# Definition talk:Supremum/Also defined as

English language note: primarily, "the" indicates uniquness; where not unique, indicates some specialness according to context. With upper bounds, that specialness may be leastness. Nonetheless, using a nonprimary meaning of a word lets more difficulty to understanding e.g. by a mind not fully knowing English. Indirigible (talk) 02:34, 15 September 2020 (UTC)

What's your point? --prime mover (talk) 05:26, 15 September 2020 (UTC)
Second sentence is false. Indirigible (talk) 08:31, 14 October 2020 (UTC)

Let me explain the philosophy behind this page (and indeed, the whole of $\mathsf{Pr} \infty \mathsf{fWiki}$).

Not everybody defines everything in mathematics in the same way. Different sources have different definitions for certain concepts. And sometimes the fashion changes so that a definition evolves over the course of history.

It is the intention of $\mathsf{Pr} \infty \mathsf{fWiki}$ to document all the variant definitions of a concept. In this case, the mainstream definition of "upper bound" of a subset is any number (or general element of an ordered set) in the superset which is greater than every element in the subset. However, there is another variant definition, which can be found in some source works, which define "upper bound" in the same way as we do on $\mathsf{Pr} \infty \mathsf{fWiki}$, that is: as the supremum. That is, they say: "The upper bound of a subset is the smallest element of the superset which is greater than all elements of the subset". This is not the mainstream definition of "upper bound". But such sources exist, and so we document what they say. And, having done so, we point out that if an element of the superset under consideration is greater than every element of the subset, but is not the smallest element of the superset, then these sources (who define "upper bound" unconventionally) do not consider such an element an "upper bound".

Whether this is "correct" or not depends upon whether you accept the non-conventional definition of "upper bound". But on $\mathsf{Pr} \infty \mathsf{fWiki}$ we do not hold that dogmatic approach. We would merely refer to it as "unconventional" rather than "incorrect".

Unfortunately, we are an English language website. A basic level of understanding of the language is necessary to be able to use it, and in order to contribute it is necessary to be fluent. If contributors become confused by certain usages, there is little we can do except to explain as best we can when such difficulties arise. --prime mover (talk) 09:34, 14 October 2020 (UTC)

I should have been clearer: given the first sentence, the second is false. The first sentence does not imply there being only one upper bound, only that the supremum is an upper bound with a contextually-determined property, e.g. leastness, that only it has. This is a valid albeit non-primary English use of "the." That that variant definition of "supremum" was meant was not clear.
If a basic understanding of English should be sufficient for reading, ambiguity should be avoided. If fluency should be necessary for writing, there is no good excuse for ambiguity. Indirigible (talk) 18:16, 14 October 2020 (UTC)
What's false about it then? --prime mover (talk) 19:08, 14 October 2020 (UTC)
The convention wherein the supremum is the upper bound does not preclude other upper bounds being. Indirigible (talk) 20:15, 14 October 2020 (UTC)
If it says "the" then it means it's unique, there is only one and so there can't be another. --prime mover (talk) 20:51, 14 October 2020 (UTC)
The page does not mention whatever is that other definition of "upper bound." Indirigible (talk) 20:20, 14 October 2020 (UTC)
I would have to go and do some research but it will take some time ploughing through my library to find whatever those sources are. --prime mover (talk) 20:51, 14 October 2020 (UTC)