Infinity

Infinity is something that is boundless, limitless, or endless. It is shown with the ∞ sign, called the infinity symbol. People have talked about infinity since the time of the ancient Greeks. In the 1600s, new math tools helped people study infinity more closely.

At the end of the 1800s, a mathematician named Georg Cantor studied infinite sets and infinite numbers. He showed that some infinite things can be bigger than others. For example, there are more points on a line than there are integers.

In math, infinity is a tool that helps us solve problems, just like any other number or shape. In physics and cosmology, scientists still wonder 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 deeply about very large and very small numbers. In math, both tiny numbers (infinitesimals) and huge numbers (infinities) follow special rules.

In real analysis, the symbol ∞, called "infinity," is used to describe something that keeps getting bigger and bigger without end. It isn’t a real number, but a way to talk about limits. For example, when we say x goes to ∞, we mean x grows forever. Infinity also helps us understand endless sums and areas that cover unlimited space.

Set theory

Main articles: Cardinality and Ordinal number

Set theory looks at special kinds of infinity. A mathematician named Georg Cantor created these ideas in the late 1800s. He called them ordinal and cardinal infinities.

The smallest ordinal infinity is about counting up forever. Cardinal infinity helps us understand how big different infinite groups can be. For example, the group of natural numbers (1, 2, 3, ...) is infinite. But there are also bigger kinds of infinity.

Cantor showed that some infinities are bigger than others. The infinity of real numbers, like all points on a line, is bigger 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. Finitism is 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. It avoids the idea of anything endless or unlimited.

Logic

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

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

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

Applications

In physics, scientists use numbers to describe things that change smoothly, like temperature, and things 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 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.

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 act 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.