By Robert Bellin, Pietro Ponzetto
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Additional resources for Mastering the modern Benoni and the Benko Gambit
To establish his earlier claim it would therefore suffice to show that the real numbers were equinumerous with the class of all countable ordinals (the Continuum Hypothesis in its full-fledged form).
As indices for those transfinite derivatives, and in 1883, in his landmark book Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a General Theory of Manifolds), he took the bold step of conceiving such indices as transfinite ordinal numbers . The notion of transfinite cardinal number evolved during the same period, as an outgrowth of Cantor’s discovery that the real numbers could not be placed in one-to-one correspondence with the integers. As his correspondence with Dedekind makes clear, that discovery took place in December 1873, at a time when his interests were turning “away from trigonometric series to an analysis of [the notion of] continuity” .
An equally important current within ancient Greek thought, especially among the Megarians and Stoics, was the tradition of dialectical disputation. It was out of that tradition that the classical paradoxes arose, including the Antinomy of the Liar, attributed in its original form (“All Cretans are liars”) to Epimenides of Crete. And it was that aspect of logic that was responsible for the subject’s Excursus 39 survival during the Middle Ages. Unlike mathematics, which languished in the West for centuries after the fall of the ancient Greek civilization, the Aristotelian and Stoic traditions in logic were kept alive by scholastic philosophers, who sought to apply logical principles in their theological debates.
Mastering the modern Benoni and the Benko Gambit by Robert Bellin, Pietro Ponzetto