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This modified function <math>\psi</math> coincides with the previous one up to (and including) <math>\psi(\psi_1(0))</math> — which is the Bachmann-Howard ordinal. But now we can get beyond this, and <math>\psi(\psi_1(0)+1)</math> is <math>\varepsilon_{\psi(\psi_1(0))+1}</math> (the next <math>\varepsilon</math>-number after the Bachmann-Howard ordinal). We have made our system ''doubly'' impredicative: to create notations for countable ordinals we use notations for certain ordinals between <math>\Omega</math> and <math>\Omega_2</math> which are themselves defined using certain ordinals beyond <math>\Omega_2</math>.
:<math>\psi(\alpha)</math> is the smallest ordinal which cannot be expressed from <math>0</math>, <math>1</math>, <math>\omega</math>, <math>\Omega</math> and <math>\Omega_2</math> using sums, products, exponentials, and the <math>\psi_1</math> and <math>\psi</math> function (to previously constructed ordinals less than <math>\alpha</math>).
i.e., allow the use of <math>\psi_1</math> only for arguments less than <math>\alpha</math> itself. With this definition, we must write <math>\psi(\Omega_2)</math> instead of <math>\psi(\psi_1(\Omega_2))</math> (although it is still also equal to <math>\psi(\psi_1(\Omega_2)) = \psi(\zeta_{\Omega+1})</math>, of course, but it is now constant until <math>\Omega_2</math>). This change is inessential because, intuitively speaking, the <math>\psi_1</math> function collapses the nameable ordinals beyond <math>\Omega_2</math> below the latter so it matters little whether <math>\psi</math> is invoked directly on the ordinals beyond <math>\Omega_2</math> or on their image by <math>\psi_1</math>. But it makes it possible to define <math>\psi</math> and <math>\psi_1</math> by ''simultaneous'' (rather than “downward”) induction, and this is important if we are to use infinitely many collapsing functions.
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