```Date: Jan 12, 2013 11:36 AM
Author: ross.finlayson@gmail.com
Subject: Re: ZF - Powerset + Choice = ZFC ?

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?Regards,Ross Finlayson
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