## Second-order logic and foundations of mathematics (2001)

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Venue: | The Bulletin of Symbolic Logic |

Citations: | 18 - 3 self |

### BibTeX

@ARTICLE{Väänänen01second-orderlogic,

author = {Jouko Väänänen},

title = {Second-order logic and foundations of mathematics},

journal = {The Bulletin of Symbolic Logic},

year = {2001},

volume = {7},

pages = {2001}

}

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### Abstract

We discuss the differences between first-order set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former. 1

### Citations

572 |
A decision method for elementary algebra and geometry
- Tarski
- 1951
(Show Context)
Citation Context ...t the structure R is the only model of RCF2. The advantage of RCF over RCF2 is that it can also be used as a tool for proving the decidability of the first-order theory of the arithmetic of the reals =-=[Tar48]-=-. We can also argue in the urlogic ZFC that the object language satisfies the Gödel Completeness Theorem, i.e. if a sentence φ of the object language is true in every model of another sentence ψ of th... |

199 |
Subsystems of Second Order Arithmetic
- Simpson
- 1999
(Show Context)
Citation Context ...out models of set theory that came with the proof, is a huge step toward understanding why CH has not yet been settled in V . Likewise, the study of Henkin models of second-order artihmetic (see e.g. =-=[Sim99]-=-) isolate reasons why some results of number theory or analysis are hard to prove. 2 Preliminary example Mathematicians argue exactly but informally. This has worked well for centuries. However, if we... |

185 |
Completeness in the theory of types
- Henkin
- 1950
(Show Context)
Citation Context ...or subsets of the universe (sometimes variables for n-ary relations as well, but this is not important in this context). The deductive calculus DED2 of secondorder logic is based on rules and axioms (=-=[Hen50]-=-) which guarantee that the ∗I am grateful to Juliette Kennedy for many helpful discussions while developing the ideas of this paper. † Research partially supported by grant 40734 of the Academy of Fin... |

71 |
The Principles of Mathematics Revisited
- Hintikka
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(Show Context)
Citation Context ...ch through all natural numbers and then perform a finite polynomial calculation on that number. 8 IF-logic Hintikka has suggested that the so-called IF-logic provides a new foundation for mathematics =-=[Hin96]-=-. IF-logic is an extension of first-order logic which is semantically equivalent with the Σ 1 1-part of second-order logic. The secondorder theories P2, A2 and ZFC2 can all be (finitely) axiomatized i... |

28 |
On extensions of elementary logic
- Lindström
- 1969
(Show Context)
Citation Context ...der sentence has for each finite number n a model (perhaps infinite) with at least n elements, then the sentence has models in all infinite cardinalities. It is remarkable that by Lindström’s Theorem =-=[Lin69]-=-, first-order logic is the only logic with this property. This extreme flexibility of first-order logic with respect to the cardinality of the universe is often held against first-order logic. This fl... |

27 |
Informal Rigour and Completeness Proofs
- Kreisel
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(Show Context)
Citation Context ...re is a model of ZFC2, which is not a model of ZFC M 2 , and if there is a Mahlo cardinal, then there is a model of ZFC M 2 , which is not a model of ZFC WC 2 . 9 This has been pointed out by Kreisel =-=[Kre67]-=-. 15sIf we add to ZFC2 the axiom “There are no inaccessible cardinals”, the resulting system is categorical. Similarly ZFC M 2 and ZFC WC 2 can be strengthened to categorical second-order theories. Wh... |

7 | Set-theoretic definability of logics - Väänänen - 1985 |

5 |
Absolute logics and L∞ω
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- 1972
(Show Context)
Citation Context ...der logic An important feature of first-order logic is that it is absolute in a strong sense. Absoluteness has a technical meaning in set theory, which first-order logic fulfills in several ways (see =-=[Bar72]-=- and [V85]). For example, the property ZFC ⊢ φ is an r.e. property of a first-order sentence φ. Therefore, if ZFC ⊢ φ is true, its formalized version holds inside any model of first-order Peano arithm... |

5 |
Reductions in the theory of types
- Hintikka
- 1955
(Show Context)
Citation Context ...d-order sentences φ in a vocabulary that contains one binary predicate symbol P . It is known that V al 2 is a highly complex subset of N. For example, V al 2 is not Σ m n for any m, n < ω ([Mon65a], =-=[Hin55]-=-). What exactly is the complexity of this set? Theorem 1 V al 2 is the complete Π2-definable 12 set of integers. Proof. Let us first observe that the predicate x = P(y) is Π1-definable. We can also Π1... |

4 |
Reductions of higher-order logic
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(Show Context)
Citation Context ...alid second-order sentences φ in a vocabulary that contains one binary predicate symbol P . It is known that V al 2 is a highly complex subset of N. For example, V al 2 is not Σ m n for any m, n < ω (=-=[Mon65a]-=-, [Hin55]). What exactly is the complexity of this set? Theorem 1 V al 2 is the complete Π2-definable 12 set of integers. Proof. Let us first observe that the predicate x = P(y) is Π1-definable. We ca... |

4 | The spectrum problem. II. Totally transcendental and in depth - Shelah - 1982 |

2 | The spectrum problem. I. ℵε-saturated models, the main gap - Shelah - 1982 |

1 |
A note on weak second order logic with variables for elementarily de relations
- Lindstrom
- 1971
(Show Context)
Citation Context ...els 〈A, S〉 where S is the set of all finite subsets of A. One can also limit S to the set of all countable subsets of A. Finally, one may limit S to the set of all first-order definable subsets of A (=-=[Lin73]-=-). 2s• There is a finite second-order axiom system RCF2 such that the field R of real numbers is the only (up to isomorphism) full model of RCF2. As the above facts demonstrate, it is highly critical ... |

1 |
Set theory and higher-order logic
- Montague
- 1963
(Show Context)
Citation Context ... of second-order predicate logic. Depending on the context, the non-logical vocabulary may consist of symbols for the arithmetic of natural numbers, arithmetic of real numbers, and so forth. Montague =-=[Mon65b]-=- gives secondorder Peano axioms Z2 for number theory, and second-order axioms RCF2 for real closed fields. For full second-order logic there is a notion of “semantical” derivation: We can derive ψ fro... |

1 |
Concerning the problem of axiomatizability of the field of real numbers in the weak second order logic
- Mostowski
- 1961
(Show Context)
Citation Context ...rder axiom system Z2 such that the semiring N of natural numbers is the only full model (up to isomorphism) of Z2. 1 Also other kinds of models have been studied. In weak second-order logic (see e.g. =-=[Mos61]-=-) one considers only models 〈A, S〉 where S is the set of all finite subsets of A. One can also limit S to the set of all countable subsets of A. Finally, one may limit S to the set of all first-order ... |

1 |
Foundations without foundationalism. The Clarendon Press Oxford
- Shapiro
- 1991
(Show Context)
Citation Context ...ary to formalize basic concepts such as the concepts of language and criteria of truth. We study two metatheories of mathematics: first-order set theory and second-order logic. It is often said (e.g. =-=[Sha91]-=-), that second-order logic is better than first-order set theory because it can in its full semantics axiomatize categorically N and R, while first-order axiomatization of set theory admits non-standa... |

1 | Absolute logics and L1 - Barwise - 1972 |

1 | Concerning the problem of axiomatizability of the of real numbers in the weak second order logic - Mostowski - 1961 |

1 | The spectrum problem. I. @ " -saturated models, the main gap - Shelah - 1982 |

1 | Set-theoretic de of logics - Vaananen - 1985 |