Re: Request for link to proof for set theory proposition
- From: mmweiss@xxxxxx
- Date: 11 Jul 2006 17:21:34 -0700
Aatu Koskensilta wrote:
Scott wrote:
Hi:
I have seen the following theorem/proposition of ZF set theory in
various places:
Theorem: (a subset V) -> (a in V)
where a is any set and V is any level. Can someone provide a link to a
proof for this theorem?
Probably not. It's false.
The converse is true though.
Say V_0 is 0, V_a+1 is powerset(V_a), and V_l is Union{V_k:k<l} for
limit l.
If x is in V_a+1 then x is a subset of V_a. Then by induction
hypothesis, every member of x is a subset of V_a. So every member of x
is a member of V_a+1.
If x is in V_l then x is in V_a for some a<l. Then by i.h. every
member of x is a member of V_a, so every member of x is a member of
V_l.
If you alternatively define the V_a as {x: rank(x)<a} then the result
follows from: if x in y then rank(x) < rank(y). The rank of a set x is
the least ordinal greater than the rank of every member of x.
(Of course, this isn't intended for Aatu's benefit.)
.
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