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Countable Ordinals
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Andrew J Bacon
science forum beginner


Joined: 30 Mar 2006
Posts: 2

PostPosted: Thu Mar 30, 2006 1:45 am    Post subject: Countable Ordinals Reply with quote
Hi,
I was wondering if anyone could answer this question for me about countable
ordinals, that is ordinals of size aleph-0.

Can they be expressed as N[w], i.e. as a polynomial in w (omega) with
coefficients in the natural numbers.

So for example: 3.w^2 + 2.w + 5 where . is ordinal product and + is ordinal
addition.

Thanks,

Andrew
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matthias@dodgeit.com
science forum Guru Wannabe


Joined: 10 Nov 2005
Posts: 148

PostPosted: Thu Mar 30, 2006 2:03 am    Post subject: Re: Countable Ordinals Reply with quote
Andrew J Bacon wrote:
Quote:
I was wondering if anyone could answer this question for me about countable
ordinals, that is ordinals of size aleph-0.

Can they be expressed as N[w], i.e. as a polynomial in w (omega) with
coefficients in the natural numbers.

So for example: 3.w^2 + 2.w + 5 where . is ordinal product and + is ordinal
addition.

No; consider the countable ordinal defined to be the sup of { omega^n
| n finite }. But you can almost do what you are asking for; look up
Cantor normal form.
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Stephen J. Herschkorn
science forum Guru


Joined: 24 Mar 2005
Posts: 641

PostPosted: Thu Mar 30, 2006 2:09 am    Post subject: Re: Countable Ordinals Reply with quote
Andrew J Bacon wrote:

Quote:
I was wondering if anyone could answer this question for me about countable
ordinals, that is ordinals of size aleph-0.

Can they be expressed as N[w], i.e. as a polynomial in w (omega) with
coefficients in the natural numbers.

So for example: 3.w^2 + 2.w + 5 where . is ordinal product and + is ordinal
addition.



No. An epsilon number is an ordinal number e such that e = w^e
(ordinal exponentiation), and epsilon_0, the first epsilon number, is
countable. See the Exercise 1.6 in Kunen.

BTW, unless they have changed conventions, 3 w^2 = w^2. Perhaps you
meant (w^2) 3?

Also, that same exercise in Kunen shows that any nonzero ordinal can be
written in "base omega." That is, if a > 0 , there exist unique
nonzero natural n, ordinals b1,..., b_n, with a >= b1 > b2 > ... >
b_n, and nonzero naturals m1,..., m_n such that a = (w^b1) m1 + ... +
(w^b_n) m_n. This is called Cantor Normal Form.

--
Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
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Andrew J Bacon
science forum beginner


Joined: 30 Mar 2006
Posts: 2

PostPosted: Thu Mar 30, 2006 11:13 am    Post subject: Re: Countable Ordinals Reply with quote
Ah thanks. I will have to have a look in this Kunen you talk of.
BTW yes, I meant (w^3).2 I tend to get them confused because we originally
defined ordinals in terms of the cross product 2x(w^3) with an ordering
giving priority to elements in 2.

--
AJB

"I wish to propose for the reader's favourable consideration a doctrine
which may, I fear, appear wildly paradoxical and subversive. The doctrine in
question is this: that it is undesirable to believe in a proposition when
there is no ground whatever for supposing it true." -- Bertrand Russell
"Stephen J. Herschkorn" <sjherschko@netscape.net> wrote in message
news:f4HWf.31$RU7.2@fe11.lga...
Quote:
Andrew J Bacon wrote:

I was wondering if anyone could answer this question for me about
countable ordinals, that is ordinals of size aleph-0.

Can they be expressed as N[w], i.e. as a polynomial in w (omega) with
coefficients in the natural numbers.

So for example: 3.w^2 + 2.w + 5 where . is ordinal product and + is
ordinal addition.


No. An epsilon number is an ordinal number e such that e = w^e
(ordinal exponentiation), and epsilon_0, the first epsilon number, is
countable. See the Exercise 1.6 in Kunen.

BTW, unless they have changed conventions, 3 w^2 = w^2. Perhaps you
meant (w^2) 3?

Also, that same exercise in Kunen shows that any nonzero ordinal can be
written in "base omega." That is, if a > 0 , there exist unique nonzero
natural n, ordinals b1,..., b_n, with a >= b1 > b2 > ... > b_n, and
nonzero naturals m1,..., m_n such that a = (w^b1) m1 + ... + (w^b_n)
m_n. This is called Cantor Normal Form.

--
Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
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