Re: Initial ordinal, forming omega 1

persres@xxxxxxxxxxxxxx a écrit :
Hello,
Thanks for your reply. I am a little puzzled by this because it
seems to me that you are constructing a set with order type 2^aleph-
not.

Read again. It is a subset of P(NxN) (which, by the way, has no order type)



What has been explained below seems to be "The existence of a set
with cardinality of the continuum".
Which has an ordcerable subset of type w_1


Where as w1 would be the first
uncountable ordinal. (Am I missing something here).

Looks like it


The first initial ordinal we can get by axiom infinity. The way I
understand w1 would be the limit of all countable ordinals.


Correct, and power set ensures this limit can be constructed *in* P(NxN)
Can
someone explain please how the definition of an initial ordinal ties
in with the construction below. Also, how can we say that this is the
first uncountable ordinal?.

It is, by construction, the limit of all countable ordinals. Draw your own conclusion...

Any help or if too long suitable
references would be very helpful.


Thanks


On 15 Oct, 03:39, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
On Tue, 14 Oct 2008 19:05:42 -0700 (PDT), pers...@xxxxxxxxxxxxxx wrote:
Hello,
I am trying to understand how initial ordinals are formed.
First of all, I believe limit ordinals are of the form w, w+w, w+w
+w, ...w^2,...w^3, ...w^w,....and so on.
Now, I am not able to understand how you can get the next initial
ordinal after w. I believe that would be w1, the first uncountable
ordinal. (Corresponding to the Aleph1).
Can some one explain how initial ordinals get formed. Through which
axiom of set theory.
Thanks
By the power set axiom we have P(NxN), the power set of the set of all
ordered pairs of naturals. Each member of P(NxN) can be thought of as a
relation on N. Some of those relations represent well-orderings. Call
that subset W.

We can then define an equivalence relation on W. Two well-orderings are
considered equivalent if they are order-isomorphic. Using the axiom of
replacement, the resulting equivalence classes can be reduced to the von
Neumann ordinals via the Mostowski collapse. The result is w1.

--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>

.

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