## Forcing in Proof Theory (2004)

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Venue: | BULL SYMB LOGIC |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Avigad04forcingin,

author = {Jeremy Avigad},

title = {Forcing in Proof Theory},

journal = {BULL SYMB LOGIC},

year = {2004},

volume = {10},

number = {3},

pages = {305--333}

}

### OpenURL

### Abstract

Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.

### Citations

710 |
An Introduction to Modal Logic
- Hughes, Cresswell
- 1972
(Show Context)
Citation Context ...52], [56], [80] for details on these points of view. I have not even touched on the use of Kripke structures to model the semantics of various modal operators. For this, see, for example, [18], [37], =-=[44]-=-. 2.4. The syntactic perspective. Up to this point, I have been discussing forcing semantics as though the underlying Kripke structures live "in the real world." But from a hard core proof-theoretic p... |

634 |
Modal Logic, An Introduction
- Chellas
- 1980
(Show Context)
Citation Context ... well. See [37, 52, 55, 80] for details on these points of view. I have not even touched on the use of Kripke structures to model the semantics of various modal operators. For this, see, for example, =-=[18, 38, 44]-=-. 2.4 The syntactic perspective Up to this point, I have been discussing forcing semantics as though the underlying Kripke structures live “in the real world.” But from a hard core prooftheoretic poin... |

470 |
P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions
- Bertot, Castéran
- 2004
(Show Context)
Citation Context ...e constructions (CIC ) Good references for intuitionistic systems in general are [6, 37]. For more information on type theory, see [28]; for the calculus of inductive constructions in particular, see =-=[7]-=-. 3.3 Reverse mathematics 1970’s, Harvey Friedman observed that by restricting the induction and comprehension principles in full axiomatic second-order arithmetic, one obtains theories that are stron... |

274 |
Set Theory - An Introduction to Independence Proofs
- Kunen
- 1980
(Show Context)
Citation Context ...ind of "poor man's model theory." Of course, it is the classical version of forcing that is essentially the notion that set theorists know and love. In standard set-theoretic constructions (see e.g., =-=[52]-=-, [71]) sets in the generic extension are named by elements of the ground model, in such a way that the relations of elementhood and identity are settled by the generic. View these names as the inhabi... |

263 |
Foundations of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ...� ϕ). We will see below that in many applications, this is sufficient. In some cases, however, it is useful to 9sbe able to use the more common form of genericity in an intuitionistic setting. Beeson =-=[13]-=- presents a version of the forcing relation that is classically but not intuitionistically equivalent to the classical forcing relation; his version has the property that the generic validity of ϕ is ... |

215 |
Bounded Arithmetic, Propositional Logic and Complexity Theory
- Krajíček
- 1996
(Show Context)
Citation Context ...eories, and therefore upper bounds on the increase in length of proof. But there have been other, perhaps more striking, applications of forcing towards obtaining lower bounds, as in [82], [1], [67], =-=[50]-=-, [78]. I will, regrettably, not discuss these methods here.308 JEREMY AVIGAD ?2. The forcing relation. 2.1. Minimal, classical, and intuitionistic logic. Proof theorists commonly distinguish between... |

126 |
Π 0 1 classes and degrees of theories
- Soare
- 1979
(Show Context)
Citation Context ...shows that every infinite tree T on {0, 1 } has a path computable in T', the Turingjump of T. Kleene's counterexample shows that we cannot always find a path computable from T. But Jockusch and Soare =-=[46]-=- have shown that we can have the next best thing: every infinite binary tree T has a path P computable in T' that is furthermore low: the Turing jump of P, P', is also computable in T'. This very eleg... |

115 |
The complexity of the pigeonhole principle
- Ajtai
- 1994
(Show Context)
Citation Context ...viewed between theories, and therefore upper bounds on the increase in length of proof. But there have been other, perhaps more striking, applications of forcing towards obtaining lower bounds, as in =-=[82, 1, 67, 50, 78]-=-. I will, regrettably, not discuss these methods here. 3sas a procedure transforming a proof of ϕ to a proof of ψ, and so on. Note that minimal logic has nothing to say about ⊥, which is therefore tre... |

107 |
Metamathematics of First-Order Arithmetic
- Hájek, Pudlák
- 1998
(Show Context)
Citation Context ...1]. Of course, these axioms can be extended with stronger hypotheses, such as “large-cardinal” axioms. For information on primitive recursive arithmetic, see [13, 36]; for first-order arithmetic, see =-=[10, 14, 17]-=-; for second-order arithmetic, see [33]; for higher-order arithmetic, see [34]. 3.2 Constructive foundations Given the history of Hilbert’s program, it should not be surprising that proof theorists ha... |

105 | The complexity of propositional proofs - Urquhart - 1995 |

101 |
Higher Recursion Theory
- Sacks
- 1990
(Show Context)
Citation Context ...forcing provides a means of saying what it means for a property to be "generically true" of a Polish space (see, for example, [49]). The point of view of the effective descriptive set theorist, as in =-=[68]-=-, lies somewhere between that of the descriptive set theorist and recursion theorist. Ideas from forcing have even been influential in computational complexity; for example, the separation of complexi... |

78 | editor. Handbook of Proof Theory
- Buss
- 1998
(Show Context)
Citation Context ...d is varied and diverse. Here I will focus specifically on the proof theory of mathematical reasoning, but even with this restriction, the field is dauntingly broad: the 1998 Handbook of Proof Theory =-=[8]-=- runs more than 800 pages, with a name index that is almost as long as this article. As a result, I can only attempt to convey a feel for the subject’s goals and methods of analysis, and help ease the... |

76 |
A mathematical incompleteness in Peano arithmetic, Handbook for mathematical logic
- Paris, Harrington
- 1977
(Show Context)
Citation Context ... in the search for natural combinatorial independences, that is, natural finitary combinatorial principles that are independent of conventional mathemati10cal methods. The Paris-Harrington statement =-=[22]-=- is an early example of such a principle. Since then, Harvey Friedman, in particular, has long sought to find exotic combinatorial behavior in familiar mathematical settings. Such work gives us glimpe... |

69 | Gödel’s functional (“Dialectica”) interpretation
- Avigad, Feferman
- 1998
(Show Context)
Citation Context ...RA, IΣ1 , RCA0 , and WKL0 are all the primitive recursive functions. In contract, one can characterize the provably total computable functions of PA and HA in terms of higher-type primitive recursion =-=[4, 35]-=-, or using principles of primitive recursion along the ordinal ε0 [34, 25]. Weaker theories of arithmetic can be used to characterize complexity classes like the polynomial time computable functions [... |

67 |
W.: Iterated Inductive Definitions and Subsystems of Analysis
- Buchholz, Feferman, et al.
- 1981
(Show Context)
Citation Context ...ation translation in cases like these. An early instance of this idea can be found in Buchholz' interpretation of theories of inductive definitions ID<, in their intuitionistic counterparts ID<,,, in =-=[17]-=-. More recently, Coquand realized that this idea could be used to interpret I Z in II'. He and Hofmann [27] then extended the interpretation to Buss' theory S2 of bounded arithmetic. Independently, in... |

61 |
Models for Smooth Infinitesimal Analysis
- Moerdijk, Reyes
- 1991
(Show Context)
Citation Context ...categorical logic that are not presented in terms of axiomatic theories, but can be perhaps turned into conservation theorems when presented in more syntactic terms. Moerdijk and Reyes' constructions =-=[60]-=- of models of smooth infinitesimal analysis provide examples; see also the discussion of intuitionistic models of nonstandard analysis at the end of Section 5.2. ?5. Point-free model theory. A central... |

56 |
Mathematical Intuitionism: Introduction to Proof Theory, volume 67
- Dragalin
- 1988
(Show Context)
Citation Context ... theory, discussed in [77]. But there is a long tradition of using algebraic semantics to prove cut elimination, often described in terms of a Kripke model or forcing relation; see, for example [16], =-=[30]-=-, [63]. The resulting proofs lie somewhere between the model-theoretic arguments and the explicitly syntactic ones: they typically have more algorithmic content than the former, but are less dependent... |

53 |
Applied Proof Theory: Proof Interpretations and their Use in Mathematics
- Kohlenbach
- 2008
(Show Context)
Citation Context ...uction is restricted to Σ1 formulas, and WKL0 , the subsystem of second-order arithmetic based on Weak König’s Lemma, are conservative over primitve recursive arithmetic for the class of Π2 sentences =-=[2, 10, 14, 19, 31, 33, 36]-=-. 85. Cut elimination or an easy model-theoretic argument shows that a restricted second-order version, ACA0 , of Peano arithmetic is a conservative extension of Peano arithmetic itself. Similarly, G... |

50 | The lengths of proofs
- Pudlák
(Show Context)
Citation Context ...ension of Peano arithmetic itself. Similarly, Gödel-Bernays-von Neumann set theory GBN , which has both sets and classes, is a conservative extension of Zermelo-Fraenkel set theory. See, for example, =-=[26, 33]-=-. In general, proofs in ACA0 may suffer an interated exponential increase in length when translated to PA; and similarly for GBN and ZF , or IΣ1 and PRA. 6. Theories of nonstandard arithmetic and anal... |

49 |
Sheaves in geometry and logic
- Lane, S, et al.
- 1992
(Show Context)
Citation Context ...of the development of forcing can be found in [61]. For the development of Kripke semantics, see [38]; for a historical overview of the connections between logic and sheaf theory, see the prologue to =-=[56]-=-. ) 2004. Association for Symbolic Logic 1079-8986/04/1003-0001/$3.90 305306 JEREMY AVIGAD forcing provides a means by which we can explicate the notion of necessary truth, or truth in all possible w... |

47 | Internal Set Theory : A New Approach to Nonstandard Analysis
- Nelson
- 1977
(Show Context)
Citation Context ...ension of some suitable universe (e.g., second- or higher-order arithmetic, or a universe of sets), and then "transfers" results back to the original, standard structure. Kreisel [51] and then Nelson =-=[62]-=- showed that one can treat this process axiomatically; for example, Nelson's Internal Set Theory is a conservative extension of ZFC with a predicate for the "standard" sets and axioms characterizing t... |

46 |
Models of Peano Arithmetic
- Kaye
- 1991
(Show Context)
Citation Context ...1]. Of course, these axioms can be extended with stronger hypotheses, such as “large-cardinal” axioms. For information on primitive recursive arithmetic, see [13, 36]; for first-order arithmetic, see =-=[10, 14, 17]-=-; for second-order arithmetic, see [33]; for higher-order arithmetic, see [34]. 3.2 Constructive foundations Given the history of Hilbert’s program, it should not be surprising that proof theorists ha... |

45 |
On the scheme of induction for bounded arithmetic formulas
- Wilkie, Paris
- 1987
(Show Context)
Citation Context ...ations between theories, and therefore upper bounds on the increase in length of proof. But there have been other, perhaps more striking, applications of forcing towards obtaining lower bounds, as in =-=[82]-=-, [1], [67], [50], [78]. I will, regrettably, not discuss these methods here.308 JEREMY AVIGAD ?2. The forcing relation. 2.1. Minimal, classical, and intuitionistic logic. Proof theorists commonly di... |

44 | Theories of finite type related to mathematical practice - Feferman - 1977 |

42 |
Ramsey's theorem and recursion theory
- Jockusch
- 1972
(Show Context)
Citation Context ... with Specker, who, in 1966, showed that RT2 fails in the recursive setting: there is a recursive coloring of pairs of natural numbers with no recursive infinite homogeneous subset. In 1970, Jockusch =-=[45]-=- presented a thorough analysis of the complexity of infinite homogeneous sets of recursive k-colorings. For example, a particular instance of one of his theorems shows that there is a recursive colori... |

41 | On the strength of Ramsey’s theorem for pairs
- Cholak, Jockusch, et al.
(Show Context)
Citation Context ...n [70]. The question as to whether RT 2 2 implies WKL0 over RCA0 is still open, as is the problem of determining the first-order consequences of RCA0 + RT 2 2. 4 Recently Cholak, Jockusch, and Slaman =-=[19]-=- made some progress with respect to determining the strength of RT 2 2. For example, using a recursiontheoretic construction, they showed: Theorem 3.4 For every 2-coloring C of pairs of natural number... |

36 |
Computability theory
- Cooper
- 2003
(Show Context)
Citation Context ...tes of knowledge over time. For the recursion theorist, forcing provides a convenient way of describing constructions in which a sequence of requirements is satisfied one at a time (see, for example, =-=[21, 56]-=-). For the model theorist, forcing is a construction that provides a suitably generic model of any inductive (∀∃) theory. From the point of view of sheaf theory, forcing provides a way of describing t... |

34 |
The point of pointless topology
- Johnstone
- 1983
(Show Context)
Citation Context ...roximated by subideals, and so on. I have neither the space nor the ability to provide an adequate overview of constructive, point-free approaches to mathematics. For that, I will refer the reader to =-=[47]-=-, [48], [35], [34], [22], [69]; for examples of the use of pointfree thinking in extracting constructive proofs from classical arguments, see [23], [24], [26], [29]. My goal in this section is, rather... |

30 |
La logique des topos
- Boileau, Joyal
- 1981
(Show Context)
Citation Context ...ent of Herbrand’s theorem and Skolem functions in [25], or the uniform method of obtaining a number of conservation results in [8]. Other proof-theoretic applications of forcing ideas are sketched in =-=[14]-=-. 5.2 Weak theories of nonstandard arithmetic The subject of nonstandard analysis has both a semantic and a syntactic side. As practiced by Abraham Robinson, nonstandard analysis is a model-theoretic ... |

30 | First-order proof theory of arithmetic
- Buss
- 1998
(Show Context)
Citation Context ...r formulas and proofs. • The methods provide information about the logic that is independent of the choice of formal system that is used to represent it. For more on the cut-elimination theorems, see =-=[10, 29, 34, 36]-=-. 3 Methods and goals 3.1 Classical foundations Recall that Hilbert’s program, broadly construed, involves representing mathematical reasoning in formal systems and then studying those formal systems ... |

29 |
Subsystems of Second-Order Arithmetic
- Simpson
- 1999
(Show Context)
Citation Context ...tic. (It is folklore that the full choice schema, and even a stronger schema of dependent choice, is interpretable in second-order arithmetic, by developing G6del's constructible hierarchy there. See =-=[72]-=-, [57] and the discussion in Sections 3.3 and 4.1 below.) Axiomatic second-order arithmetic is often termed "analysis" because, by coding real numbers and continuous functions as sets of natural numbe... |

27 |
Formal spaces
- Fourman, Grayson
- 1982
(Show Context)
Citation Context ... subideals, and so on. I have neither the space nor the ability to provide an adequate overview of constructive, point-free approaches to mathematics. For that, I will refer the reader to [47], [48], =-=[35]-=-, [34], [22], [69]; for examples of the use of pointfree thinking in extracting constructive proofs from classical arguments, see [23], [24], [26], [29]. My goal in this section is, rather, to describ... |

26 |
Fragments of arithmetic
- Sieg
- 1985
(Show Context)
Citation Context ...uction is restricted to Σ1 formulas, and WKL0 , the subsystem of second-order arithmetic based on Weak König’s Lemma, are conservative over primitve recursive arithmetic for the class of Π2 sentences =-=[2, 10, 14, 19, 31, 33, 36]-=-. 85. Cut elimination or an easy model-theoretic argument shows that a restricted second-order version, ACA0 , of Peano arithmetic is a conservative extension of Peano arithmetic itself. Similarly, G... |

25 | Some points in formal topology
- Sambin
(Show Context)
Citation Context ... on. I have neither the space nor the ability to provide an adequate overview of constructive, point-free approaches to mathematics. For that, I will refer the reader to [47], [48], [35], [34], [22], =-=[69]-=-; for examples of the use of pointfree thinking in extracting constructive proofs from classical arguments, see [23], [24], [26], [29]. My goal in this section is, rather, to describe some application... |

23 |
Dynamical methods in algebra: effective Nullstellensätze
- Coste, Lombardi, et al.
- 2001
(Show Context)
Citation Context ...or that, I will refer the reader to [47], [48], [35], [34], [22], [69]; for examples of the use of pointfree thinking in extracting constructive proofs from classical arguments, see [23], [24], [26], =-=[29]-=-. My goal in this section is, rather, to describe some applications of "point-free" ideas to constructivizing model-theoretic arguments. Many model-theoretic constructions are based on either the comp... |

23 |
Dalen: Constructivism in Mathematics: An Introduction, volume I
- Troelstra, van
- 1988
(Show Context)
Citation Context ...ty. Moreover, Kripke semantics offers natural ways of modeling logics where terms are only partially defined, which is to say, they may fail to denote existing objects. For extensions like these, see =-=[80]-=-. In passing from minimal to intuitionistic logic, we added the clause * pt l. But all we really need is that I -+ p is forced, and, furthermore, it suffices to make sure that this is the case when (p... |

22 |
Neubegründung derMathematik.Erste Miteilung’,in Gesammelte Abhandlungen, vol. III
- Hilbert
- 1935
(Show Context)
Citation Context ...it were; and the same time it becomes possible to draw a sharp and systematic distinction in mathematics between the formulae and formal proofs on the one hand, and the contentual ideas on the other. =-=[16]-=- Gödel’s second incompleteness theorem shows that any “unobjectionable” portion of mathematics is insufficient to establish its own consistency, let alone the consistency of any theory properly extend... |

21 |
Interpreting classical theories in constructive ones
- Avigad
(Show Context)
Citation Context ...ecently, Coquand realized that this idea could be used to interpret IΣ1 in IΣ i 1 . He and Hofmann [27] then extended the interpretation to Buss’ theory S 1 2 of bounded arithmetic. Independently, in =-=[4]-=-, I extended the interpretation to bounded arithmetic, as well as to subsystems of second-order arithmetic based on Σ 1 1 choice and various fragments of admissible set theory. Let us consider Σ1 indu... |

21 |
Subsystems of set Theory and Second Order Number Theory
- Pohlers
- 1998
(Show Context)
Citation Context ...of a theory in terms of the computable orderings that it proves to be well-ordered. The stronger a theory is, the more powerful the principles of transfinite induction it can prove. See, for example, =-=[24, 25, 34]-=-. Alternatively, one can focus on a theory’s computational strength. Suppose a theory T proves a statement of the form ∀x∃yR(x, y), where x and y range over the natural numbers, and R is a computation... |

20 |
Combinatorics in subsystems of second order arithmetic
- Hirst
(Show Context)
Citation Context ...oring C of pairs of natural numbers, there is an infinite homogeneous set H that is low2 in C, i.e., H" <T C". Using a forcing analogue of the same methods, they obtained the following: 4Jeffry Hirst =-=[43]-=- has shown that RCAo + RT2 implies 12 collection; it is conceivable that the first-order consequences of RCAo + RT2 are exactly those of I I plus X2 collection.320 JEREMY AVIGAD THEOREM 3.5. RCAo + I... |

20 | The complexity of propositional proofs - Segerlind - 2007 |

18 | A feasible theory for analysis
- Ferreira
- 1994
(Show Context)
Citation Context ...as been used to shed light on other aspects of weak K6nig's lemma. For example, Simpson and Smith [73] extended Harrington's argument to a n2 conservative extension of elementary arithmetic. Ferreira =-=[32]-=- obtained analogous results for a theory of polynomial-time computable arithmetic, and Fernandes [31] has recently extended this to obtain a conservation theorem for a principle of strict nI reflectio... |

18 | Advanced Topics in Types and Programming Languages
- Pierce
- 2002
(Show Context)
Citation Context ... proof systems that combine aspects of both programming and proving. The references in Section 3.2 above provide logical perspectives on constructive type theory. For a computational perspective, see =-=[23]-=-. 4.4 Automated reasoning and formal verification Another domain where proof-theoretic methods are of central importance is in the field of automated reasoning and formal verification. In computer sci... |

16 | Formalizing forcing arguments in subsystems of secondorder arithmetic
- Avigad
- 1996
(Show Context)
Citation Context ...a of dependent choice, is interpretable in second-order arithmetic, by developing Gödel’s constructible hierarchy there. See [72, 57] and the discussion in Sections 3.3 and 4.1 below.) 3 When I wrote =-=[3]-=-, I was not sensitive to these issues; although I was working with classical logic, I used Beeson’s version of the forcing clause for implication where the usual version would have worked just as well... |

16 | A new method of establishing conservativity of classical systems over their intuitionistic version
- Coquand, Hofmann
- 1999
(Show Context)
Citation Context ...ion of theories of inductive definitions ID<, in their intuitionistic counterparts ID<,,, in [17]. More recently, Coquand realized that this idea could be used to interpret I Z in II'. He and Hofmann =-=[27]-=- then extended the interpretation to Buss' theory S2 of bounded arithmetic. Independently, in [3], I extended the interpretation to bounded arithmetic, as well as to subsystems of second-FORCING IN P... |

16 | 1978]: Two Applications of Logic to Mathematics
- Takeuti
(Show Context)
Citation Context ...k. In fact, there is a long tradition of showing that one can get pretty far with restricted subsystems. Such research extends from the work of Weyl [81] and Hilbert and Bernays [42], through Takeuti =-=[76]-=-, to contemporary work in the "reverse mathematics" program by Simpson, Friedman, and many others [72]. In the reverse mathematics tradition, one drops the schema of comprehension in favor of weaker s... |

15 |
Intuitionistische Untersuchungen der formalistischen Logik
- Kuroda
- 1951
(Show Context)
Citation Context ...see below that Cohen's original "strong forcing" relation is best understood in terms of a slick variant of the double-negation translationFORCING IN PROOF THEORY 309 known as the Kuroda translation =-=[53]-=-. For any formula o, let K denote the result of doubly-negating atomic formulae, and adding a double negation after each universal quantifier. Although pK is not always equivalent to opN in minimal lo... |

15 |
A model for intuitionistic non-standard arithmetic
- Moerdijk
- 1995
(Show Context)
Citation Context ...ple, [59], [9]. The constructions in [59] are inspired by more general sheaf-theoretic constructions of models of not only nonstandard analysis, but synthetic differential geometry: see, for example, =-=[58]-=-, [60], [65]. 5.3. Eliminating Skolem functions. A Skolem axiom has the form Vx, y (P(x, y) - O (x, f (x))), where f is a new function symbol introduced to denote a "Skolem function" for cp. Intuitive... |

15 | The Seventeen Provers of the World
- Wiedijk
- 2006
(Show Context)
Citation Context ...se it effectively, and replacing proof search with calculation wherever possible. For more information on automated reasoning, see [15, 27]. For more information on formally verified mathematics, see =-=[39]-=-, or the the December 2008 issue of the Notices of the American Mathematical Society, which was devoted to formal proof. 114.5 Proof complexity Finally, the field of proof complexity combines methods... |

13 |
Eliminating definitions and Skolem functions in first-order logic
- Avigad
- 2001
(Show Context)
Citation Context ...s. This is a very meager requirement, and, since partial functions can be represented as sequences of ordered pairs, any “sequential” theory of arithmetic suffices. Modulo the details (carried out in =-=[6]-=-), we then have the following partial answer to Pudlák’s question: Theorem 5.3 One can eliminate Skolem axioms in polynomial time from any theory in which one has a suitable coding of finite partial f... |