Another important part of the proof (elided in the current Wikipedia article) is showing that the "minus 1" operation, after mapping to omegas, results in an ordinal number that is less than the previous one. Note that, after subtracting one from the finite number, writing the resulting finite number in hereditary base notation may result in a very different "shape" than the previous number, compare by analogy 420000 and 419999. We'll need to invoke exactly how hereditary base notation is constructed and its resulting properties, as well as the definition of "less than" for transfinite ordinal numbers. Previous note.

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