Already vs. Almost — What's the Difference?

By this or a specified time

In set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set. The notion of "negligible" depends in the context, and may mean "of measure zero" (in a measure space), "countable" (when uncountably infinite sets are involved), or "finite" (when infinite sets are involved).For example: The set S = { n ∈ N | n ≥ k } {\displaystyle S=\{n\in \mathbb {N} \,|\,n\geq k\}} is almost N {\displaystyle \mathbb {N} } for any k {\displaystyle k} in N {\displaystyle \mathbb {N} } , because only finitely many natural numbers are less than k {\displaystyle k} .

So soon

Not quite; very nearly

(Informal) Used as an intensive

Slightly short of; not quite; nearly

Prior to some specified time, either past, present, or future; by this time; previously.

Very close to, but not quite.

So soon.

(mathematics) Up to, except for a negligible set where negligible is not universally but contextually defined.

(US) An intensifier used to emphasize impatience or express exasperation.

A null set; except for a set of measure 0.

Prior to some specified time, either past, present, or future; by this time; previously.

(informal) Something or someone that doesn't quite make it.

Prior to a specified or implied time;

Nearly; well nigh; all but; for the greatest part.

(of actions or states) slightly short of or not quite accomplished; `near' is sometimes used informally for `nearly' and `most' is sometimes used informally for `almost';