axiom of power set and the axiom schema of replacement are independent of each other. Neither is deducible from the other plus the remaining axioms of ZFC.
