Set theory and the continuum hypothesis
Introduction to set theory and topology
Other axiomatizations of set theory, such as those of vonNeumann-Bernays-Gödel (NBG), or Morse-Kelley (MK), allow also fora formal treatment of proper classes.
Set Theory And The Continuum Hypothesis ..
The AC was, for a long time, a controversial axiom. On the onehand, it is very useful and of wide use in mathematics. On the otherhand, it has rather unintuitive consequences, such as theBanach-Tarski Paradox, which says that the unit ball can bepartitioned into finitely-many pieces, which can then be rearranged toform two unit balls. The objections to the axiom arise from the factthat it asserts the existence of sets that cannot be explicitlydefined. But Gödel's 1938 proof of its consistency, relative tothe consistency of ZF, dispelled any suspicions left about it.
set theory - Independence of the continuum hypothesis …
We study Baire category for subsets of 2^omega that aredownward-closed with respect to the almost-inclusion ordering (on thepower set of the natural numbers, identified with 2^omega). We showthat it behaves better in this context than for general subsets of2^omega. In the downward-closed context, the ideal of meager sets isprime and b-complete (where b is the bounding number), while thecomplementary filter is g-complete (where g is the groupwise densitycardinal). We also discuss other cardinal characteristics of thisideal and this filter, and we show that analogous results for measurein place of category are not provable in ZFC.
PDF 55,26MB Set Theory And The Continuum Hypothesis …
In order to avoid the paradoxes and put it on a firm footing, settheory had to be axiomatized. The first axiomatization was due toZermelo (1908) and it came as a result of the need to spell out thebasic set-theoretic principles underlying his proof of Cantor'sWell-Ordering Principle. Zermelo's axiomatization avoids Russell'sParadox by means of the Separation axiom, which is formulated asquantifying over properties of sets, and thus it is a second-orderstatement. Further work by Skolem and Fraenkel led to theformalization of the Separation axiom in terms of formulas offirst-order, instead of the informal notion of property, as well as tothe introduction of the axiom of Replacement, which is also formulatedas an axiom schema for first-order formulas (see next section). Theaxiom of Replacement is needed for a proper development of the theoryof transfinite ordinals and cardinals, using transfinite recursion(see ). It is also needed toprove the existence of such simple sets as the set of hereditarilyfinite sets, i.e., those finite sets whose elements are finite, theelements of which are also finite, and so on; or to prove basicset-theoretic facts such as that every set is contained in atransitive set, i.e., a set that contains all elements of its elements(see Mathias 2001 for the weaknesses of Zermelo set theory). A furtheraddition, by von Neumann, of the axiom of Foundation, led to thestandard axiom system of set theory, known as the Zermelo-Fraenkelaxioms plus the Axiom of Choice, or ZFC.