Date: Jan 12, 2013 11:36 AM
Subject: Re: ZF - Powerset + Choice = ZFC ?

On Jan 8, 6:45 pm, "Ross A. Finlayson" <>
> On Jan 7, 7:13 pm, "Ross A. Finlayson" <>
> wrote:

> > On Jan 7, 2:30 pm, Dan Christensen <>
> > 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 at

> > 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?


Ross Finlayson