How is infinity used in mathematics

One of the more common context is limits. This is kind of a notation shorthand though, since it is NOT correct to say that larger numbers are closer to infinity than smaller numbers. In most cases where infinity is written into an expression, e. Another way "infinity" is used is to describe the size of sets. There are an infinite number of integers, and also an infinite number of even integers, and also an infinite number of prime integers, not to mention rational numbers fractions , or even the set of all polynomials.

The size of these sets can be described using Cardinal numbers. It may be interesting to see one of the first fully rigorous accounts of the geometric sense of infinity. In Euclidean geometry, it was realized that there was a duality between points and lines — it was realized that many theorems in the field of projective geometry were duals of one another: you could take one theorem, swap the notion of "point" and "line", fix up some details, and the result would be the other theorem.

As a simple example of this idea, "through every two distinct points there is exactly one line" becomes "on every two distinct lines there is exactly one point", which fixes up to "every pair of distinct lines is either parallel or has exactly one point of intersection". The breakthrough came up algebraically. Now, every pair of Euclidean parallel lines does meet at a unique point at infinity. There is one additional line, the "line at infinity", which passes through all of the points at infinity.

But algebraically, it's clear that infinity is not special; it behaves just like any other place on the projective plane.

And this isn't just an esoteric thing; projective coordinates are very important for doing geometry, particularly when working with perspective e. Also when doing algebra geometrically! If you repeat this construction with the line rather than the plane, you get the projective line.

You can naturally extend arithmetic; i. There are incredibly many infinities in mathematics. When we talk about infinity in mathematics, we almost always talk about a size cardinality of a set. Sets can have finite size, but infinite sizes exist as well. There are other, larger infinite sizes. Other concepts of infinity arise when we are looking at limits.

We can ask ourselves the question where this sequence goes to spoilers, it's infinity. Intuitively, we want to know about the infinitieth "element" of the sequence. Formally, we call that number the limit of the sequence. Well, it depends on what domain you are working in. Infinity has multiple meanings, I will try to answer it from a set theoretic perspective. In more formal language, we a bijective function is defined as a function being both injective a function that maps distinct elements to distinct elements and surjective a function in which all elements in the codomain has exactly one element in the domain that maps to that particular element.

Two sets are said to have the same cardinality "size" , if there exists such an injective function between them. In a sense you can view such a bijection as a function that uniquely pairs every element in the domain with another element in the codomain, which is why we use this definition.

Countable infinite set: If we can "enumerate" the elements i. A natural question is: Does uncountable infinities exists? This is equivalent to asking "Are there different infinities?

That's a contradiction! In fact, we can generalize this, to say that there exist no bijection from a set to its power set the set of all its subset. Notice something? Sign up to join this community. The best answers are voted up and rise to the top. Galileo himself did not see how to carry out such a modification of these notions; this was to be the task of Georg Cantor, some years later. One of the reasons that Galileo felt it necessary to come to some sort of terms with the actual infinite was his desire to treat space and time as continuously varying quantities.

But this variable t that grows continuously from, say, zero to ten is apeiron, both in the sense that it takes on arbitrary values, and in the sense that it takes on infinitely many values. This view of position as a function of time introduced a problem that helped lead to the founding of the Calculus in the late s. The problem was that of finding the instantaneous velocity of a moving body, whose distance x from its starting point is given as a function f t of time. It turns out that to calculate the velocity at some instant t o , one has to imagine measuring the speed over an infinitely small time interval dt.

The quantity dt is called an infinitesimal, and obeys many strange rules. If dt is added to a regular number, then it can be ignored, treated like zero. But, on the other hand, dt is regarded as being different enough from zero to be usable as the denominator of a fraction. So is dt zero or not? Adding finitely many infinitesimals together just gives another infinitesimal. But adding infinitely many of them together can give either an ordinary number, or an infinitely large quantity.

Bishop Berkeley found it curious that mathematicians could swallow the Newton-Leibniz theory of infinitesimals, yet balk at the peculiarities of orthodox Christian doctrine.

Wherein It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith. The use of infinitely small and infinitely large numbers in calculus was soon replaced by the limit process. But it is unlikely that the Calculus could ever have developed so rapidly if mathematicians had not been willing to think in terms of actual infinities.

In the past fifteen years, Abraham Robinson's non-standard analysis has produced a technique by which infinitesimals can be used without fear of contradiction. Robinson's technique involves enlarging the real numbers to the set of hyperreal numbers, which will be discussed in Chapter 2. After the introduction of the limit process, calculus was able to advance for a long time without the use of any actually infinite quantities. But as mathematicians tried to get a precise description of the continuum or real line, it became evident that infinities in the foundations of mathematics could only be avoided at the cost of great artificiality.

Mathematicians, however, still hesitated to plunge into the world of the actually infinite, where a set could be the same size as a subset, a line could have as many points as a line half as long, and endless processes were treated as finished things. It was George Cantor who, in the late s, finally created a theory of the actual infinite which by its apparent consistency, demolished the Aristotelian and scholastic "proofs" that no such theory could be found. Although Cantor was a thoroughgoing scholar who later wrote some very interesting philosophical defenses of the actual infinite, his point of entry was a mathematical problem having to do with the uniqueness of the representation of a function as a trigonometric series.

To give the flavor of the type of construction Cantor was working with, let us consider the construction of the Koch curve shown in Figure 4. The Koch curve is found as the limit of an infinite sequence of approximations. The first approximation is a straight line segment stage 0. The middle third of this segment is then replaced by two pieces, each as long as the middle third, which are joined like two sides of an equilateral triangle stage 1. At each succeeding stage, each line segment has its middle third replaced by a spike resembling an equilateral triangle.

Now, if we take infinity as something that can, in some sense, be attained, then we will regard the limit of this infinite process as being a curve actually existing, if not in physical space, then at least as a mathematical object. The Koch curve is discussed at length in Benoit Mandelbrot's book, Fractals, where he explains why there is reason to think of the Koch curve in its infinite spikiness as being a better model of a coastline than any of its finitely spiky approximations.

Cantor soon obtained a number of interesting results about actually infinite sets, most notably the result that the set of points on the real line constitutes a higher infinity than the set of all natural numbers.

That is, Cantor was able to show that infinity is not an all or nothing concept: there are degrees of infinity. This fact runs counter to the naive concept of infinity: there is only one infinity, and this infinity is unattainable and not quite real. Cantor keeps this naive infinity, which he calls the Absolute Infinite, but he allows for many intermediate levels between the finite and the Absolute Infinite. These intermediate stages correspond to his transfinite numbers.

In the next section we will discuss the possibility of finding physically existing transfinite sets. We will then look for ways in which such actual infinities might exist mentally. Finally we will discuss the Absolute, or metaphysical, infinite. This threefold division is due to Cantor, who, in the following passage, distinguishes between the Absolute Infinite, the physical infinities, and the mathematical infinities:. The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent other-worldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number, or order type.

I wish to make a sharp contrast between the Absolute and what I call the Transfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite. There are three ways in which our world appears to be unbounded and thus, perhaps, infinite.

It seems that time cannot end. It seems that space cannot end. And it seems that any interval of space or time can be divided and subdivided endlessly. We will consider these three apparent physical infinities in three subsections. Suppose that the human race was never going to die out -- that any given generation would be followed by another generation.

Would we not then have to admit that the number of generations of man is actually infinite? Aristotle argued against this conclusion, asserting that in this situation the number of generations of man would be but potentially infinite; that is, infinite only in the sense of being inexhaustible.

He maintained that at any given time there would only have been some finite number of generations, and that it was not permissible to take the entire future as a single whole containing an actual infinitude of generations.

It is my opinion that this sort of distinction rests on a view of time that has been fairly well discredited by modern relativistic physics. In order to agree with Aristotle that, although there will never be a last generation, there is no infinite set of all the generations, we must believe that the future does not exist as a stable, definite thing.

For if we have the future existing in a fixed way, then we have all of the infinitely many future generations existing "at once. But one of the chief consequences of Einstein's Special Theory of Relativity is that it is space-time that is fundamental, not isolated space which evolves as time passes.

I will not argue this point in detail here, but let me repeat that on the basis of modern physical theory we have every reason to think of the passage of time as an illusion.

Past, present, and future all exist together in space-time. So the question of the infinitude of time is not one that is to be dodged by denying that time can be treated as a fixed dimension such as space. The question still remains: is time infinite? If we take the entire space-time of our universe, is the time dimension infinitely extended or not? Fifty, or even twenty, years ago it would have been natural to assert that our universe has no beginning or end and that time is thus infinite in both directions.

But recently it has become an established fact that the universe does have a beginning in time known as the Big Bang. The Big Bang took place approximately 15 billion years ago. At that time our universe was the size of a point, and it has been expanding ever since. What happened before the Big Bang? It is at least possible to answer, "Nothing. This is a subtle distinction, but a useful one.

If we think of time as being all the points greater than or equal to zero, then there is a first instant: zero. But if we think of time as being all the points strictly greater than zero, then there is no first instant. But in any case, if we think of time as not existing before the Big Bang, then there are certainly not an infinite number of years in our past. And what about the future? There is no real consensus on this. Many cosmologists feel that our universe will eventually stop expanding and collapse to form a single huge black hole called the Big Stop or the Gnab Gib; others feel that the expansion of the universe will continue indefinitely.

If the universe really does start as a point and eventually contract back to a point, is it really reasonable to say that there is no time except for the interval between these points? What comes before the beginning and after the end?

One response is to view the universe as an oscillating system, which repeatedly goes through expansions and contractions. This would reintroduce an infinite time, which could, however, be avoided. The way in which one would avoid infinite time in an endlessly oscillating universe would be to adopt a belief in what used to be called "the eternal return.

The idea is that a finite universe must return to the same state every so often, and that once the same state has arisen, the future evolution of the universe will be the same as the one already undergone. The doctrine of eternal recurrence amounts to the assumption that Figure From R. Rucker, Geometry, Relativity, and the Fourth Dimension. There is a simpler model of an oscillating universe with circular time, which can be called toroidal space-time. In toroidal space-time we have an oscillating universe that repeats itself after every cycle.

Note, however, that if the universe really expands forever, then it cannot ever repeat itself, as the average distance between galaxies is a continually increasing quantity that never returns to the same value. We now turn to a consideration of the possibility of spatial infinities. The potential versus actual infinity distinction is sometimes used to try to scotch this question at the outset.

Immanuel Kant, for instance, argues that the world cannot be an infinite whole of coexisting things because "in order therefore to conceive the world, which fills all space, as a whole, the successive synthesis of the parts of an infinite world would have to be looked upon as completed; that is, an infinite time would have to be looked upon as elapsed, during the enumeration of all coexisting things.

Kant's point is that space is in some sense not already really there -- that things exist together in space only when a mind perceives them to do so. If we accept this, then it is true that an infinite space is something that no finite mind can know of after any finite amount of time.

But one feels that the world does exist as a whole, in advance of any efforts on our part to see it as a unity. And if we take all of space-time, it certainly does not seem to be meaningless to ask whether the spatial extent of space-time is infinite or not. In De Rerum Natura, Lucretius first gave the classic argument for the unboundedness of space: "Suppose for a moment that the whole of space were bounded and that someone made his way to its uttermost boundary and threw a flying dart.

So great was their revulsion against the apeiron that Parmenides, Plato, and Aristotle all held that the space of our universe is bounded and finite, having the form of a vast sphere. When faced with the question of what lies outside this sphere, Aristotle maintained that "what is limited, is not limited in reference to something that surrounds it. In modern times we have actually developed a way to make Aristotle's claim a bit more reasonable.

As Lucretius realized, the weak point in the claim that space is a finite sphere is that such a space has a definite boundary. But there is a way to construct a three-dimensional space which is finite and which does not have boundary points: simply take the hypersurface of a hypersphere. Such a space is endless but not infinite. To understand how something can be endless but not infinite, think of a circle.

A fly can walk around and around the rim of a glass without ever coming to a barrier or stopping point, but none the less he will soon retrace his steps. Again, the surface of the Earth is a two-dimensional manifold which is finite but unbounded unbounded in the sense of having no edges. You can travel and travel on the Earth's surface without ever coming to any truly impassible barrier. The reason that the two-dimensional surface of the Earth is finite but unbounded is that it is bent, in three-dimensional space, into the shape of a sphere.

In the same way, it is possible to imagine the three-dimensional space of our universe as being bent, in some four-dimensional space, into the shape of a hypersphere. It was Bernhard Riemann who first realized this possibility in There is, however, a traditional belief that anticipates the hypersphere. This tradition, described in the essay, "The Fearful Sphere of Pascal," by Jorge Luis Borges, is summarized by the saying attributed to the legendary magician Hermes Trismegistus that "God is an intelligible sphere, whose center is everywhere and whose circumference is nowhere.

To see why this is so, consider the fact that if space is hyperspherical, then one can cover all of space by starting at any point and letting a sphere expand outwards from that point. The curious thing is that if one lets a sphere expand in a hyperspherical space, there comes a time when the circumference of the sphere turns into a point and disappears. This fact can be grasped by considering the analogous situation of the sequence of circular latitude lines on the spherical surface of the earth.

Aristotle had believed that the world was a series of nine spheres centered around the Earth. The last of these crystalline spheres was called the Primum Mobile and lay beyond the sphere upon which were fastened all of the stars other than the sun, which was attached to the fourth sphere.

In the Paradisio, Dante is led out through space by Beatrice. Beyond these nine spheres lie nine spheres of angels, corresponding to the nine spheres of the world. Beyond the nine spheres of angels lies a point called the Empyrean, which is the abode of God. The puzzling thing about Dante's cosmos as it is drawn in Figure 14 is that here the Empyrean appears not to be a point, but rather to be all of space except for the interior of the last sphere of angels.

But this can be remedied if we take space to be hyperspherical! In Figure 15 I have drawn the model we obtain if we take the diagram on the last page and curve it up into a sphere with a point-sized Empyrean.

In the same way, the three-dimensional model depicted by the first picture can be turned into the finite unbounded space of the second picture if we bend our three-dimensional space in such a way that all of the space outside our last angelic sphere is compressed to a point.

This whole notion of hyperspherical space was not consciously developed until the mid-nineteenth century. In the Middle Ages there was a general and uncritical acceptance of Aristotle's view of the universe -- without Dante's angelic spheres.

Lucretius, of course, had insisted that space is infinite, and there were many other thinkers, such as Nicolas of Cusa and Giordano Bruno, who believed in the infinitude of space. Some kept to the Aristotelian world system, but suggested that there were many such setups drifting around; others opted for a looser setup under which stars and planets are more or less randomly mixed together in infinite space.

Bruno strongly advocated such viewpoints in his writings, especially his dialogue of , "On the Infinite Universe and Worlds. In , a wealthy Venetian persuaded Bruno to come from Frankfurt to teach him "the art of memory and invention. His host had been working closely with the ecclesiastical authorities, who considered Bruno a leading heretic or heresiarch. Bruno was turned over to the Inquisition.

For nine years Bruno was interrogated, tortured, and tried, but he would not give up his beliefs; early in he was burned at the stake in the Roman Piazza Campo di Fiori. Bruno's example caused Galileo to express himself a good deal more cautiously on scientific questions in which the Church had an interest. Whether or not our space is actually infinite is a question that could conceivably be resolved in the next few decades.

Assuming that Einstein's theory of gravitation is correct, there are basically two types of universe: i a hyperspherical closed and unbounded space that expands and then contracts back to a point; ii an infinite space that expands forever. It is my guess that case i will come to be most widely accepted, if only because the notion of an actually infinite space extending out in every direction is so unsettling.

The fate of the universe in case i is certainly more interesting, since such a universe collapses back to an infinitely dense space-time singularity that may serve as the seed for a whole new universe.

In case ii , on the other hand, we simply have cooling and dying suns drifting further and further apart in an utterly empty black immensity. Even though I am basically pro-infinity, my emotions lie with the hyperspherical space. But is there any way of finding a spatial infinity here? Well, what about that four-dimensional space in which our hyperspherical universe is floating? Many would dismiss this space as a mere mathematical fiction.

This widely held position is really a more sophisticated version of Aristotle's claim that what is limited need not be limited with reference to something outside itself.

But what if one chooses to believe that the four-dimensional space in which our universe curves is real? We might imagine a higher 4-D four-dimensional world called, let us say, a duoverse. The duoverse would be 4-D space in which a number of hyperspheres were floating.

The hypersurface of each of the hyperspheres would be a finite, unbounded 3-D universe. Thus, a duoverse would contain a number of 3-D universes, but no inhabitant of any one of these universes could reach any one of the others, unless he could somehow travel through 4-D space.

By lowering all the dimensions by one, one can see that this situation is analogous to a universe that is a 3-D space in which a number of spheres are floating. The surface of each sphere or planet is a finite, unbounded 2-D space; and no one can get from one planet surface to another planet surface without travelling through 3-D space. Following the Hermetic principle, "As above, so below," one is tempted to believe that the duoverse we are in is actually a finite and unbounded 4-D space the 4-D surface of a 5-D sphere in 5-D space , and that there are a number of such duoverses drifting about in a 5-D triverse.

This could be continued indefinitely. One is reminded of those Eastern descriptions of the world as a disk resting on the backs of elephants, who stand upon a turtle, who stands upon a turtle, who stands upon a turtle, who stands upon a turtle, etc. Note that in that particular sort of cosmos there is only one universe, one duoverse, one triverse, and so on. But in the kind of infinitely regressing cosmos that I have drawn in Figure 18, we have infinitely many objects at each level.

Note also that to get from star A to star B one would have to move through 5-D space to get to a different duoverse. It is a curious feature of such a cosmos that, although there are an infinite number of stars, no one n -dimensional space has more than a finite number of them. The question we are concerned with here is whether or not space is infinitely large. There seem to be three options: i There is some level n for which n -dimensional space is real and infinitely extended. The situation where our three-dimensional space is infinitely large falls under this case.

The situation where our three-dimensional space is finite and unbounded, and the reality of four-dimensional space denied, falls under this case. In this case we either have an infinite number of universes, duoverses, etc.

So is space infinite? It seems that we can insist that at some dimensional level it is infinite; adopt the Aristotelian stance that space is finite at some level beyond which nothing lies; or accept the view that there is an infinite sequence of dimensional levels. In this last case we already have a qualitative infinity in the dimensionality of space, and we may or may not have a quantitative infinity in terms, say, of the total volume of all the 3-D spaces involved.

In this subsection I will discuss the existence of the infinity in the small, as opposed to the infinity in the large, which has just been discussed. Since a point has no length, no finite number of points could ever constitute a line segment, which does have length. So it seems evident that every line segment, or, for that matter, every continuous plane segment or region of space, must consist of an infinite number of points.

By the same token, any interval of time should consist of an infinite number of instants; and any continuous region of space-time would consist of an infinite number of events event being the technical term for a space-time location, i. It is undeniable that a continuous region of mathematical space has an infinite number of mathematical points.

Right now, however, we are concerned with physical space. We should not be too hasty in assuming that every property of the abstract mathematical space we use to organize our experiences is an actual property of the concrete physical space we live in. But what is "the space we live in"? If it is not the space of mathematical physics, is it the space of material objects? Is it the space of our perceptions? In terms of material objects or of perceptions, points do not really exist; for any material or perceptual phenomenon is spread over a certain finite region of space-time.

So when we look for the infinity in the small in matter, we do not ask whether matter consists of an infinity of unobservable mass-points, but, rather, whether matter is infinitely divisible. A commitment to avoiding the formless made it natural for Greek atomists such as Democritus to adopt a theory of matter under which the seemingly irregular bodies of the world are in fact collections of indivisible, perfectly formed atoms.

The four kinds of atoms were shaped, according to Plato, like four of the regular polyhedra. There is one other polyhedron, the twelve-sided dodecahedron, and this was thought somehow to represent the Universe with its twelve signs of the zodiac.

For the atomists, it was as if the world were an immense Lego set, with four kinds of blocks. The diverse substances of the world -- oil, wood, stone, metal, flesh, wine, and so on -- were regarded as being mixtures of the four elemental substances: Earth, Air, Fire, and Water.

Thus, gold was regarded by Plato as being a very dense sort of Water, and copper was viewed as gold with a small amount of Earth mixed in. The alchemists and early chemists adopted a similar system, only the number of elemental substances became vastly enlarged to include all homogeneous substances, such as the various ores, salts, and essences.

The fundamental unit here was the molecule. A new stage in man's conception of matter came when it was discovered that if an electric current is passed through water, it can be decomposed into hydrogen and oxygen. Eventually, the vast diversity of existing molecules was brought under control by regarding molecules as collections of atoms.

Soon some ninety different types of atoms or chemical elements were known. A new simplification occurred when it was discovered, by bombarding a sheet of foil with alpha rays, that an atom consists of a positive nucleus surrounded by electrons. Shortly after this the neutron was discovered, and the physical properties of the various atoms were accounted for by regarding them as collections of protons, neutrons, and electrons.

Over the last half century it has been learned, by using particle accelerators, that there are actually many types of "elementary particles" other than the neutron, electron, and proton.

The situation in high-energy physics today is as follows. A few particles -- electrons, neutrinos, and muons -- seem to be absolutely indivisible. These particles are called leptons. All others -- protons, neutrons, mesons, lambdas, etc. The historical pattern in the investigation of matter has been the explanation of diverse substances as combinations of a few simpler substances. Diversity of form replaces diversity of substance.

So it is no surprise that it has been proposed that the great variety of divisible particles that exist can be accounted for by assuming that these particles are all built up out of quarks. Zeno of Elea born circa B. Of all Zeno's paradoxes, the most famous is his paradox of the Tortoise and Achilles.

In the paradox, a tortoise challenges the Greek hero Achilles to a race, providing the tortoise is given a small head start. The tortoise argues he will win the race because as Achilles catches up to him, the tortoise will have gone a bit further, adding to the distance.

In simpler terms, consider crossing a room by going half the distance with each stride. First, you cover half the distance, with half remaining. The next step is half of one-half, or a quarter.

Three quarters of the distance is covered, yet a quarter remains. Although each step brings you closer, you never actually reach the other side of the room. Or rather, you would after taking an infinite number of steps. Mathematicians use a symbol for pi because it's impossible to write the number down. Pi consists of an infinite number of digits. It's often rounded to 3. One way to think about infinity is in terms of the monkey theorem.

According to the theorem, if you give a monkey a typewriter and an infinite amount of time, eventually it will write Shakespeare's Hamlet. While some people take the theorem to suggest anything is possible, mathematicians see it as evidence of just how improbable certain events are.

A fractal is an abstract mathematical object, used in art and to simulate natural phenomena. Written as a mathematical equation, most fractals are nowhere differentiable. When viewing an image of a fractal, this means you could zoom in and see new detail. In other words, a fractal is infinitely magnifiable. The Koch snowflake is an interesting example of a fractal.

The snowflake starts as an equilateral triangle. For each iteration of the fractal:. The process may be repeated an infinite number of times. The resulting snowflake has a finite area, yet it is bounded by an infinitely long line.

Infinity is boundless, yet it comes in different sizes. The positive numbers those greater than 0 and the negative numbers those smaller than 0 may be considered to be infinite sets of equal sizes. Yet, what happens if you combine both sets?