Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. (of a subset) the greatest element of the containing set that is smaller than or equal to all elements of the subset. The infimum may or may not be a member of the subset.
Etymologies
Examples

(The degree of an universally quantified formula xÏ is defined as the infimum of truth degrees of instances of Ï; similarly

The greatest lower bound is also sometimes called the "infimum", abbreviated "inf".

If either one of these functions, say θF / a, is influenced by some information that is free in the above sense (i.e., not a function of A's choice of directions and events Fearlier than that choice), then there must be an an earliest ( "infimum") Ftime t0 after which all such information is available to a. Since the nonfree information is also available at t0, all these information bits, free and nonfree, must have a value 0 or 1 to enter as arguments in the function θF / a.

Now, IIRC it is Linde himself who has pointed out that one can push infimum of worldlines unboundedly back, making the process physically "eternal" in both directions.
If We Live in a Multiverse, How Many Are There?  Universe Today

It is the difference between choosing the minimum of the set of found fossils and searching the infimum of the set of possible fossils.

That is we consider a confidence region of fits based on the statistics and we plot the supremum (least upper bound) and infimum (greatest lower bound) of all these fits.

I remember the time when he did not venture to offend me by a word: now he is at liberty to do me an actual injury. sed viden? fortuna humana fingit artatque ut lubet: me, qui liber fueram servom fecit, e summo infimum; qui imperare insueram, nunc alterius imperio obsequor. et quidem si, proinde ut ipse fui imperator familiae, habeam dominum, non verear ne iniuste aut graviter mi imperet.

Again, the railroad might be absolved from the charge of violating natural justice if the lower rates which it extended to the monopoly did not fall below the lowest level (pretium infimum) of justice, while the charges exacted from the independent shippers did not exceed the highest level (pretium summum) sanctioned by justice.

CASSIA (Sturtii), tomentosoincana, foliis 4jugis foliolis lanceolatolinearibus planis: glandula depressa inter par infimum, racemo corymboso paucifloro cum pedunculo suo folium paulo superante v. aequante, calyce tomentoso.

In general a set with upper bounds need not have a supremum; a set with lower bounds need not have an infimum.
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