On Jan 21, 10:07 am, Zuhair <zaljo...@gmail.com> wrote: > Harvey Friedman had presented several formulations of theories using > some concepts in theology relative to which ZFC is provable to be > consistent! So some kind of mentioning of the supernatural (or what is > mutually interpretable with it!) is needed to prove ZFC's > consistency. > > See:http://www.cs.nyu.edu/pipermail/fom/2013-January/016881.html > > Doesn't that say that mathematics following ZFC is only grounded in > Mythology driven principles! > > Doesn't that mean that ZFC based mathematics is too imaginary that > even if consistent still it is based and rooted in fantasy that cannot > really meet reality! > > Why should one follow such a system so grounded? > > With the scientific triumph it looks that mathematics is ought to be > revolutionized to parallel science instead of absolutist tendencies of > the old era of Religion and mythology as it seems to be currently > rooted in?! > > Zuhair
ZF's universe in a naive set theory would be the Russell set and contain itself.
Why do people have such a hard time believing that the specification of the set of all sets that don't contain themselves has as a sputnik the set itself, contra the specification? Irrational numbers are regularly modeled as convergent sequences of rationals that aren't rational. The simple definition of division brings forth from the integers non-integer rationals. The infinitely-straight-sided polygon has just the one curving side.
The universe of all sets would be its own powerset, this is Cantor's paradox. The arrow reaches its target, this is Zeno's paradox. ZF's universe would contain itself, this is Russell's paradox.
So, then with regards to Freidman's realm of "Real Objects", here of ZF or ZFC or ZFC with some large cardinals (that as regular would be cardinals in ZF(C), and as irregular could be their own powersets), then to the ultra-infinitism that there is a Universe and God sorts it out, has that God sorts out that the Universe, mathematically, is its own powerset.
And, for any being in existence, the Universe is definitely a "Real Object", for all in it.