Godel actually proved that there is no complete and consistent axiomatic formulation of arithmetic: in particular, there is no set of axioms for the structure of the natural numbers which both (1) allows the deduction of every true theorem in arithmetic, and (2) does not also allow the deduction of any false theorem in arithmetic. This is Godel's Theorem.

A*rith"me*tic (#), n. [OE. arsmetike, OF. arismetique, L. arithmetica, fr. Gr. (sc. ), fr. arithmetical, fr. to number, fr. number, prob. fr. same root as E. arm, the idea of counting coming from that of fitting, attaching. See Arm. The modern Eng. and French forms are accommodated to the Greek.]

1.

The science of numbers; the art of computation by figures.

2.

A book containing the principles of this science.

Arithmetic of sines, trigonometry. -- Political arithmetic, the application of the science of numbers to problems in civil government, political economy, and social science. -- Universal arithmetic, the name given by Sir Isaac Newton to algebra.

 

© Webster 1913.

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