Abstract
This chapter develops the theory of natural numbers based on Dedekind–Peano Axioms, also known as Peano Arithmetic Peano Arithmetic. The basic theory of ratios (positive rational numbers) is also developed. It concludes with a section on formal definition by primitive recursion.
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Notes
- 1.
“God created the natural numbers, all else is work of man,” said Kronecker.
- 2.
This can be done more rigorously using the method of definition by primitive recursion due to Dedekind, R.—nnnDedekind, covered in the last section of this chapter.
- 3.
Phrase of Landau, E.—nnnEdmund Landau [47].
- 4.
The notion can be defined for (the positive elements of) any ordered field, where it will hold if and only if the field is Archimedean. A Dedekind partition L, U satisfying the condition of Theorem 134 is sometimes called a Dedekind partition!Scott cut—nnn Scott cut.
References
R. Dedekind. The nature and meaning of numbers (1888). In Essays on the Theory of Numbers [12]. English translation of “Was sind und was sollen die Zahlen?”, Vieweg, 1888.
E. Landau. Foundations of Analysis. Chelsea Publishing Company, 1966.
G. Peano. The principles of arithmetic, presented by a new method (1889). In From Frege to Gödel [78], pages 83–97. English translation of “Arithmetices principia, nova methodo exposita”, Turin, 1889.
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Dasgupta, A. (2014). The Dedekind–Peano Axioms. In: Set Theory. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-8854-5_2
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DOI: https://doi.org/10.1007/978-1-4614-8854-5_2
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