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Gödel’s Second Theorem and the Provability of God’s Existence

Abstract

According to a common view, belief in God cannot be proved and is an issue that must be left to faith. Kant went even further and argued that he can prove this unprovability. But any argument implying that a certain sentence is not provable is challenged by Gödel’s second theorem (GST). Indeed, one trivial consequence of GST is that for any formal system F that satisfies certain conditions and for every sentence K that is formulated in F it is impossible to prove, from F, that K is not provable. In the article, I explore the general issue of proving the unprovability of the existence of God. Of special interest is the question of the relation between the existence of God and the conditions that F must satisfy in order to allow for its subjection to GST.

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Notes

  1. Rigorous formulations and proofs for GST can be found in many textbooks. For a clear exposition of the connection between modal logic and provability see [3, chapter 27].

  2. For didactical reasons I have avoided venturing into provability logic.

  3. The [3, p. 232] formulation is the following: Let T be a consistent, axiomatizable extension of P. Then the consistency sentence for T is not provable in T. We can strengthen the theorem by requiring that T extend a theory weaker than PA (− P in Boolos’ formulation), e.g., GST holds even for theories T that contain the arithmetic that can represent all the recursive primitive functions. But, I don’t think it is important to delve into this issue here.

  4. I am avoiding here the issue of the definition of God. At the end of the article I shall expand on definitions of God that require second order language.

  5. It is a shorthand of the formula “\({Prf_{F}}\)(x, y)” that expresses provability in F using the language of F (when F satisfies the conditions for GST).

  6. This line was adopted by [9, p. 64] in showing that ontological arguments are not successful.

  7. The question of whether they can be proven from a richer system is always there, but in the case of proof of the existence of God, there is an interesting possibility that I will elaborate upon after presenting the next point.

  8. Note that the term “possible argument” has no formal definition. Every proof for the existence of God presented by a philosopher may count as a possible argument.

  9. This would require that F be strong enough to represent its syntax and its notion of provability. If PA is interpretable in F, this would be more than enough.

  10. According to Steiner’s careful study of the applicability of mathematics [12], we must conclude that human beings occupy a unique place in the universe. One can imagine different proofs that build on this mystery towards the conclusion that there is a creator. I am not assuming here that the arguments that follow these routes are valid! But we want F to be rich enough to allow for the formulations of such arguments.

  11. Noteworthy, Wigner [13] ties the unreasonable effectiveness of mathematics in physics with the consistency of mathematics. In this respect even the consistency of mathematics may serve as a supposition for the existence of God. If so, we have an answer to the question raised in note 4, even if we add the unprovable assumption to a more sophisticated system, now that our system is different from the system in which we began, the consistency of the new system, presents a seemingly new evidence of the existence of God to be taken into account.

  12. For an introduction to SOL [3, chapter 22].

  13. There are different formulations of the ontological argument that require neither second order logic nor a modal logic. [7] contains a formulation of the ontological argument that is formulated in free logic for definite descriptions and a genuine existence predicate ’E!x’ which is not defined in terms a quantified formula of the form ‘y’. [8] simplifies the proof even further. When studying Anselm’s version of the ontological arguments, things are more complicated. An analysis of reconstructions of Anselm’s argument shows that higher order logic is required in order to capture his assumptions. See [11].

  14. A thorough study of this issue within Kant’s transcendental idealism must give special attention to the fact that reason leads to antinomies, and its inconsistency is provable.

References

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Correspondence to Meir Buzaglo.

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Buzaglo, M. Gödel’s Second Theorem and the Provability of God’s Existence. Log. Univers. 13, 541–549 (2019). https://doi.org/10.1007/s11787-019-00235-z

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Keywords

  • Gödel’s second theorem
  • Theology
  • Unprovability
  • Kant
  • Al-Ghazâlî

Mathematics Subject Classification

  • Primary 03B35
  • Secondary 03B25
  • 03B42
  • 01A30
  • 01A50