Proofs of Impossibility

  • Pavel Pudlák
Part of the Springer Monographs in Mathematics book series (SMM)


This chapter is about results that show the impossibility to derive some theorems from particular sets of axioms and the impossibility to decide some problems using algorithms. In order to put the things into a historical context, we start with classical results from geometry and algebra. We explain Gödel’s incompleteness theorems and sketch their proofs. We show the algorithmic undecidability of the halting problem. We present concrete theorems that are independent of the axioms of Peano Arithmetic and some stronger theories. In the last section, we explain the method of forcing, which is used to prove independence results in set theory.


Peano Arithmetic Incompleteness Theorem Goodstein Sequence Paris-Harrington Theorem Finite Ramsey Theorem 
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© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Pavel Pudlák
    • 1
  1. 1.ASCRPragueCzech Republic

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