Independence (mathematical Logic) - Independence Results in Set Theory

Independence Results in Set Theory

Many interesting statements in set theory are independent of Zermelo-Fraenkel set theory (ZF). The following statements in set theory are known to be independent of ZF, granting that ZF is consistent:

  • The axiom of choice
  • The continuum hypothesis and the generalised continuum hypothesis
  • The Suslin conjecture

The following statements (none of which have been proved false) cannot be proved in ZFC to be independent of ZFC, even if the added hypothesis is granted that ZFC is consistent. However, they cannot be proved in ZFC (granting that ZFC is consistent), and few working set theorists expect to find a refutation of them in ZFC.

  • The existence of strongly inaccessible cardinals
  • The existence of large cardinals
  • The non-existence of Kurepa trees

The following statements are inconsistent with the axiom of choice, and therefore with ZFC. However they are probably independent of ZF, in a corresponding sense to the above: They cannot be proved in ZF, and few working set theorists expect to find a refutation in ZF. However ZF cannot prove that they are independent of ZF, even with the added hypothesis that ZF is consistent.

  • The Axiom of determinacy
  • The axiom of real determinacy
  • AD+

Read more about this topic:  Independence (mathematical Logic)

Famous quotes containing the words independence, results, set and/or theory:

    “Traditionally in American society, men have been trained for both competition and teamwork through sports, while women have been reared to merge their welfare with that of the family, with fewer opportunities for either independence or other team identifications, and fewer challenges to direct competition. In effect, women have been circumscribed within that unit where the benefit of one is most easily believed to be the benefit of all.”
    —Mary Catherine Bateson (b. 1939)

    “A distinction of property results from that very protection which a free Government gives to unequal faculties of acquiring it.”
    —James Madison (1751–1836)

    “Alas, poor Yorick! I knew him, Horatio: a fellow of infinite jest, of most excellent fancy.... Where be your jibes now, your gambols, your songs, your flashes of merriment that were wont to set the table on a roar?”
    —William Shakespeare (1564–1616)

    “every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
    —Thomas Nagel (b. 1938)