The practice of refusing infinite values for meas-urable quantities does not come from a priori orideological motivations, but rather from moremethodological and pragmatic motivations.[ci-tation needed]One of the needs of any physical and scientifictheory is to give usable formulas that correspondto or at least approximate reality.As an example if any object of infinite gravita-tional mass were to exist, any usage of the for-mula to calculate the gravitational force wouldlead to an infinite result, which would be of nobenefit since the result would be always the sameregardless of the position and the mass of theother object. The formula would be useful neither to computethe force between two objects of finite mass norto compute their motions.If an infinite mass object were to exist, any objectof finite mass would be attracted with infiniteforce (and hence acceleration) by the infinitemass object, which is not what we can observein reality.
The practice of refusing infinite values for meas- Sometimes infinite result of a physical quan-urable quantities does not come from a priori or tity may mean that the theory being used toideological motivations, but rather from more compute the result may be approaching themethodological and pragmatic motivations.[ci- point where it fails. This may help to indicatetation needed] One of the needs of any physical the limitations of a theory.and scientific theory is to give usable formulas This point of view does not mean that infinitythat correspond to or at least approximate real- cannot be used in physics.ity. For convenience’s sake, calculations, equations,As an example if any object of infinite gravita- theories and approximations often use infinitetional mass were to exist, any usage of the for- series, unbounded functions, etc., and maymula to calculate the gravitational force would involve infinite quantities. Physicists howeverlead to an infinite result, which would be of no require that the end result be physically mean-benefit since the result would be always the same ingful. In quantum field theory infinities ariseregardless of the position and the mass of the which need to be interpreted in such a way asother object. to lead to a physically meaningful result, a pro- cess called renormalization.
Cantor defined two kinds of infinite Generalizing finite and the ordinary infinite sequences whichnumbers: ordinal numbers and cardi- are maps from the positive integers leads to mappings fromnal numbers. ordinal numbers, and transfinite sequences.Ordinal numbers may be identifiedwith well-ordered sets, or counting Cardinal numbers define the size of sets, meaning how manycarried on to any stopping point, in- members they contain, and can be standardized by choos-cluding points after an infinite num- ing the first ordinal number of a certain size to represent theber have already been counted. cardinal number of that size.
DISTANCE However, there are some theoretical circum- stances where the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the gen- eral theory of relativity allow for finite mass distributions of zero size, and thus infinite density.This is an example of what is called amathematical singularity, or a pointwhere a physical theory breaks down.This does not necessarily mean thatphysical infinities exist; it may meansimply that the theory is incapable ofdescribing the situation properly. Twoother examples occur in inverse-squareforce laws of the gravitational forceequation of Newtonian gravity andCoulomb’s law of electrostatics. At r=0these equations evaluate to infinities.