### Abstract

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories.

Original language | English (US) |
---|---|

Pages (from-to) | 389-420 |

Number of pages | 32 |

Journal | Synthese |

Volume | 103 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1995 |

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### ASJC Scopus subject areas

- Social Sciences(all)

### Cite this

*Synthese*,

*103*(3), 389-420. https://doi.org/10.1007/BF01089734

**Finite mathematics.** / Lavine, Shaughan M.

Research output: Contribution to journal › Article

*Synthese*, vol. 103, no. 3, pp. 389-420. https://doi.org/10.1007/BF01089734

}

TY - JOUR

T1 - Finite mathematics

AU - Lavine, Shaughan M

PY - 1995/6

Y1 - 1995/6

N2 - A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories.

AB - A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories.

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UR - http://www.scopus.com/inward/citedby.url?scp=34249761869&partnerID=8YFLogxK

U2 - 10.1007/BF01089734

DO - 10.1007/BF01089734

M3 - Article

VL - 103

SP - 389

EP - 420

JO - Synthese

JF - Synthese

SN - 0039-7857

IS - 3

ER -