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According to current logic theories, the solution to the cardinality of the continuum, that is the question of how big |R| is, is that it can be any value from [math]\aleph_1[/math] to [math]\aleph_ω[/math], because you can use forcing to create "new" reals between the existing ones as many times as you like, and the question of whether or not the "new" reals are valid is undecidable.
Why is this taken as the solution instead of a sign that our understanding of mathematical logic is inadequate?



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