15 Matching Annotations
  1. Oct 2024
    1. Cohen (Independence of the Axiomof Choice; The Independence of the Continuum Hypothesis I, I1) completed theproof of independence for each by showing neither could be deduced from theexisting axioms (by showing the negation of each could consistently be added tothe Zermelo—Fraenkel axiom scheme). See P. J. Cohen (Set Theory and theContinuum Hypothesis) for a discussion of these results and his intuition about thecontinuum hypothesis. Another expository reference is Cohen (IndependenceResults in Set Theory).

      In 1963 Paul Cohen completed the work of Gödel by proving the independence of the axiom of choice and the continuum hypothesis from the Zermelo-Fraenkel set theory axioms. He did this by showing that neither could be deduced from the existing axioms and specifically by showing that the negation of each could be added to ZF consistently.

    2. Godel (The Consistency of the Axiom of Choice and of the Generalized Con-tinuum Hypothesis with the Axioms of Set Theory) proved in 1940 that additionof either the axiom of choice or the continuum hypothesis to existing set theoreticaxioms would not produce a contradiction.

      Gödel's breakthrough in 1940 was to prove that one could extend the axioms of set theory to include the axiom of choice or the continuum hypothesis without introducing contradictions.

  2. Jan 2024
  3. Feb 2022
  4. Jul 2021
  5. Feb 2021
    1. the 2 hardest problems in computer science are essentially the 2 hardest problems of life in general, as far as humans and information are concerned.
    2. The non-determinism is why cache invalidation — and that other hard problem, naming things — are uniquely and intractably hard problems in computer science. Computers can perfectly solve deterministic problems. But they can’t predict when to invalidate a cache because, ultimately, we, the humans who design and build computational processes, can’t agree on when a cache needs to be invalidated.
    3. I like the answers already given, but I think they both could use an even more top-level, generalized explanation.
    4. Cache invalidation is hard because: Everything in life we want to know, changes.Those changes are non-deterministic.
  6. Aug 2020
  7. Apr 2020
  8. Jan 2020
    1. You can also fit generalized additive models (Chapter @ref(polynomial-and-spline-regression)), when linearity of the predictor cannot be assumed. This can be done using the mgcv package:
  9. Jan 2017