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2 results

  • REALIZING REALIZABILITY RESULTS WITH CLASSICAL CONSTRUCTIONS

  • ASAF KARAGILA
  • Journal:
    Bulletin of Symbolic Logic / Volume 25 / Issue 4 / December 2019
    Published online by Cambridge University Press:
    19 December 2019, pp. 429-445
    Print publication:
    December 2019

  • J. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.

  • ITERATING SYMMETRIC EXTENSIONS

  • ASAF KARAGILA
  • Journal:
    The Journal of Symbolic Logic / Volume 84 / Issue 1 / March 2019
    Published online by Cambridge University Press:
    14 March 2019, pp. 123-159
    Print publication:
    March 2019

  • The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.