On Jan 8, 6:45 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Jan 7, 7:13 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> > wrote: > > > On Jan 7, 2:30 pm, Dan Christensen <Dan_Christen...@sympatico.ca> > > wrote: > > > > On Monday, January 7, 2013 4:51:47 PM UTC-5, david petry wrote: >
> > > > Once again, I recommend that you read Nik Weaver's article. > > > > IIUC, he hopes to do mathematics (e.g. real analysis) without sets (or any equivalent notion). If he wants to be taken seriously, he should just go ahead and do so. I have tried to do so for a number of years myself to no avail. I don't much care for the ZF axioms of regularity and infinity myself. I haven't found any use for them in my own work, and haven't incorporated them (or any equivalent) in my own simplified set theory. But I really don't see how you can do foundational work without a powerset axiom. > > > > Dan > > > Download my DC Proof 2.0 software athttp://www.dcproof.com > > > Doesn't it follow from pairing and union? > > > Basically we know that any element of a set is a set via union. > > > Then, each of those as elements as a singleton subset, is a set, as > > necessary via inductive recursion and building back up the sets. > > > Via pairing, the two-elements subsets are sets, via pairing, the three > > element sets are sets, ..., via induction, each of the subsets are > > sets. > > > Then all the subsets are each sets, via pairing, that's a set. > > > It follows from pairing and union. > > Then, here it seems that there is a ready enough comprehension of the > elements of the set, composed in each way, composing a set, that would > be the set of subsets of a set, or its powerset, without axiomatic > support, instead as a theorem of pairing and union, and into strata > with countable or general choice. > > In a set theory, then what sets have powersets that don't exist via > union and pairing? In a theory without well-foundedness perhaps those > that are irregular, yet then the transitive closure would simply be > irregular too and the powerset would be simply enough constructed (via > induction and for the infinite, transfinite induction). > > So, what sets have powersets not constructible as the result of > induction over union and pairing, in ZF - Powerset or ZFC - Powerset? >
So, why axiomatize powerset if it's not an independent axiom?