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Formal Proofs and Decidibility

In: Finsler Set Theory: Platonism and Circularity

Author

Listed:
  • David Booth

    (Three Fold Foundation 307)

  • Renatus Ziegler

    (Mathematisch-Astronomische Sektion am Geotheanum)

Abstract

In order to demonstrate the consistency of certain axiom systems, Hilbert makes use of a theory of mathematical proof in which the proof must be thought of as rigorously formalized in concrete symbols (see Hilbert [1922], [1923], [1926], Bernays [1922], Ackermann [1924]). “A proof is an array which must be graphically represented in its entirety” (Hilbert [1923, 152]). He adds: “A formula shall be said to be provable if it is either an axiom, or arises by substitution into an axiom, or is the concluding formula of a proof” (ibid., 152–153). The aim, then, is to show that, in a given axiom system, a contradictory formula (formalized in the same way) can with certainty never be proven. Axiom systems for which this can be demonstrated are said to be “consistent” (ibid., 157 and [1926, 179]). In the following where the formalization is quite general, such systems will be called formally consistent.

Suggested Citation

  • David Booth & Renatus Ziegler, 1996. "Formal Proofs and Decidibility," Springer Books, in: David Booth & Renatus Ziegler (ed.), Finsler Set Theory: Platonism and Circularity, pages 50-55, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-9031-1_4
    DOI: 10.1007/978-3-0348-9031-1_4
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