Matematiikan ja tilastotieteen laitoksen kurssisivualue

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Introduction to large cardinals, fall 2014

First lecture September 23.


Visiting Prof. Boban Velickovic (Paris VII)


10 sp.


Advanced studies


Elements of set theory



The axioms of large cardinals postulate the existence of very large objects, but they have an impact on classical mathematical objects, such as the real line.  They form an essentially linear hierarchy which measures the consistency strength of consistency of all mathematical theories.  Large cardinals play a central role in the Goedel program program whose goal is to extend the standard axiomatic framework ZFC of set theory in order to resolve problems that are undecidable on the basis of ZFC alone. 
In this course we present the basics of the theory of large cardinals
  • Inaccessible, Mahlo, weakly and strongly compact, measurable cardinals
  • Partition relations and generalizations of Ramseys theorem, indiscernibles, 0#
  • Saturated ideals and iterated ultrapowers Ideal saturated
  • Infinite games and determinacy
  • Very large cardinals and the Holy Grail of set theory

Course outline

  • The measure problem, measurable cardinals.
  • Trees and partitions weakly and strongly compact cardinals
  •  The constructible universe L, elementary embeddings the Ehrenfeucht-Mostowski models, 0#.
  • Iterated ultra powers and the model L[U].
  • Very large cardinals: supercompact, huge cardinals, etc. 


Weeks 39-42 and 44-51, Tuesday 10-12 in room C124 and Thursday 10-12 in room C122. Two hours of exercise classes per week.

First lecture September 23.




  • T. Jech, Set Theory: The Third Millennium Edition, revised and expanded (Springer Monographs in Mathematics), 2006
  • K. Kunen, Set Theory: An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics, Volume 102, North Holland, 1980;
  • A. Kanamori, The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994
  • A. Levy,  Basic  set  theory  (Springer  Verlag), 1979


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1.Friday 14-16 C123 Lauri Keskinen 
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