Re: First order theory of rationals

On May 23, 7:52 am, "pers...@xxxxxxxxxxxxxx" wrote:
Hello,
I am reading peasno's axioms. I understand they only include
natural numbers.
What is the first order theory that includes rational numbers.

The rational numbers are an ordered field.

My questions are as follows -
1) How do I extend peano's axioms to include negative integers.

The integers are a ring with unit.

2) Then how do we extend it for division and the rational numbers.


This is not always done by "extending".

It is usually more convenient to just start all over again at the
higher level of complexity. If you are trying to encode everything
in set theory then it is almost necessary.

3) And finally, all of real numbers.

The real numbers can be thought of as the "complete" ordered field
(the one including all its limit points), or the "Archimedean" ordered
field.

Is there something weaker than ZF.

Well, yes, coming from algebra as opposed to from set theory,
the axioms for rings and fields are much weaker than the
axioms for ZF.

Something like peano's axioms.

Try this:
http://www.pitt.edu/~dickinsm/1020-2071/integerAxioms.pdf
.

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