Introduction
What is essential in the system of natural numbers?
In modern logic, natural numbers are most often defined
as finite ordinals or cardinals. A natural number is the cardinality
(intuitively, number of elements) of a finite set. The basic operations like
forming the successor of a number are introduced after the definition of
numbers. There is also the axiomatic approach, which defines some properties
for numbers (Peano axioms) instead of any definition for what the numbers
really are. The cardinality approach can be proved to satisfy the axioms.
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Lecture
In this paragraph, we will develop the Peano Axioms and use them to provide a completely formal definition of the natural numbers N. In what follows, it is best to train yourself to assume nothing and use only statements that are known to be true via axioms or statements that follow from these axioms. We will formalize the notion of equality and then present the Peano axioms.
In this paragraph, we will develop the Peano Axioms and use them to provide a completely formal definition of the natural numbers N. In what follows, it is best to train yourself to assume nothing and use only statements that are known to be true via axioms or statements that follow from these axioms. We will formalize the notion of equality and then present the Peano axioms.
7 Peano Axioms by Roberto Pelayo: pdf
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In this section we introduce axioms for the
natural numbers. These axioms are properties, satisfied by the natural numbers, which we will agree
to accept. We will extend the set of natural numbers later to include the
integers, the rational numbers and then the real numbers. These sets will all
be defined using the natural numbers
and additional definitions, but we will not need further axioms to complete the
task of defining the real numbers.
Operations defined on N by Andrew Pollington: pdf
More understanding
The Dedekind, Peano Axioms by D. Joyce: pdf
A Theory of Natural Numbers by Richard B. Angell: pdf
Difficulties of the set of natural numbers by Qiu Kui Zhang: pdf
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