Re: A simple question?


MoeBlee wrote:
zuhair wrote:
Hi,

For w={ 0,1,2,3,....} , is P(w) a well ordered set?

You mean the power set of omega?

Your question is not complete.

Under what axioms?

A set is well ordered if there is a well ordering of the set.

I do not think this is correct. Well-ordered can only apply to a
totally ordered set. For example, the real numbers are not well
ordered, but there is a well-ordering of the real numbers which is not
the standard ordering.
I think the real question here is: what ordering are you assigning to
P(w)?

With the axiom of choice added to the ZF axioms, every set has a well
ordering.

As I understand, ZF without the axiom of choice cannot prove that the
power set of omega has a well ordering.

MoeBlee

.

Relevant Pages

  • Re: A simple question?
    ... You mean the power set of omega? ... Under what axioms? ... claim that the standard ordering is the well ordering. ...
    (sci.math)
  • Re: A simple question?
    ... You mean the power set of omega? ... Under what axioms? ... claim that the standard ordering is the well ordering. ...
    (sci.math)
  • Re: Reals without infinity
    ... version of Peano's Axioms as an intitial premise. ... Cartesian product, power set and subsets axioms. ... Cartesian product, power set and subset axioms, as well as the ...
    (sci.math)
  • Re: Attempts to Refute Cantors Uncountability Proof?
    ... and see that indeed each step is justified by the basic axioms ... Then there is no surjective function from X to ... Let f be a function from X to its power set, ... What are the axioms of your set theory? ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... It would make more sense to first remove unobservables from Physics. ... Why would you want to say anything about the axioms beyond stating the ... >want to know why you think the power set is interesting. ... Except that I can't, BECAUSE OF THOSE COMMUTATION ...
    (sci.math)