## Types in logic and mathematics before 1940 (2002)

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Venue: | Bulletin of Symbolic Logic |

Citations: | 10 - 5 self |

### BibTeX

@ARTICLE{Kamareddine02typesin,

author = {Fairouz Kamareddine and Twan Laan and Rob Nederpelt},

title = {Types in logic and mathematics before 1940},

journal = {Bulletin of Symbolic Logic},

year = {2002}

}

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### Abstract

Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λ-calculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced

### Citations

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(Show Context)
Citation Context ... We write ##C for the original calculus of Church as presented in [14]. Note that this is di#erent from the calculus # # used in frameworks like the Barendregt cube and the pure type systems found in =-=[3]-=-. c # 2002, Association for Symbolic Logic 1079-8986/02/0802-0001/$7.10 185 186 FAIROUZ KAMAREDDINE, TWAN LAAN, AND ROB NEDERPELT describe the simple theory of types that resulted from simplifications... |

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(Show Context)
Citation Context ... logic and mathematics at the end of the 19th and the beginning of the 20th century. But it was not the only method developed for this purpose. Another tool was the fine-tuning of Cantor's Set Theory =-=[9, 10]-=- by Zermelo [73], and the iterative conception of set (see [7]) that resulted from the foundation axiom of Zermelo-Fraenkel's set theory ZF. Although it was clear that in ZF, the foundation axiom does... |

65 |
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(Show Context)
Citation Context ...cs. Since 1940, functions have remained one of the main objects of study for type theorists. The historical remarks in this article have been taken from various resources. The most important ones are =-=[6, 19, 68, 44, 53, 69, 72]-=-. In Section 2 we discuss the prehistory of type theory. We first argue that the concept of types has always been present in mathematics, though nobody 7 The first two accounts of avoiding the paradox... |

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(Show Context)
Citation Context ...adoxes, it was added as a technical refinement. The separation axiom which replaced the unrestricted comprehension axioms is the one responsible for avoiding the paradoxes. This axiom goes as follows =-=[23, 2]-=-: (Comprehension) For each open well-formed formula #, #y #x [(x # y) ## #(x)] where y is not free in #(x). This unrestricted comprehension leads to a paradox by taking #(x) to be (x # x): #y #x [(x #... |

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(Show Context)
Citation Context .... Controversial results had appeared in analysis. Many of these controversies were solved by the work of Cauchy. For instance, he introduced a precise definition of convergence in his Cours d'Analyse =-=[11]-=-. Due to the more exact definition of real numbers given by Dedekind [21], the rules for reasoning with real numbers became even more precise. In 1879, Frege published his Begri#sschrift [25], in whic... |

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(Show Context)
Citation Context ... the 20th century. But it was not the only method developed for this purpose. Another tool was the fine-tuning of Cantor's Set Theory [9, 10] by Zermelo [73], and the iterative conception of set (see =-=[7]-=-) that resulted from the foundation axiom of Zermelo-Fraenkel's set theory ZF. Although it was clear that in ZF, the foundation axiom does not help in avoiding the paradoxes, it was added as a technic... |

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(Show Context)
Citation Context ...true". The paradox was widely known in antiquity. For instance, it is referred to in the Bible (Titus 1:12). It is based on the confusion between language and meta-language. The Burali-Forti para=-=dox ([8]-=-, 1897) is the first of the modern paradoxes. It is a paradox within Cantor's theory on ordinal numbers. 14 Cantor's paradox on the largest cardinal number occurs in the same field. It must have been ... |

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10 |
zur Begründung der transfiniten Mengenlehre
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(Show Context)
Citation Context ... logic and mathematics at the end of the 19th and the beginning of the 20th century. But it was not the only method developed for this purpose. Another tool was the fine-tuning of Cantor's Set Theory =-=[9, 10]-=- by Zermelo [73], and the iterative conception of set (see [7]) that resulted from the foundation axiom of Zermelo-Fraenkel's set theory ZF. Although it was clear that in ZF, the foundation axiom does... |

9 | editors. Philosophy of Mathematics - Benacerraf, Putnam - 1983 |

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8 |
A set of postulates for the foundation of logic (1
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(Show Context)
Citation Context ...ter was needed and # happened to have been chosen. Moreover, Curry had told him that Church had a manuscript in which there were many occurrences of # already in 1929, so three years before the paper =-=[12]-=- appeared. TYPES IN LOGIC AND MATHEMATICS BEFORE 1940 197 Grundgesetze in which he provided a very detailed and correct description of the paradox. The derivation goes as follows (using the same argum... |

8 | Letter to Russell. English translation - Frege - 1902 |

7 | 1893) Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet - Frege |

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4 | Russell’s Hidden Substitutional Theory - Landini - 1998 |

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3 | A Correspondence between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems - Kamareddine, Laan |

3 | Mathematical Logic and the Foundations of Mathematics - Kneebone - 1963 |