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dmmI have finally read the 1967 paper by Dana Scott, "A Proof of the Independence of the Continuum Hypothesis"
www2.karlin.mff.cuni.cz/~krajicek/scott67.pdf
That's so much better than Cohen's original approach. With Cohen's result, not only his "forcing technique" is difficult, but Cohen had to work with countable models, so a reader was left with the impression that something was "morally wrong" (ощущение что фокусник вытащил кролика из шляпы). Yes, logic is typically about finite texts over a finite alphabet, so one can often build countable models of this and that, but does the result actually shed any light on the nature of cardinalities, or only on the limitations of our formal methods?
Here the approach is very different and much more "elementary". First, one rewrites the Continuum Hypothesis as a property of subsets of real numbers (for any subset X of reals, either there is a surjective function from natural numbers onto X, or there is a surjective function from X onto reals).
Then one considers models where statements are valued not in [0,1], but in a complete Boolean algebra. Namely one considers a set Omega, a sigma-algebra of its subsets, and a countably additive probability measure over that. One considers the "sigma-ideal" of subsets having measure zero and factors the original sigma-algebra over that sigma-ideal to avoid algebraic pathologies. This factor serves as our complete Boolean algebra B of logical values.
Then it turns out that real random variables over B are a B-valued model of the theory of real numbers.
Then one takes the set Omega of sufficiently high cardinality (higher than continuum), and one can use a subset of Omega of an intermediate cardinality to build a "subset X of those real random variables", such that one can't "surjectively map naturals onto X", and one can't "surjectively map X onto the whole set of those random variables" (pages 18-20). So one gets a model where that rewrite of the Continuum Hypothesis is false.
This work and its neighborhood is also a start of the whole approach to "generalized fuzzy mathematics", where one uses Boolean algebras or Heyting algebras or even more general structures as spaces of logical values.
www2.karlin.mff.cuni.cz/~krajicek/scott67.pdf
That's so much better than Cohen's original approach. With Cohen's result, not only his "forcing technique" is difficult, but Cohen had to work with countable models, so a reader was left with the impression that something was "morally wrong" (ощущение что фокусник вытащил кролика из шляпы). Yes, logic is typically about finite texts over a finite alphabet, so one can often build countable models of this and that, but does the result actually shed any light on the nature of cardinalities, or only on the limitations of our formal methods?
Here the approach is very different and much more "elementary". First, one rewrites the Continuum Hypothesis as a property of subsets of real numbers (for any subset X of reals, either there is a surjective function from natural numbers onto X, or there is a surjective function from X onto reals).
Then one considers models where statements are valued not in [0,1], but in a complete Boolean algebra. Namely one considers a set Omega, a sigma-algebra of its subsets, and a countably additive probability measure over that. One considers the "sigma-ideal" of subsets having measure zero and factors the original sigma-algebra over that sigma-ideal to avoid algebraic pathologies. This factor serves as our complete Boolean algebra B of logical values.
Then it turns out that real random variables over B are a B-valued model of the theory of real numbers.
Then one takes the set Omega of sufficiently high cardinality (higher than continuum), and one can use a subset of Omega of an intermediate cardinality to build a "subset X of those real random variables", such that one can't "surjectively map naturals onto X", and one can't "surjectively map X onto the whole set of those random variables" (pages 18-20). So one gets a model where that rewrite of the Continuum Hypothesis is false.
This work and its neighborhood is also a start of the whole approach to "generalized fuzzy mathematics", where one uses Boolean algebras or Heyting algebras or even more general structures as spaces of logical values.
no subject
Date: 2026-04-28 04:51 am (UTC)Thank you!