Accessible Independence Results for Peano Arithmetic

@article{Kirby1982AccessibleIR,
  title={Accessible Independence Results for Peano Arithmetic},
  author={Laurence Kirby and Jeff B. Paris},
  journal={Bulletin of The London Mathematical Society},
  year={1982},
  volume={14},
  pages={285-293}
}
  • L. Kirby, J. Paris
  • Published 1 July 1982
  • Mathematics
  • Bulletin of The London Mathematical Society
Recently some interesting first-order statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely number-theoretic in character (as opposed to metamathematical or combinatorial). The methods used to prove it, however, are combinatorial. We also give another independence result (unashamedly combinatorial in character) proved by the same methods. 
Independence of Ramsey theorem variants using ε 0 ∗
We discuss the nite adjacent Ramsey theorem, one of the most recent independence results in Peano Arithmetic, and show some fascinating connections with two of the earliest examples of naturalExpand
Independent Combinatoric Worm Principles for First Order Arithmetic and Beyond
In this thesis we study Beklemishev’s combinatorial principle Every Worm Dies, EWD which although true, it is unprovable in Peano Arithmetic (PA). The principle talks about sequences of modalExpand
Combinatorial Statements Independent of Arithmetic
When Peano’s first order axioms of arithmetic (P) were originally formulated it was generally felt that these axioms summed up all that was obviously true about the natural numbers (ℕ) with additionExpand
Independence of Ramsey theorem variants using ε 0 , Draft
We discuss the Vnite adjacent Ramsey theorem, one of the most recent independence results in Peano Arithmetic, and show some fascinating connections with two of the earliest examples of naturalExpand
Nonprovability of Certain Combinatorial Properties of Finite Trees
Abstract In this paper we exposit some as yet unpublished results of Harvey Friedman. These results provide the most dramatic examples so far known of mathematically meaningful theorems of finiteExpand
Mathematical Incompleteness Results in First-Order Peano Arithmetic: A Revisionist View of the Early History
  • Saul Kripke
  • Mathematics
  • History and Philosophy of Logic
  • 2021
In the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey’s theorem is celebrated as the first result of a long ‘search’ for a purely mathematical incompleteness result inExpand
More Intensional Versions of Rice's Theorem
TLDR
A generalisation of Rice’s Theorem concerning equivalence classes of programs is proved and it is shown how it can be used to study intensional properties such as time and space complexity. Expand
A short proof of two recently discovered independence results using recursion theoretic methods
Recently L. A. S. Kirby and J. Paris showed that a theorem of R L. Goodstein cannot be proved in Peano's Arithmetic. We give an alternative short proof of their result, based only on well establishedExpand
SOME TRANSFINITE INDUCTION DEDUCTIONS
This paper develops the ordinal numbers and transfinite induction, then demonstrates some interesting applications of transfinite induction. One such application is the proof that there is a set inExpand
Unprovability results involving braids
We construct long sequences of braids that are descending with respect to the standard order of braids (‘Dehornoy order’), and we deduce that, contrary to all usual algebraic properties of braids,Expand
...
1
2
3
4
5
...

References

SHOWING 1-7 OF 7 REFERENCES
Some Independence Results for Peano Arithmetic
  • J. Paris
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1978
TLDR
A purely model theoretic method for obtaining independence results for Peano's first order axioms (P) is outlined and provides for the first time elementary combinatorial statements about the natural numbers which are not provable in P. Expand
A Hierarchy of Cuts in Models of Arithmetic
In this paper we show that it is possible to classify most of the natural families of cuts considered to date in terms of a single hierarchy. This classification gives conservation and independenceExpand
On the Restricted Ordinal Theorem
TLDR
Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument, by utilising transfinitely induction to prove that certain sequences of reduction processes, enumerated by ordinals less than e, are finite. Expand
SOLOVAY, 'Rapidly growing Ramsey functions
  • Ann. of Math.,
  • 1981
Review: R. L. Goodstein, On the Restricted Ordinal Theorem