Set-theoretic definition of natural numbers
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A set-theoretic definition of the natural numbers was proposed by John von Neumann. It allows the discussion of natural numbers in a system based (as modern mathematics is) on axiomatic set theory.
Neumann proposed the following definition:
- Define the empty set to be zero.
- Define the successor of n as n ∪ {n}
The set N of all natural numbers is then guaranteed to exist by axiom of infinity. It can easily be shown that the above definition satisfies the Peano axioms. It also (in contrast to some alternative definitions) has the property that each natural number n is a set with exactly n elements: {0,1,2,...,n-1}
See Peano arithmetic.
References
- What are the Natural Numbers? (http://www.maths.may.ie/staff/gmg/nn.ps) (Post Script file), Gary McGuire