Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incompleteness Theorem almost to Robinson's Arithmetic Q

Dan E. Willard
The Journal of Symbolic Logic
Vol. 67, No. 1 (Mar., 2002), pp. 465-496
Stable URL: http://www.jstor.org/stable/2695021
Page Count: 32
  • Read Online (Free)
  • Download ($10.00)
  • Subscribe ($19.50)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incompleteness Theorem almost to Robinson's Arithmetic Q
Preview not available

Abstract

Let us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13], [18]. We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π1 sentence, valid in the standard model of the Natural Numbers and denoted as V, such that if α is any finite consistent extension of Q + V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property. Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ0, as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].)

Page Thumbnails

  • Thumbnail: Page 
465
    465
  • Thumbnail: Page 
466
    466
  • Thumbnail: Page 
467
    467
  • Thumbnail: Page 
468
    468
  • Thumbnail: Page 
469
    469
  • Thumbnail: Page 
470
    470
  • Thumbnail: Page 
471
    471
  • Thumbnail: Page 
472
    472
  • Thumbnail: Page 
473
    473
  • Thumbnail: Page 
474
    474
  • Thumbnail: Page 
475
    475
  • Thumbnail: Page 
476
    476
  • Thumbnail: Page 
477
    477
  • Thumbnail: Page 
478
    478
  • Thumbnail: Page 
479
    479
  • Thumbnail: Page 
480
    480
  • Thumbnail: Page 
481
    481
  • Thumbnail: Page 
482
    482
  • Thumbnail: Page 
483
    483
  • Thumbnail: Page 
484
    484
  • Thumbnail: Page 
485
    485
  • Thumbnail: Page 
486
    486
  • Thumbnail: Page 
487
    487
  • Thumbnail: Page 
488
    488
  • Thumbnail: Page 
489
    489
  • Thumbnail: Page 
490
    490
  • Thumbnail: Page 
491
    491
  • Thumbnail: Page 
492
    492
  • Thumbnail: Page 
493
    493
  • Thumbnail: Page 
494
    494
  • Thumbnail: Page 
495
    495
  • Thumbnail: Page 
496
    496