Re: The Power Set Reduced

On Feb 14, 5:42 pm, Chris Menzel <cmen...@xxxxxxxxxxxxxxxxxxxx> wrote:
On Wed, 14 Feb 2007 15:46:20 GMT, Frederick Williams
<Frederick_Willi...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> said:

MoeBlee wrote:

... some people do
hold that the axioms of set theory are evident upon consideration of
the cumulative hierarchy.

Doesn't the cumulative hierarchy assume that, if X is a set, PX is one
also?

Yes. The hierarchy is defined formally as follows (where P is the
powerset operator):

V(0) = 0
V(n+1) = P(V(n)) U V(n)
V(m) = U{V(n) | n < m}, for limit ordinals m

The "U V_n" part of the second clause is unnecessary in the hierarchy of
pure sets where we start with the empty set, but it emphasizes the
cumulative nature of the structure nicely, in that we pull everything
from previous levels along with us as we move to higher levels in the
hierarchy.

Nice. That makes it even simpler. If I understand correctly, we have:

Let:
V(0) = 0
V(n+) = PV(n) u V(n)
V(m) = U{V(n) | n<m}, for limit ordinals m.

Let:
W(0) = 0
W(n+) = PW(n)
W(m) = U{W(n) | n<m}, for limit ordinals m.

By induction, V(k) = W(k), for all ordinals k.

MoeBlee




.

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