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Are "lower bound" and "upper bound" unambigous? What if one's in the negative axis?

Consider e.g.

$$\{-n: n \in \mathbb{N} \}$$

This has no lower bound, if one consider lower to mean towards $- \infty$. However, what if one defined that the set is ordered so that one counts like $0,-1, -2, ...$. Then one could conceive that $0$ is the lower bound from the perspective of the index set. Not from the number greatness perspective though.

So is "lower bound" unambigous? Is my index set example "valid"? Perhaps it's not and I should rather rely on natural number axioms (Peano) and how they define order?

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The definition of the concept "lower bound of a subset" involves statements about the ordering of elements. If you change the way the order is defined then you change which things are lower bounds. But there's no ambiguity in the definition.

The integers come equipped with a natural ordering defined from the Peano axioms. That's the one meant when lower bounds are discussed with no further qualification.

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