Time Schedule:
Bob Alan Dumas
PHIL 570
Seattle Campus
Prerequisite: PHIL 470.
Class description
The Independence of the Continuum Hypothesis: Very soon after proving that the set of real numbers (the continuum) is bijective with the power set of the natural numbers, Cantor conjectured that every subset of the real numbers is at most countable or is bijective with the set of real numbers. This is the origin of the well-known continuum hypothesis (CH): The continuum is of the least uncountable cardinality. In 1938, Godel defined the constructible universe (L) and proved that if Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is consistent, then ZFC + V=L (ZFC + “every set is constructible”) is consistent as well. Indeed, “inside” any model of ZFC would reside a model of ZFC + V=L (namely the constructible sets of that model of ZFC). He also proved that CH holds in any model of ZFC + V=L. Godel had previously proved that any collection of axioms in first-order logic powerful enough to do arithmetic was logically incomplete, so ZFC was known to be logically incomplete. Could CH be the first really interesting example of a logically independent set-theoretic statement? In 1963, Cohen developed the method of forcing – a way to construct new models of ZFC using topological techniques. With this method Cohen was able to prove that if ZFC is consistent, then ZFC with the failure of CH is consistent. Hence CH is logically independent of ZFC. In this seminar, we will prove the independence of CH. This will be a detailed treatment in which we will prove dozens of theorems, most of which are of independent interest. We will cover portions of the first 14 chapters of Set Theory, by Thomas Jech, keeping our focus on those results necessary for the two part proof of the independence of CH. Grades will be determined by two interesting and difficult problem sets, due on the seventh week and finals week. Prerequisite: PHIL 470. TEXT: Set Theory, The Third Millennium edition,Thomas Jech.
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