Content deleted Content added
Line 195:
Indeed, there is no reason to stop at two levels: using <math>\omega+1</math> new cardinals in this way, <math>\Omega_1,\Omega_2,\ldots,\Omega_\omega</math>, we get a system essentially equivalent to that introduced by Buchholz,<ref name="Buchholz"/> the inessential difference being that since Buchholz uses <math>\omega+1</math> ordinals from the start, he does not need to allow multiplication or exponentiation; also, Buchholz does not introduce the numbers <math>1</math> or <math>\omega</math> in the system as they will also be produced by the <math>\psi</math> functions: this makes the entire scheme much more elegant and more concise to define, albeit more difficult to understand. This system is also sensibly equivalent to the earlier (and much more difficult to grasp) "ordinal diagrams" of Takeuti<ref>Takeuti, 1967 (Ann. Math.)</ref> and <math>\theta</math> functions of Feferman: their range is the same (<math>\psi_0(\varepsilon_{\Omega_\omega+1})</math>, which could be called the Takeuti-Feferman–Buchholz ordinal, and which describes the [[ordinal analysis|strength]] of [[Second-order arithmetic#Stronger systems|<math>\Pi^1_1</math>-comprehension]] plus [[bar induction]]).
<!-- You can also have <math>\Omega</math> or more cardinals, in fact as many as nesting of the ψ function allows:
:<math>\psi_\nu(\alpha)</math> is the smallest ordinal which cannot be expressed from <math>0</math>, <math>1</math>, <math>\omega</math>, and all ordinals smaller than <math>\Omega_\nu</math> using sums, products, exponentials, and the <math>\psi_\kappa</math> functions (for <math>\kappa</math> being a previously constructed ordinal) to previously constructed ordinals less than <math>\alpha</math>.
<math>\Omega_0</math> being 0 here, abbreviate <math>\psi_0</math> as <math>\psi</math>. The limit of this system is <math>\psi(\text{The first ordinal }\alpha\text{ such that }\Omega_\alpha = \alpha)</math>. -->
=== A "normal" variant ===
| |||