https://en.m.wikipedia.org/wiki/Finitism

https://en.m.wikipedia.org/wiki/Primitive_recursive_arithmetic

https://en.m.wikipedia.org/wiki/Elementary_function_arithmetic

https://en.m.wikipedia.org/wiki/Dyck_language

https://plato.stanford.edu/entries/geometry-finitism/

https://en.m.wikipedia.org/wiki/Rational_trigonometry

Check Troelstra and Granstrom for references to Goodstein. Compare with Robert Grassmann

[Troelstra_A_S]_History_of_Constructivism_in_the_2.pdf (Also re Kolmogorov, CRM, RM, BCM etc)

See also Mayberry and Klein on ancient Greek. Mayberry also mentions Vopenka Alternative Set Theory (semisets).

Troelstra’s “History of Constructivism in the 20th Century” is in:

(Lecture Notes in Logic 36) Juliette Kennedy, Roman Kossak (eds.)-Set Theory, Arithmetic, and Foundations of Mathematics_ Theorems, Philosophies-Cambridge University Press (2011).pdf

@book{book:857712,

title = {Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies},

author = {Juliette Kennedy, Roman Kossak (eds.)},

publisher = {Cambridge University Press},

isbn = {1107008042,9781107008045},

year = {2011},

series = {Lecture Notes in Logic 36},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=afa30ac52f529ea4ba7f0ec3ff18ec9f}

}

(Encyclopedia of mathematics and its applications) John P. Mayberry-The foundations of mathematics in the theory of sets-Cambridge University Press (2001).djvu

@book{book:857567,

title = {The foundations of mathematics in the theory of sets},

author = {John P. Mayberry},

publisher = {Cambridge University Press},

isbn = {0521770343, 9780521770347, 0521172713, 9780521172714},

year = {2001},

series = {Encyclopedia of mathematics and its applications},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=44fd18d2c6da1241a24e7d7cc76f8404}

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@article{Goodstein-1945,

doi = {10.1112/plms/s2-48.1.401},

title = {Function Theory in an Axiom-Free Equation Calculus},

author = {Goodstein, R. L.},

publisher = {Oxford University Press},

journal = {Proceedings of the London Mathematical Society},

issnp = {0024-6115},

issne = {1234-5678},

year = {1945},

month = {01},

day = {01},

volume = {s2-48},

issue = {1},

page = {401–434},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1112/plms/s2-48.1.401},

}

@book{book:290445,

title = {Recursive Number Theory: A Development of Recursive Arithmetic in a Logic-Free Equation Calculus},

author = {Goodstein R.L.},

publisher = {NH},

isbn = {0444533850,9780444533852},

year = {1957},

series = {Studies in Logic and the Foundations of Mathematics, Volume 20},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=98615897AC8DA94B2523B3B177A7389B}

}

@inbook{Goodstein-1957,

doi = {10.1016/s0049-237x(08)71966-9},

title = {[Studies in Logic and the Foundations of Mathematics] Recursive Number Theory – A Development of Recursive Arithmetic in a Logic-Free Equation Calculus Volume 20 || Preface},

author = {Goodstein, R.L.},

isbn = {9780444533852},

year = {1957},

page = {vii–ix},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1016/s0049-237x(08)71966-9},

}

@article{R-1973,

doi = {10.1002/malq.19730190802},

title = {On Limiting the Applications of the Uniqueness Rules in The Equation Calculus},

author = {R. Louis Goodstein},

publisher = {John Wiley and Sons},

journal = {Mathematical Logic Quarterly},

issnp = {0942-5616},

issne = {1521-3870},

year = {1973},

volume = {19},

issue = {8-10},

page = {115–116},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1002/malq.19730190802},

}

@book{book:786896,

title = {Strict Finitism and the Logic of Mathematical Applications},

author = {Feng Ye (auth.)},

publisher = {Springer Netherlands},

isbn = {9400713460,9789400713468},

year = {2011},

series = {Synthese Library 355},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=3037B345B71B9DAC3207D077E7820B56}

}

@inbook{Skolem-1955,

doi = {10.1016/s0049-237x(09)70300-3},

title = {[Studies in Logic and the Foundations of Mathematics] Mathematical Interpretation of Formal Systems Volume 16 || Peano’s Axioms and Models of Arithmetic},

author = {Skolem, Th.},

isbn = {9780444533814},

year = {1955},

page = {1–14},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1016/s0049-237x(09)70300-3},

}

Plato: Parmenides 149a7-c3. A Proof by Complete Induction?

(Perspectives in Logic) Stephen G. Simpson-Subsystems of Second Order Arithmetic-Cambridge University Press (2009).pdf

http://www.personal.psu.edu/t20/sosoa/

@book{book:1122291,

title = {Subsystems of Second Order Arithmetic},

author = {Stephen G. Simpson},

publisher = {Cambridge University Press},

isbn = {978-0-521-88439-6,978-0-511-57985-1},

year = {2009},

series = {Perspectives in Logic},

edition = {2nd},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=9085e826cd1783a5246ad0ac4f534725}

}

## 2 thoughts on “Finitism”