# Reversing building irrationals on the rationals

• Aug 3rd 2010, 09:45 AM
wonderboy1953
Reversing building irrationals on the rationals
Has anyone tried to start off with irrational numbers as a foundation, then derive the rational numbers from them?
• Aug 3rd 2010, 11:55 PM
CaptainBlack
Quote:

Originally Posted by wonderboy1953
Has anyone tried to start off with irrational numbers as a foundation, then derive the rational numbers from them?

The axiomatic definition of the reals already contains "0", "1", "+" and ".", so the natural numbers can be generated easily from these and the other axioms, once you have the naturals the rationals can be constructed by the usual process.

CB
• Aug 4th 2010, 01:06 PM
wonderboy1953
Quote:

Originally Posted by CaptainBlack
The axiomatic definition of the reals already contains "0", "1", "+" and ".", so the natural numbers can be generated easily from these and the other axioms, once you have the naturals the rationals can be constructed by the usual process.

CB

It's an interesting fact CB.

Historically speaking the discovery of the rational numbers preceded the irrational numbers (a secret society was said to have killed someone who revealed the discovery of an irrational number or numbers as that cult feared that the harmony of the rationals would be disrupted). It would seem illogical that the reverse can happen, but to my knowledge nothing has been said about this so I'm exploring this area to see what someone knows about this.