Skip to main content
SpringerLink
Log in

Fast-Collapsing Theories

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexander, S., The theory of several knowing machines, Doctoral dissertation. The Ohio State University, 2013.

  2. Alexander, S., A machine that knows its own code, Studia Logica, 2013. doi:10.1007/s11225-013-9491-6.

  3. Alexander, S., (preprint), Self-referential theories, (Submitted).

  4. Benacerraf P.: God, the Devil, and Gödel. The Monist 51, 9–32 (1967)

    Article  Google Scholar 

  5. Carlson T. J.: Ordinal arithmetic and \({\Sigma_1}\) -elementarity. Archive for Mathematical Logic 38, 449–460 (1999)

    Article  Google Scholar 

  6. Carlson T. J.: Knowledge, machines, and the consistency of Reinhardt’s strong mechanistic thesis. Annals of Pure and Applied Logic 105, 51–82 (2000)

    Article  Google Scholar 

  7. Carlson, T. J., Sound epistemic theories and collapsing knowledge, Slides from the Workshop on The Limits and Scope of Mathematical Knowledge at the University of Bristol, 2012.

  8. Lucas J. R.: Minds, machines, and Gödel. Philosophy 36, 112–127 (1961)

    Article  Google Scholar 

  9. Penrose R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Oxford (1989)

    Google Scholar 

  10. Putnam H.: After Gödel. Logic Journal of the IGPL 14, 745–754 (2006)

    Article  Google Scholar 

  11. Reinhardt W.: Absolute versions of incompleteness theorems. Noûs 19, 317–346 (1985)

    Article  Google Scholar 

  12. Shapiro, S., Epistemic and intuitionistic arithmetic, in S. Shapiro (ed.), Intensional Mathematics, North-Holland, Amsterdam, 1985, pp. 11–46.

  13. Simpson, S., Subsystems of Second Order Arithmetic, Cambridge University Press, Cambridge, Second edition, 2009.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, USA

    Samuel A. Alexander

Authors
  1. Samuel A. Alexander
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samuel A. Alexander.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alexander, S.A. Fast-Collapsing Theories. Stud Logica 103, 53–73 (2015). https://doi.org/10.1007/s11225-013-9537-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-013-9537-9

Keywords

Search

Navigation