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: > > On Monday, January 7, 2013 1:45:40 PM UTC-8, Dan Christensen wrote: > > > > On Monday, January 7, 2013 3:59:48 PM UTC-5, david petry wrote: > > > > > On Monday, January 7, 2013 11:14:57 AM UTC-8, Dan Christensen wrote: > > > > > > On Monday, January 7, 2013 8:50:09 AM UTC-5, david petry wrote > > > > > > > An article by Nic Weaver is worth a read: > > > > > > >http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.1680v1.pdf > > > > > I suppose it is surprising to classically trained mathematicians, but it is not necessary to define a set of all continuous functions, nor even a set of all real numbers, to develop the mathematics used in science. > > > > That WOULD be surprising if it were true. > > > 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.