Forum:FOST

I found there are two major ordinals related to the FOST:

1) The ordinal $$\alpha$$ so that $$f_\alpha(n)$$ has the same growth rate as Ra(n), using natural way of defining fundamental sequences. Formal defintion of "growth rate" could be found here. For naturalness of fundamental sequences we can use the following restriction: no finite-to-finite function is allowed and $$\omega[n]$$ sets to n. So, for example, the definition of sequence $$\omega*2[n] = \omega+n^2$$ is not natural because it contains the finite-to-finite function f(n) = n^2.

2) The smallest ordinal which cannot be defined in FOST. Since it operates with infinite sets, it can define not only $$f_\omega(n)$$, $$f_{\varepsilon_0}(n)$$ and $$f_{\Gamma_0}(n)$$, but also $$\omega$$, $$\varepsilon_0$$ and $$\Gamma_0$$ as well, and there will be also ordinals which are undefinable by FOST.

Any suggestions about what ordinal is larger? Ikosarakt1 (talk ^ contribs) 10:22, April 27, 2014 (UTC)