Infinity

Infinity is something which is boundless, limitless, or endless. It is denoted by ∞, called the infinity symbol. From the time of the ancient Greeks, the philosophical nature of infinity has been the subject of debate. In the 17th century, with the introduction of the infinity symbol and infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities.

At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. For example, the infinite number of points on a line is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity. The mathematical concept of infinity and the manipulation of infinite sets are widely used in many areas of mathematics.

In physics and cosmology, it is still an open question whether the universe is spatially infinite or not.

History

Ancient cultures had many ideas about infinity. The ancient Indians and the Greeks thought about infinity as a big idea, but they did not define it clearly like we do today.

One of the earliest Greek thinkers, Anaximander, talked about something "unbounded" or "infinite." Later, Aristotle talked about two kinds of infinity: one that could grow forever and one that could not actually exist. There were also puzzles, like one made by Zeno of Elea, who asked if a fast runner like Achilles could ever catch up to a slower tortoise that had a head start. People thought about this puzzle for a very long time before they had better ways to understand it.

Calculus

Gottfried Leibniz, one of the creators of infinitesimal calculus, thought a lot about infinite numbers and how they can be used in math. For Leibniz, both very small numbers (infinitesimals) and very large numbers (infinities) were perfect ideas that followed special rules.

In real analysis, the symbol ∞, called "infinity," is used to show when something gets bigger and bigger without stopping. It isn’t a real number, but a way to talk about limits. For example, when we say x goes to ∞, we mean x keeps growing forever. Infinity also helps us understand sums that go on endlessly and areas that cover infinite space.

Set theory

Main articles: Cardinality and Ordinal number

Set theory explores special kinds of infinity developed by mathematician Georg Cantor in the late 1800s. Cantor created ideas called ordinal and cardinal infinities. The smallest ordinal infinity relates to counting up forever, while cardinal infinity helps us understand the size of different infinite collections. For example, the set of natural numbers (1, 2, 3, ...) is infinite, but there are also larger kinds of infinity.

Cantor showed that some infinities are bigger than others. The infinity of real numbers, like all possible points on a line, is larger than the infinity of natural numbers. This is because we can match each natural number to a point on the line, but there will always be points left over. Mathematicians still study even bigger kinds of infinity today.

Geometry

Until the late 1800s, people rarely talked about infinity in geometry. They thought of a line as something you could make longer and longer, but never truly endless. They also didn't think a line was made of endless points — just a place where points could go.

One special area, called projective geometry, added something called points at infinity to help explain how things look far away, like parallel lines seeming to meet in the distance. This made studying lines easier because all lines, even parallel ones, would meet at these special points.

Today, we see lines as having endless points. Some special kinds of spaces can even have endless sizes, and some shapes, like the Koch snowflake, can have endless edges but still take up a certain amount of space.

Finitism

Some mathematicians, like Leopold Kronecker, were unsure about using the idea of infinity in the late 1800s. This led to a way of thinking called finitism, which is a part of bigger ideas in math philosophy such as constructivism and intuitionism.

Finitism believes that only things that can be fully counted or built step-by-step should be used in math, avoiding the idea of anything endless or unlimited.

Logic

In logic, an infinite regress argument is a special kind of argument that shows a thesis might be flawed because it leads to an endless series that either doesn't exist or wouldn't work as intended.

In first-order logic, important theorems like the compactness theorem and the Löwenheim–Skolem theorems help build non-standard models that have particular infinite features.

Main article: Infinite regress
Main articles: Compactness theorem, Löwenheim–Skolem theorems, Non-standard models

Applications

In physics, scientists use numbers to describe things that can change smoothly, like temperature, and numbers we can count, like apples. They also think about ideas like an endless wave, even though we can't create one in experiments.

Many years ago, people like Giordano Bruno wondered if the universe itself might be infinite. Today, scientists still ask if there are endless stars or if space goes on forever. They study the universe's shape and look at ancient light from stars to try to find answers. Some ideas even suggest there might be an infinite number of universes!

In computing, special values called "infinity" are used when numbers get too big or when dividing by zero. Programmers can use these values in their code for tasks like sorting or searching. They can also create loops that run forever by never giving a condition to stop them.

Arts, games, and cognitive sciences

Perspective artwork uses the idea of vanishing points, which are like mathematical points at infinity, to show space and distance in paintings. Artist M.C. Escher often used the idea of infinity in his artwork.

There are also versions of chess played on an unbounded board, called infinite chess. Cognitive scientist George Lakoff thinks about infinity in math and science as a kind of metaphor.