Variants of ZFC

This page contains a few notes about ZFC and related axiom systems. The facts mentioned here may be well-known.
  1. Replacement and transitive closure
    The usual definition of "transitive closure" of a set as a certain countable union needs an instance of the replacement axiom. Is that really necessary? Could one not define the transitive closure in a different way, e.g. as an intersection of certain transitive sets? Also, in the most natural model of ZC (ZFC minus replacement), transitive closure always exists, so does it follow from ZC?
    The (easy) answer is no. The short note tcl gives 2 models of ZC in which tcl does not exist. (Also available in dvi and PostScript format.)
  2. Set theory with a universal set. If we drop the axiom of foundation and require replacement and separation (Aussonderung) only for hereditarily well-founded sets, can we consistently postulate the existence of a universal set, and the existence of complements of every set?
    I do not know the answer, but it is not too difficult to see that this is possible if we also restrict the power set axiom to hereditarily well-founded sets. (Available in pdf, tex, dvi and PostScript format.)

    Note: There is a book by E.Forster that discusses several variants of "set theory with a universal set", most of them in the context of NF.

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