Ordinal collapsing function: Difference between revisions

Revision as of 22:57, 30 April 2022

Revision as of 22:59, 30 April 2022

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It is instructive (although not exactly useful) to make <math>\psi</math> less powerful.

If we alter the definition of <math>\psi</math> above to omit exponentiation from the repertoire from which <math>C(\alpha)</math> is constructed, then we get <math>\psi(0) = \omega^\omega</math> (as this is the smallest ordinal which cannot be constructed from <math>0</math>, <math>1</math> and <math>\omega</math> using addition and multiplication only), then <math>\psi(1) = \omega^{\omega^2}</math> and similarly <math>\psi(\omega) = \omega^{\omega^\omega}</math>, <math>\psi(\psi(0)) = \omega^{\omega^{\omega^\omega}}</math> until we come to a fixed point which is then our <math>\psi(\Omega) = \varepsilon_0</math>. We then have <math>\psi(\Omega+1) = {\varepsilon_0}^\omega</math> and so on until <math>\psi(\Omega 2) = \varepsilon_1</math>. Since multiplication of <math>\Omega</math>'s is permitted, we can still form <math>\psi(\Omega^2) = \phi_2varphi_2(0)</math> and <math>\psi(\Omega^3) = \phi_3varphi_3(0)</math> and so on, but our construction ends there as there is no way to get at or beyond <math>\Omega^\omega</math>: so the range of this weakened system of notation is <math>\psi(\Omega^\omega) = \phi_\omega(0)</math> (the value of <math>\psi(\Omega^\omega)</math> is the same in our weaker system as in our original system, except that now we cannot go beyond it). This does not even go as far as the Feferman–Schütte ordinal.

If we alter the definition of <math>\psi</math> yet some more to allow only addition as a primitive for construction, we get <math>\psi(0) = \omega^2</math> and <math>\psi(1) = \omega^3</math> and so on until <math>\psi(\psi(0)) = \omega^{\omega^2}</math> and still <math>\psi(\Omega) = \varepsilon_0</math>. This time, <math>\psi(\Omega+1) = \varepsilon_0 \omega</math> and so on until <math>\psi(\Omega 2) = \varepsilon_1</math> and similarly <math>\psi(\Omega 3) = \varepsilon_2</math>. But this time we can go no further: since we can only add <math>\Omega</math>'s, the range of our system is <math>\psi(\Omega\omega) = \varepsilon_\omega = \phi_1varphi_1(\omega)</math>.

In both cases, we find that the limitation on the weakened <math>\psi</math> function comes not so much from the operations allowed on the ''countable'' ordinals as on the ''uncountable'' ordinals we allow ourselves to denote.