Georg Cantor's theory on infinities flow from the inductive method. That's the method by which you say if some function of "i" is true for i=1, and if you can show that where the function for "i" is true then f(i+1) is also true, then the thesis is true for the whole {i} field.
The inductive method gave rise to Karl Popper's metaphysic. Not all philosophers approve induction.
Cantor never challenged induction as such. What Cantor had done was to define and to find some "j" which doesn't exist in the inductive field of countable numbers. His "j" was 2∞-4. He'd shown up induction's mathematical limits.
Bertrand Russell famously proposed another limit to inductive reasoning, as applied to the natural world:
We know that all these rather crude expectations of uniformity are liable to be misleading. The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.
Russell suggests that the chicken was unaware of the general fate of the domesticated junglefowl, and of the applications of "Bayes' Theorem" - or if you like of Bayes-Price-Laplace. You might think you're in an average day of your life. But are you in the average day of your species' life?
Induction, then, is a Platonic ideal, or a Pythagorean one; best applicable to the mathematics of the possible. For the real world, you need B-P-L.
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