Ordinal collapsing function: Difference between revisions

Revision as of 20:27, 18 September 2024

Revision as of 20:27, 18 September 2024

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==== First impredicative values ====

Again, <math>\psi(\Omega) = \zeta_0</math>. However, when we come to computing <math>\psi(\Omega+1)</math>, something has changed: since <math>\Omega</math> was ("artificially") added to all the <math>C(\alpha)</math>, we are permitted to take the value <math>\psi(\Omega) = \zeta_0</math> in the process. So <math>C(\Omega+1)</math> contains all ordinals which can be built from <math>0</math>, <math>1</math>, <math>\omega</math>, <math>\Omega</math>, the <math>\phi_1varphi_1\colon\alpha\mapsto\varepsilon_\alpha</math> function ''up to <math>\zeta_0</math>'' and this time also <math>\zeta_0</math> itself, using addition, multiplication and exponentiation. The smallest ordinal not in <math>C(\Omega+1)</math> is <math>\varepsilon_{\zeta_0+1}</math> (the smallest <math>\varepsilon</math>-number after <math>\zeta_0</math>).

We say that the definition <math>\psi(\Omega) = \zeta_0</math> and the next values of the function <math>\psi</math> such as <math>\psi(\Omega+1) = \varepsilon_{\zeta_0+1}</math> are [[Impredicativity|impredicative]] because they use ordinals (here, <math>\Omega</math>) greater than the ones which are being defined (here, <math>\zeta_0</math>).