User blog comment:Vel!/1+w/@comment-5982810-20141011203204/@comment-2033667-20141011205047


 * 1) \(1 + \omega = \omega\). (by definition of ordinal addition)
 * 2) \(\omega[n] = n\). (Wainer hierarchy)
 * 3) Assume that \((\alpha + \beta)[n] = \alpha + \beta[n]\).
 * 4) Substitute \(\alpha = 1\) and \(\beta = \omega\) in (1) to get \((1 + \omega)[n] = 1 + \omega[n]\).
 * 5) We may substitute (1) into (4) to get \(\omega[n] = 1 + \omega[n]\)
 * 6) \(n = 1 + n\) by (1).
 * 7) Substitute \(n = 0\) and \(0 = 1\).
 * 8) This is a contradiction, therefore the assumption in (3) is false.
 * 9) Conclusion: \((\alpha + \beta)[n] \neq \alpha + \beta[n]\).