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The law of trichotomy still isn't covered. It can be split into two parts: at most one of the three cases can occur, and at least one of the three cases occurs. more examples on math forum; The first can be stated as an axiom of addition as
It is not the case that x = x + y.
And that says it is not the case that x > x. The other half requires the axiom
For each x and y, either x = y or there is some z such that x + z = y, or there is some z such that x = y + z.
With these axioms, all the properties of magnitudes needed in the first few books of the Elements can be proved. For instance, we can prove
If 2x = 2y, then x = y.
more examples on free online math tutoring.
The law of trichotomy still isn't covered. It can be split into two parts: at most one of the three cases can occur, and at least one of the three cases occurs. more examples on math forum; The first can be stated as an axiom of addition as
It is not the case that x = x + y.
And that says it is not the case that x > x. The other half requires the axiom
For each x and y, either x = y or there is some z such that x + z = y, or there is some z such that x = y + z.
With these axioms, all the properties of magnitudes needed in the first few books of the Elements can be proved. For instance, we can prove
If 2x = 2y, then x = y.
more examples on free online math tutoring.
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