And why do you care?
See Article History Infinity, the concept of something that is unlimited, endless, without bound. Three main types of infinity may be distinguished: Mathematical infinities occur, for instance, as the number of points on a continuous line or as the size of the endless sequence of counting numbers: Spatial and temporal concepts of infinity occur in physics when one asks if there are infinitely many stars or if the universe will last forever.
In a metaphysical discussion of God or the Absolute, there are questions of whether an ultimate entity must be infinite and whether lesser things could be infinite as well.
One of the earliest appearances of infinity in mathematics regards the ratio between the diagonal and the side of a square. In modern mathematics this discovery is expressed by saying that the ratio is irrational and that it is the limit of an endless, nonrepeating decimal series.
To avoid the use of actual infinity, Eudoxus of Cnidus c. The issue of infinitely small numbers led to the discovery of calculus in the late s by the English mathematician Isaac Newton and the German mathematician Gottfried Wilhelm Leibniz.
Newton introduced his own theory of infinitely small numbers, or infinitesimalsto justify the calculation of derivativesor slopes. In order to find the slope that is, the change in y over the change in x for a line touching a curve at a given point x, yhe found it useful to look at the ratio between dy and dx, where dy is an infinitesimal change in y produced by moving an infinitesimal amount dx from x.
Infinitesimals were heavily criticized, and much of the early history of analysis revolved around efforts to find an alternate, rigorous foundation for the subject.
The use of infinitesimal numbers finally gained a firm footing with the development of nonstandard analysis by the German-born mathematician Abraham Robinson in the s. A more direct use of infinity in mathematics arises with efforts to compare the sizes of infinite setssuch as the set of points on a line real numbers or the set of counting numbers.
Mathematicians are quickly struck by the fact that ordinary intuitions about numbers are misleading when talking about infinite sizes. Medieval thinkers were aware of the paradoxical fact that line segments of varying lengths seemed to have the same number of points. For instance, draw two concentric circles, one twice the radius and thus twice the circumference of the other, as shown in the figure.
Intuition suggests that the outer circle should have twice as many points as the inner circle, but in this case infinity seems to be the same as twice infinity. Galileo demonstrated that the set of counting numbers could be put in a one-to-one correspondence with the apparently much smaller set of their squares.
He similarly showed that the set of counting numbers and their doubles i.
First Cantor rigorously demonstrated that the set of rational numbers fractions is the same size as the counting numbers; hence, they are called countable, or denumerable.
Of course this came as no real shock, but later that same year Cantor proved the surprising result that not all infinities are equal.
To compare sets, Cantor first distinguished between a specific set and the abstract notion of its size, or cardinality. Unlike a finite set, an infinite set can have the same cardinality as a proper subset of itself.
Cantor used a diagonal argument to show that the cardinality of any set must be less than the cardinality of its power set—i. In general, a set with n elements has a power set with 2n elements, and these two cardinalities are different even when n is infinite.
The transfinite cardinals include aleph-null the size of the set of whole numbersaleph-one the next larger infinityand the continuum the size of real numbers. The continuum problem is the question of which of the alephs is equal to the continuum cardinality.
A concise introduction to Cantor's mathematics of infinite sets. Infinite Reflections, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 () 1– Grime, James. "Infinity is bigger than you think". Numberphile. Brady Haran. Most everyone is familiar with the infinity symbol, the one that looks like the number eight tipped over on its side. Infinity sometimes crops up in everyday speech as a superlative form of the word many. Jan 30, · Teaching the Mathematics of Infinity. By Patrick Honner the winner is the person who names the largest number. In “Closest to 0,” players take turns choosing rational numbers (which include both integers and fractions, like 3/4 and 7/8); the winner is the player who names the number closest to zero. I think the best.
In the early s a thorough theory of infinite sets was developed. CH is known to be undecidable on the basis of the axioms in ZFC. Set theorists continue to explore ways to extend the ZFC axioms in a reasonable way so as to resolve CH. Physical infinities The science of physical infinities is much less developed than the science of mathematical infinities.
The main reason is simply that the status of physical infinities is quite undecided. In physics one might look for infinities in space, time, divisibility, or dimensionality. Although some have speculated that three-dimensional space is infinite, cosmologists generally believe that the universe is curved in such a way as to make it finite but unbounded—akin to the surface of a sphere.
Some theories of cosmology view the universe as being embedded in a higher-dimensional superspace, which could perhaps be infinite in extent.
In the light of the big-bang model of the origin of the universe, cosmologists generally believe that the universe has a finitely long past; whether it might have an endless future is an open question.Limits at Infinity and Limits of Sequences The Area Problem 11 0 , 0 6 Section , Example 3 If becomes arbitrarily close to a unique number as approaches from Chapter 11 Limits and an Introduction to Calculus.
Most everyone is familiar with the infinity symbol, the one that looks like the number eight tipped over on its side. Infinity sometimes crops up in everyday speech as a superlative form of the word many.
Home › Math › A Quirky Introduction To Number Systems Everyone’s got quirks. Me, I like finding new ways to think about problems, and I’ve started seeing numbers in a new way. list of important mathematicians This is a chronological list of some of the most important mathematicians in history and their major achievments, as well as some very early achievements in mathematics for which individual contributions can not be acknowledged.
A Finite History of Infinity An Exploration and Curriculum of the Paradoxes and Puzzles of Infinity By Amy Whinston Under the direction of Dr. John S. Caughman. A concise introduction to Cantor's mathematics of infinite sets.
Infinite Reflections, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 () 1– Grime, James. "Infinity is bigger than you think". Numberphile.