## Epsilon-substitution method for the ramified language and # 1 -comprehension rule (1999)

Venue: | Logic and Foundations of Mathematics |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Mints99epsilon-substitutionmethod,

author = {Grigori Mints and Sergei Tupailo},

title = {Epsilon-substitution method for the ramified language and # 1 -comprehension rule},

booktitle = {Logic and Foundations of Mathematics},

year = {1999},

pages = {107--130},

publisher = {Kluwer Academic Publishers}

}

### OpenURL

### Abstract

We extend to Ramified Analysis the definition and termination proof of Hilbert’s ɛsubstitution method. This forms a base for future extensions to predicatively reducible subsystems of analysis. First such system treated here is second order arithmetic with ∆1 1-comprehension rule.

### Citations

231 | Une extension de l’interpretation de Gödel, à l’analyse, et son application l’élimination des coupures dans l’analyse et la théorie des types - Girard - 1971 |

102 |
Proof Theory
- Schütte
- 1977
(Show Context)
Citation Context ...ension rule: ∀x(∃YA↔∀ZB) F[λx∃YA]→∃XF A, B being arithmetic formulas. This system is predicative and can be embedded into ramified analysis of level less than ωω . However, the embedding presented in =-=[19]-=- uses an infinitary system, and simulating it in our framework would lead to an infinite system of critical ɛformulas. We use a modification which results in a finite system of critical ɛ-formulas hav... |

74 |
Grundlagen der Mathematik
- Hilbert, Bernays
- 1939
(Show Context)
Citation Context ...ystems of analysis. First such system treated here is second order arithmetic with ∆1 1-comprehension rule. Introduction The epsilon substitution method is based on the language introduced by Hilbert =-=[7]-=-. The main non-boolean construction of this language is ɛxF [x], read as “an x satisfying the condition F [x]”. When x is numerical, it is interpreted as the least x satisfying F [x]”. Existential and... |

51 | Die Widerspruchsfreiheit der reinen Zahlentheorie - Gentzen - 1936 |

41 | Théorie des ensembles - Bourbaki - 1958 |

25 | On the interpretation of non-finitist proofs - Kreisel - 1952 |

21 | Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Mathematische Annalen 93 - Ackermann - 1924 |

17 |
Probleme der Grundlegung der Mathematik, Math.Ann
- Hilbert
- 1929
(Show Context)
Citation Context ...ed by Hilbert was to prove termination of the sequence S0,S1,S2,... after a finite number of steps for any system of critical formulas (1). After von Neumann’s [17] attack on this (see below) Hilbert =-=[6]-=- stated further problems: 2sfind a proof of termination for pure number theory, for analysis, and for analysis with the axiom of choice, when each of these systems is suitably reformulated in the epsi... |

12 | Epsilon Substitution Method for Elementary Analysis, Archive for Math. Logic - Mints, Tupallo, et al. - 1996 |

11 |
Zur Hilbertschen Beweistheorie
- Neumann
- 1927
(Show Context)
Citation Context ...hich is false under Si. The problem stated by Hilbert was to prove termination of the sequence S0,S1,S2,... after a finite number of steps for any system of critical formulas (1). After von Neumann’s =-=[17]-=- attack on this (see below) Hilbert [6] stated further problems: 2sfind a proof of termination for pure number theory, for analysis, and for analysis with the axiom of choice, when each of these syste... |

7 | The Substitution Method - Tait - 1965 |

3 |
Strong termination for the epsilon substitution method
- Mints
- 1996
(Show Context)
Citation Context ...UE. 2 6 The proof of termination This section is similar to the last section of [15]. We slightly simplify the termination proof. In fact it is possible to prove strong termination along the lines of =-=[13]-=-, but this is not done here. In this section S, Sn, ... always denote ɛ-substitutions with {e ∈ dom(S) : S(e)=?}=∅. For each pair (e, u) we set rk((e, u)) := rk(e). Definition 24 Let S be an ɛ-substit... |

3 |
Introduction to Methamathematics
- Kleene
- 1952
(Show Context)
Citation Context ...ained from d ′ by the rule ∆. The endpiece of a DA-derivation is its part from the endformula up to and including the conclusions of lowermost rules ∆ . Definition 28 d is a pure variable derivation (=-=[16]-=-), if the eigenvariable of any rule ∃→ occurs only above its conclusion. Pure variable derivation is non-redundant if each free variable is an eigenvariable or occurs in the endsequent. Each derivatio... |

2 | Epsilon Substitution Method for the Theory of Hereditarily Finite - Mints |

2 | Functionals defined by transfinite recursion - Tait - 1965 |

1 | Consistency Proof for Arithmetic (Russian - Mints |

1 |
Substitution Method and Gentzen-type Systems
- Mints
- 1991
(Show Context)
Citation Context ... formulas s = t → (ɛxF [x, s] = ɛxF [x, t]) are consequences of the equality axioms of RAɛ. Formulas∀x(F↔G)→ɛxF = ɛxG are not axioms (and in general are not derivable) but can be easily included (cf. =-=[12]-=-). Lemma 4 RAɛ is closed under the substitution rule: if F is derivable, u = ɛηH and lev(ξ) =lev(η), then F [ξ/u] is derivable. Proof: All axioms and inference rules of RAɛ are closed under substituti... |