Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T02:35:13.977Z Has data issue: false hasContentIssue false

A system of completely independent axioms for the sequence of natural numbers

Published online by Cambridge University Press:  12 March 2014

Shianghaw Wang*
Affiliation:
The National University of Peking, Kunming, Yunnan, China

Extract

Peano's five axioms for the sequence of natural numbers run as follows: (1) 1 is a number, (2) Any number α has a unique sequent a′ (3) Different numbers have different sequents, (4) 1 is not a sequent, and (5) The principle of mathematical induction.

Now, in a system of axioms, it seems desirable, in the first place, to avoid symbols for particular individuals and symbols for functions, and, secondly, to secure the property of complete independence. In fact, Peano's system can be replaced by an equivalent one which fulfills both requirements. In such a system, the symbol “1” and the function “sequent” will not appear, although they may be introduced afresh through definitions. Besides, the principle of mathematical induction will no more play the rôle of an axiom, because, under the second requirement, it can hardly be stated in a simple and elegant form.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1943

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)