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Continuum Hypothesis

Georg CantorGeorg Cantor’s conjecture, the Continuum Hypothesis

Without equations, this states that for any set of real numbers, S, one of three things happen:

  1. S is finite.
  2. S has a 1-1 correspondence to the integers.
  3. S has a 1-1 correspondence to the reals.

There is nothing in between the integers and reals. Kurt Goedel showed that this was consistent with set theory, and Paul Cohen showed that its negation was consistent. In other words, CH is an undecidable proposition of Zermelo-Frankel set theory (and of ZFC, ZF with the Axiom Of Choice). Ditto for the Generalized Continuum Hypothesis. — Eric Jablow

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Number as Archetype

Carl Jung
Kurt GodelBy Robin Robertson

Excerpts:

  • … Because human beings are capable of counting (“one, two, three…”), we imagine that is how numbers were arrived at.
  • … The story seems to demonstrate that a crow (or at least the crow in the story) has a sense of “one”, “two”, “three”, and “many”.
  • … In brief, one corresponds to a stage of non-differentiation; two—polarity or opposition; three—movement toward resolution, as expressed, e.g., in the Christian trinity.

Carl JUngThis paper has been adapted from the final chapter of Jungian Archetypes: Jung, Godel and the History of Archetypes, Nicolas-Hay, 1996, with the permission of Nicolas-Hay. Copyright Nicolas-Hay Publishers.

The sequence of natural numbers turns out to be unexpectedly more than a mere stringing together of identical units: it contains the whole of mathematics and everything yet to be discovered in this field. — Carl Jung[1]

It has turned out that (under the assumption that modern mathematics is consistent) the solution of certain arithmetical problems requires the use of assumptions essentially transcending arithmetic; i.e., the domain of the kind of elementary indisputable evidence that may be most fittingly compared with sense perception. — Kurt Godel[2]

 

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