How can a statement like “*5 is greater than 4*” be quantified? Or is it just obvious? If we know anything about mathematics we know that there is no way we can assume something as obvious, there must be an axiomatical contruct that puts a rigorous meaning on “*5 is greater than 4*“.

The first attempt would be to say that 5=4+1 so 5 is “1 more” than 4 so must be bigger. This translates to 5-4=1: “*5 is greater than 4 because 5-4 is positive*“. Careful! 4=5+(-1) so 4-5=-1: “*4-5 is negative*“. But what does positive and negative mean? Easy? Positive is greater than zero… At this point a stronger construct is needed:

**Definition: **Call a set *positive* if for all

- Given either , or

If we think carefully, this definition concurs exactly with that of the naive notion of positive. So we can say that “*5 is greater than 4 because 5-4 is positive.”*

**Definition: **Given , is said to be *greater than* , , if is positive.

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