## Formalizing forcing arguments in subsystems of second-order arithmetic (1996)

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Venue: | Annals of Pure and Applied Logic |

Citations: | 17 - 8 self |

### BibTeX

@ARTICLE{Avigad96formalizingforcing,

author = {Jeremy Avigad},

title = {Formalizing forcing arguments in subsystems of second-order arithmetic},

journal = {Annals of Pure and Applied Logic},

year = {1996},

volume = {82},

pages = {165--191}

}

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### Abstract

We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen conservation theorems of Harrington and Brown-Simpson, giving an effective proof that W KL+0 is conservative over RCA0 with no significant increase in the lengths of proofs. 1

### Citations

500 |
Recursively Enumerable Sets and Degrees
- Soare
- 1987
(Show Context)
Citation Context ...ount of Harrington’s argument for the conservation of W KL0 over RCA0 has never been published, so we’ll describe it here briefly. The result has its origins in the low-basis theorem of [7] (see also =-=[15]-=- or [5]), but it can be phrased as a forcing argument, as is done in [12]. Let L2 be the language of second-order arithmetic. An L2 structure M consists of a first-order part, henceforth denoted by |M... |

334 |
Set Theory. An Introduction to Independence Proofs
- Kunen
- 1980
(Show Context)
Citation Context ...though the conditions, names, and atomic clauses will vary, this last part remains the same, and so we give the general framework here. (Much of our approach follows the presentation in [2]; see also =-=[8]-=-.) Before we define any forcing notion we need to have identified from within the base theory what it means for P to be a condition, which we’ll write Condition(P ), and what it means for a condition ... |

132 |
Π01 classes and degrees of theories
- Soare
- 1972
(Show Context)
Citation Context ...wledge, an account of Harrington’s argument for the conservation of W KL0 over RCA0 has never been published, so we’ll describe it here briefly. The result has its origins in the low-basis theorem of =-=[7]-=- (see also [15] or [5]), but it can be phrased as a forcing argument, as is done in [12]. Let L2 be the language of second-order arithmetic. An L2 structure M consists of a first-order part, hencefort... |

25 |
On the length of proofs of finitistic consistency statements in first order theories
- Pudlák
- 1986
(Show Context)
Citation Context ... one in RCA0, there is at most a polynomial increase in length. We’ll assume that our proof systems are standard Hilbert-style proof systems. For the “length of a proof” we employ the measure used in =-=[9]-=- which counts the number of symbols involved. Suppose one has proven a Π 1 1 sentence ∀Xϕ(X) in W KL+0 (we’ll assume for simplicity that there is only one universal second-order quantifier). We’ll exa... |

11 | The Baire Category Theorem in Weak Subsystems of Second-Order Arithmetic
- Brown, Simpson
- 1993
(Show Context)
Citation Context ... This leaves open the question of whether one can obtain a conservation result for W KL0 over RCA0 with a more restrictive bound on the increase in the lengths of proofs. Meanwhile, Brown and Simpson =-=[3]-=- used forcing methods again to extend Harrington’s conservation result to the theory W KL+0, which adds to W KL0 an additional axiom scheme (BCT ) ∀n∀σ∃τ ⊇ σϕ(n, τ) → ∃f∀n∃mϕ(n, f[m]), where σ and τ a... |

8 |
Interpretability and fragments of arithmetic
- Hájek
- 1993
(Show Context)
Citation Context ... KL+0 of a Π 1 1 formula ϕ, then f(d) codes a proof of ϕ in RCA0, and the length of f(d) is less than p(the length of d). 11 Final comments and acknowledgments The work here was done independently of =-=[4]-=-, in which P. Hájek shows that W KL0 has no significant speedup over IΣ1 in proving arithmetic sentences. Working with the language of recursion theory, he shows that one is able to abandon much of th... |

8 |
Subsystems of Z2 and Reverse Mathematics, appendix to
- Simpson
- 1986
(Show Context)
Citation Context ...y sequences that is closed under initial segments, and a path through T is a set X such that every initial segment of the characteristic function of X is in T . For more information see, for example, =-=[12, 13, 14]-=-. Both RCA0 and W KL0 were first defined by Harvey Friedman, who gave a model-theoretic proof that in fact W KL0 is conservative over P RA (Primitive Recursive Arithmetic) for Π 0 2 sentences, in the ... |

7 |
Fragments of arithmetic, Annals of Pure and
- Sieg
- 1985
(Show Context)
Citation Context ...vative over RCA0 for Π 1 1 formulas. (RCA0 is interpretable in the fragment of Peano arithmetic IΣ1, and the conservation of the latter system over P RA for Π 0 2 sentences has long been known.) Sieg =-=[10, 11]-=- later gave an effective version of Friedman’s result using cut-elimination, a Herbrand analysis, and Howard’s notion of hereditary majorizability for primitive recursive terms. Because Sieg’s proof u... |

7 | On the strength of König’s duality theorem of countable bipartite graphs
- Simpson
- 1994
(Show Context)
Citation Context ...y sequences that is closed under initial segments, and a path through T is a set X such that every initial segment of the characteristic function of X is in T . For more information see, for example, =-=[12, 13, 14]-=-. Both RCA0 and W KL0 were first defined by Harvey Friedman, who gave a model-theoretic proof that in fact W KL0 is conservative over P RA (Primitive Recursive Arithmetic) for Π 0 2 sentences, in the ... |

4 |
Herbrand Analyses,” Archive for
- Sieg
- 1991
(Show Context)
Citation Context ...vative over RCA0 for Π 1 1 formulas. (RCA0 is interpretable in the fragment of Peano arithmetic IΣ1, and the conservation of the latter system over P RA for Π 0 2 sentences has long been known.) Sieg =-=[10, 11]-=- later gave an effective version of Friedman’s result using cut-elimination, a Herbrand analysis, and Howard’s notion of hereditary majorizability for primitive recursive terms. Because Sieg’s proof u... |

4 |
Subsystems of Second Order Arithmetic, in preparation [25
- Simpson
- 1981
(Show Context)
Citation Context ...y sequences that is closed under initial segments, and a path through T is a set X such that every initial segment of the characteristic function of X is in T . For more information see, for example, =-=[12, 13, 14]-=-. Both RCA0 and W KL0 were first defined by Harvey Friedman, who gave a model-theoretic proof that in fact W KL0 is conservative over P RA (Primitive Recursive Arithmetic) for Π 0 2 sentences, in the ... |

3 |
Proof-Theoretic Investigations of Subsystems of SecondOrder Arithmetic
- Avigad
- 1995
(Show Context)
Citation Context ... two conservation results, one due to Harrington and the other due to Brown and Simpson, involving subsystems of second-order arithmetic. These results orginally appeared in the author’s dissertation =-=[1]-=-, where more details can be found. RCA0 denotes the weak base theory in the language of second-order arithmetic consisting of the quantifier-free defining axioms for the operations S, +, and ×; induct... |

3 |
Fragments of First and Second Order Arithmetic and
- Ignjatovic
- 1990
(Show Context)
Citation Context ...fs between the two systems. In fact, one can show that though there is no significant increase in the lengths of proofs between RCA0 and IΣ1, there is a superexponential increase between IΣ1 and P RA =-=[6]-=-. This leaves open the question of whether one can obtain a conservation result for W KL0 over RCA0 with a more restrictive bound on the increase in the lengths of proofs. Meanwhile, Brown and Simpson... |