Talk:First uncountable ordinal

Ikosarakt, please define order type of function. LittlePeng9 (talk) 20:10, July 9, 2013 (UTC)


 * I removed the sentence. \(\omega_1\) is the googologist's infinity. FB100Z &bull; talk &bull; contribs 20:22, July 9, 2013 (UTC)
 * Order type of f(n) is such \(\alpha\) so that \(f_\alpha(n)\) has growth rate comparable to f(n). But I can guess why you ask it. Ikosarakt1 (talk ^ contribs) 10:50, July 10, 2013 (UTC)
 * What is order type of arithmetic mean of f_a and f_a+1? LittlePeng9 (talk) 09:33, July 11, 2013 (UTC)
 * In FGH, we can compare only unary functions. If the function is binary, we get it unary taking all arguments to the one value. Arithmetic mean of n and n is n, and so we get that it grows slower than f_0(n). By the way, arithmetic mean of n and n is also H_0(n) in Hardy hierarchy, and we know that \(H_{\omega^\alpha}(n) \approx f_\alpha(n)\) for all alpha < e_0. We can take 0 = omega^(-infinity) and say that ar. mean has order type \(f_\infty(n)\). Ikosarakt1 (talk ^ contribs) 09:47, July 11, 2013 (UTC)
 * I meant that if we take any (fixed) \(\alpha\), for example \(\omega\), and we take \(f(n)={f_\omega+f_{\omega+1}\over 2}\), then what is order type of f(n)? LittlePeng9 (talk) 10:57, July 11, 2013 (UTC)
 * We get \(f(n) \approx f_{\omega+1}(n)\) (adding \(f_{\omega}(n)\) and dividing by 2 are too weak functions to affect the growth rate), and thus f(n) has order type \(\omega+1\). Ikosarakt1 (talk ^ contribs) 11:06, July 11, 2013 (UTC)

Does anyone know what ordinals beyond \(\omega_1\) have fundamental sequences? I have deduced a few rules:


 * If \(\alpha > \beta\) and \(\beta\) has a fundamental sequence, then \(\alpha + \beta\) has a fundamental sequence.
 * If \(\alpha > \beta\) and \(\beta\) has a fundamental sequence, then \(\alpha \times \beta\) has a fundamental sequence.
 * If \(\alpha\) has a fundamental sequence, then \(\omega_\alpha\) has a fundamental sequence.

FB100Z &bull; talk &bull; contribs 20:25, July 9, 2013 (UTC)

\(\omega_\omega\) has fundamental sequence \(\omega_1,\omega_2,\omega_3,\cdots\). Ikosarakt1 (talk ^ contribs) 10:52, July 10, 2013 (UTC)

The list is much, much longer, for example \(\varepsilon_{\Omega+1}\), \(\Gamma_{\Omega+1}\). The thing I wanted also to point out is that fundamental sequence doesn't need to be countable. Every limit ordinal has fundamental sequence of size equal to its cofinality. But definitions may vary. LittlePeng9 (talk) 21:00, July 9, 2013 (UTC)