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Logical Foundations of Mathematics and Computational Complexity pp 255–364Cite as

Proofs of Impossibility

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Part of the book series: Springer Monographs in Mathematics ((SMM))

Abstract

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.

“He is right,” Watson nodded. “The essential thing about mathematics is that it gives aesthetic pleasure without coming through the senses.”

Rudy Rucker, A New Golden Age

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Notes

  1. 1.

    Note, however, that e π is known to be transcendental.

  2. 2.

    See [150, 287].

  3. 3.

    Another reason for stating the rules explicitly is that it has been shown that using these instruments in a special way one can solve the trisection of the angle.

  4. 4.

    Note that Gauss claimed that he also had a proof.

  5. 5.

    This is not a precise definition, but it suffices for this rough description of the theory.

  6. 6.

    Finite groups are often presented as groups of symmetries of some structures; when introducing groups I mentioned the group of symmetries of the regular triangle. A 5 can be presented as the group of symmetries of the regular icosahedron; therefore it is also called the icosahedral group.

  7. 7.

    An example of an important structure with a decidable set of axioms is \((\mathbb {R};+,\cdot,\leq)\), the structure of real numbers with operations + and ⋅ and the relation ≤, see the Theory of real closed fields on page 91.

  8. 8.

    We used the soundness in the proof above because the argument is simpler when this assumption is present.

  9. 9.

    It would be more appropriate to call it Σ 1 completeness, but I will stick to the traditional notation.

  10. 10.

    See [125], page 458. ‘Rational integers’ are the usual whole numbers. Since the concept of integers can also be defined in number fields which are different from the field of rational numbers, he used the specification ‘rational’.

  11. 11.

    Recall that in this book a theory is always axiomatized by a decidable set of axioms.

  12. 12.

    See page 643 for the definition of ATR 0.

  13. 13.

    In set theory there is a concept of strong cardinals, but this has nothing in common with strong cuts. Strong cuts are rather related to weakly compact cardinals.

  14. 14.

    When we talk about fields, we prefer to say ‘transcendental’, instead of ‘free’.

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Authors and Affiliations

  1. ASCR, Prague, Czech Republic

    Pavel Pudlák

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  1. Pavel Pudlák
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Main Points of the Chapter

Main Points of the Chapter

  • Many important results in mathematics have the form of a proof of impossibility.

  • Very often proofs of impossibility are difficult and require the use of abstract concepts, even though the problems themselves may have elementary formulations.

  • The proofs of Gödel’s incompleteness theorems are based on a formula resembling the liar paradox. It uses the ability of the language of logic to express self-referential sentences. The unprovable sentence is equivalent to the consistency of the theory.

  • A similar argument is used to prove that some problems are algorithmically unsolvable. One can also deduce unprovability of some sentences from proofs of undecidability of related problems.

  • Algorithmically unsolvable problems can be found in various branches of mathematics including number theory and geometry.

  • One can show unprovability of some concrete mathematical problems in Finite Set Theory. The proof method, however, requires that the problems encode very fast growing functions.

  • There is a very efficient method, the method of forcing invented by Paul Cohen, using which one can prove many independence results in set theory, including the independence of the Continuum Hypothesis. The method is applicable only to sentences about infinite sets.

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Pudlák, P. (2013). Proofs of Impossibility. In: Logical Foundations of Mathematics and Computational Complexity. Springer Monographs in Mathematics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00119-7_4

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