natural number theory

Counting numbers, and all numbers, can form an axiomatic theory {natural number theory}.

axioms

Start with null set and with two axioms. Numbers correspond to two sets of previously constructed numbers, with no member of left set greater than or equal to members of right set. Number is less than or equal to another number, if and only if no member of first-number left set is greater than or equal to second number and no member of second-number right set is less than or equal to first number.

axioms: procedure

First, make null set both right and left set, to obtain number zero. Then make right set null and left set contain number zero to obtain number one. Continue to build all counting numbers. All other numbers can derive from counting numbers.

equivalence

Equivalence or one-to-one correspondence can construct counting numbers. Start with null set. Define zero as cardinal number of elements in null set and equivalent sets. Define one as number of elements in set that contains only zero as element. Define two as number of elements in set that contains only elements zero and one. Define N as number of elements in set that contains only elements zero through N.

numbers in general

Axiomatizing natural numbers allows axiomatizing all number types. The number "one" belongs to a set. Set members have one and only one successor. If two successors are equal, then members are equal. The number "one" is not any number's successor. If subset contains "one" and another number, then number successor is in subset and subset is same as whole set.

integers

Positive integers can be an axiomatic system. Undefined terms are "one", "number", and "successor of". Dedekind-Peano postulates can construct the positive integers.

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Date Modified: 2022.0224