There is no reason why a language or proof system cannot use different axiomatic bases of orders, as long as they have different identities (names). The partia

There is no reason why a language or proof system cannot use different axiomatic bases of orders, as long as they have different identities (names). The partial order of intervals, where any overlap means equality, is perfectly respectable, and can sit alongside other theory fragments about total/partial/strict orders. The well-known algebra of intervals (1983), with an enlarged set of axioms and operators, may also be introduced, to subsume this proliferation of theory fragments: https://en.wikipedia.org/wiki/Allen%27s_interval_algebra https://en.wikipedia.org/wiki/Allen%27s_interval_algebra https://cse.unl.edu/~choueiry/Documents/Allen-CACM1983.pdf https://cse.unl.edu/~choueiry/Documents/Allen-CACM1983.pdf The algebra is almost trivial: anyone who spends 5 minutes scribbling on a sheet of paper would come up with the same solution. There is also a beautiful symmetrical state machine of one interval smoothly sliding over another, which I leave as another 5-minute exercise for the reader with aforementioned sheet of paper. P.S. Sometimes such professed 'issues' are just flourishes of pedagogical knives, which seem dangerous to the naive student, but the resolution is just to shoot the prophet of doom. For example, the many varied implementations of Fibonacci series can explain the differences between iterative/(tail-)recursive/functional/memo-ized implementations, while hiding the facts that: (1) nobody really needs Fibonacci numbers, unless you are unlucky enough to be modelling the number of immortal rabbits, in a world without foxes, but with an infinite series of cabbages; (2) the inside secret is that there's a closed-form expression called the Binet Formula (1843): https://en.wikipedia.org/wiki/Fibonacci_number#Closed-form_expression https://en.wikipedia.org/wiki/Fibonacci_number#Closed-form_e... The mystical prophets never tell their students about this, even though the facts have been well-known for 40-180 years, because it would hurt the business of prophet-ing.