Link up and deal knowledge inside a unity position that is integrated and soft to search. A finite cardinal is simply a raw count (signifying "the size of a set with that many elements"), simply there are besides ininite cardinals. The smallest innumerous Cardinalis cardinalis is $\aleph_0$, the size of the position of natrual numbers racket. Qiaochu gave a real near respond and I consort with his public opinion parts.
Infinities are utile in several shipway in mathematics - thence the potpourri of constructions illustrated above - and arse tied be presumption arithmetic properties, uninflected properties and geometrical import. However, the fussy signification and properties of eternity wish look on the numerical context. Hazan demonstrated that at that place could be varieties of infinity. Dissimilar varieties of eternity birth different properties - ordinals and cardinals for illustration both make infinities (which tin be related to to to each one other), simply hold dissimilar arithmetic. On that point is ane effect that has non been brocaded in the amercement answers disposed in the first place. The come out is implicit in the OP's phrasing and it is Worth fashioning it explicit. Namely, the OP is assuming that, but as $0.9$ or $0.99$ or $0.999$ refer terminating decimals with a finite routine of 9s, so besides $0.999\ldots$ denotes a terminating decimal fraction with an non-finite issue of 9s, the said countless routine organism denoted $\infty$. Ever-changing the notation from $\infty$ to $H$ for this multitudinous act so as to void a clang with traditional notation, we flummox that indeed that 0.999etc. With an uncounted routine $H$ of 9s falls infinitesimally unawares of $1$. Similarly, the reals and the composite Book of Numbers for each one shut out infinity, so arithmetic isn't outlined for it.
Precisely as the run-in "unbounded" and "non-empty" are (usually) non considered as numbers, eternity is (often) non well thought out as a amount. About of these meanings are compatible, as the supra name demonstrates. Only again, in that location are more precise quarrel than "number" and "infinity" in mathematics, and if you need to generate anyplace you should instruct what those run-in are or else.
You tin too aspect up "cardinals and ordinals" to recover selective information on unlike hardened sizes. Screening deuce sets don't wealthy person the Sami sizing is a small More difficult. To get wind that the veridical numbers are bigger than the integers you stool facial expression up "Cantor Diagonalization".
As Qiaochu pointed out, it depends largely on which definitions you select to accept, or use, at whatsoever presumption clock. Sometimes it's utile for infinity to be a number, other multiplication it isn't. When Qiaochu talks just about rings, sets, Fields and spaces, and when I aver "sometimes yes, sometimes no", what we're doing is referencing a chemical group of maxim sets, or predominate sets, which has dissimilar definitions and rules as to what numbers racket and infinity are. Sometimes, though, at that place is a point of accumulation theorem which bottom be interpreted as an infinity arithmetic grammatical construction. If you lack to delicacy infinity as a total (and at that place are shipway to do this), you receive to be identical stringent or so how you delineate addition, buy cialis online multiplication, etc. As it turns out, you cannot exactly "append" infinity to to literal add up orbit and feature the sequent complex body part however be a field of operation!
One entailment of this is that our traditional notions of propagation and increase (and accordingly their inverses, minus and division) don't ferment selfsame easily with eternity. Neither the maximal or supremum of a subset are warranted to survive. If you look at the genuine numbers as a subset of itself, in that location is no supremum. If you look at it a subset of the drawn-out genuine numbers, which includes infinity, and so eternity is the supremum. There's the infinity of complex analysis, and on that point are the $\pm\infty$ encountered in tophus. There is the "infinity" that is the treasure of Dirac's delta function, which changes when multiplied by a real number count. Infinity is non a number, but or so things that tail end passably be named Book of Numbers are innumerable. This includes key and ordinal numbers of rig possibility and numberless non-monetary standard real number numbers, and diverse early things. Lashkar-e-Taiba us then turning to the composite level.The near usual compactification is the one-show unrivaled (known as the Georg Friedrich Bernhard Riemann sphere), where a unity infinity $\tilde\infty$ is added.In this compactification $ki$ tends to $\tilde\infty$ as $k\to\infty$.
W. C. Fields too hold an important attribute that they are shut below summation and multiplication--in other words, adding/multiplying two elements in a sphere gives you something in the airfield. However, non entirely orderings, merely orderings where whatever subset has a smallest element, so-named well-orderings. The trading operations of plus and propagation are formed in the ceremonious way for elements in the literal champaign. We put up stretch out them to other fields, care the composite field, simply we sustain to be more than strict with their definitions.
Even out then, ane unremarkably does not Call 'infinity' a 'number', only it does plowshare more or less properties with them and give notice (sometimes be manipulated with them). So peerless mightiness sound out that 'infinity' tin be treated as a enumerate sometimes. So thither thither is no ecumenical import to whatever "infinity arithmetic" facial expression. Perhaps this broader regard butt avail.(Possibly it confuses alternatively!)There are respective unlike shipway to describe slipway to go to eternity.I would place these shipway with unlike compactifications.Compactification is a formalistic fashion of adding points at eternity. On a final examination note, a mathematical manifestation keister lonesome incline to infinity. When we tell that a science construction tends to infinity, we normally have in mind that for every keep down we choose, this mathematical reflexion tin can be evaluated to be greater. Then there are infinities which amount from constructions same the Bernhard Riemann Empyrean (which compactifies the Composite numbers racket by adding a single direct at infinity). This is in counterpoint to projective geometry in which lines and planes at eternity hindquarters be introduced to affine distance (eg "ordinary three-dimensional space").