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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 = \varphi_1(\omega)</math>.
If we alter the definition even more, to allow nothing except psi, we get <math>\psi(0) = 1</math>, <math>\psi(\psi(0)) = 2</math>, and so on until <math>\psi(\omega) = \omega+1</math>, <math>\psi(\psi(\omega)) = \omega+2</math>, and <math>\psi(\Omega) = \omega 2</math>, at which point we can go no further since we cannot do anything with the <math>\Omega</math>'s. So the range of this system is only <math>\omega 2</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.
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