Re: A simple question?


Randy Poe wrote:
zuhair wrote:
All of what I am saying is that if a set subsets its power set, then
its power set also subsets its power.

What does "subsets" mean as a verb? Normally, subset is a noun.

Does "A subsets B" mean "A is a subset of B"?


In symboles:

x subsets P(x) -> P(x) subsets P(P(x) ). do anybody have an objection
to this?

If "x subsets P(x)" means "x is a subset of P(x)", then it is not
in general true. For instance, if x is the set {1}, then the subsets
of P(x) are {}, {{1}}, neither of which is x.

Accordingly we say that x= {1} is not a set which subsets its power
set.

I call every set that subsets its power set a " subpower".

accordingly x={1} is not a subpower, because as you demonstrated above,
it doesn't subset its power set.

My definition of cardinality that is: The equivalence class of
subpowers , under equivalence relation "bijection". is therefore not
Frege's definition of cardinality. which is what it is intended to be,
ie not the same as Frege's definition of cardinality.

if every x subsets its power set, then there is no usefullness in me
stating the above definition as a new one, since by then it will only
be another way of stating Frege's definition of cardianlity.

To make U understand the matters more , lets take a more complex
example , that is my definition of ordinals . which is a set that
subsets its power set and every member of it also subsets its power
set.

x is an ordinal <-> x subsets P(x) /\ Am:mex -> m subsets P(m).

This also can be writtin in the following manner.

x is an ordinal <-> x: Amex,Anem,Azen -> nex /\ zem

In ZFRC , this definition is equivalent to the standard definition of
ordinals.

This was proved by Dave Seaman.

But without the axiom of regularity , it is not equivalent to the
standard definition.

Have a nice time excersizing what I have mentioned.

Zuhair

- Randy

what do you mean general.

x subsets P(x) <-> x:Amex,Anem -> nex.

Of course not every x fulfills this definition. That what this
definition is made for.

Zuhair

.

Relevant Pages

  • Re: Equivalent sets and equivalence relations
    ... be equivalent if there exists a one-to-one correspondence between ... Then, equivalence in this is sense, for subsets of some ... set X is an equivalence relation in the power set P. ... "translate" the general definition for the ...
    (sci.math)
  • Equivalent sets and equivalence relations
    ... I'm having trouble understanding the following: ... be equivalent if there exists a one-to-one correspondence between ... Then, equivalence in this is sense, for subsets of some ... set X is an equivalence relation in the power set P. ...
    (sci.math)
  • Re: Cardinality without choice
    ... Every a in y is a transitive set bijective to z. ... Now we know for sure that Y is a proper class(when x=/=), ... The beauty of this definition of cardinality of a set, ... is that because the power set of every ...
    (sci.math)
  • Re: Semantics of First-Order Languages
    ... Okay. ... The language has one 2-place predicate symbol '=', ... greater than aleph0 and bX the power set of the set of natural numbers ... But perhaps you meant for 'b' to map to the CARDINALITY of the power ...
    (sci.logic)
  • Re: Resolving the paradoxes of set theory
    ... A = {bush,{cheney, rumsfeld}} ... >> With a hundred years of hindsight, it must be said that the power set ... Mathematical analysis can be developed without using the ... > Well, pretty obviously, in all cases, the cardinality of your new power ...
    (sci.logic)