The short answer is no; infinity, as a concept, exists in mathematics, although the phrase "actual infinity" is largely outdated, having been replaced by the concept of infinite sets.
Here's a breakdown of why and how infinity exists in mathematics:
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Infinity as a Concept: Infinity represents something without any limit. It's not a real number you can find on a number line, but rather a concept used to describe quantities or values that grow without bound.
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Infinite Sets: In set theory, infinite sets are well-defined and studied extensively. Examples include the set of all natural numbers {1, 2, 3,...}, the set of all real numbers, and the set of all points on a line. These are concrete mathematical objects that mathematicians work with regularly.
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Limits: Calculus relies heavily on the concept of limits. We can say that a function "approaches infinity" as its input grows without bound. This is a precise way of describing the function's behavior without treating infinity as a specific number. For example, the limit of 1/x as x approaches 0 from the positive side is positive infinity.
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Different Sizes of Infinity: Georg Cantor demonstrated that not all infinities are the same size. The set of natural numbers is "countably infinite," while the set of real numbers is "uncountably infinite," meaning there's a fundamentally larger infinity of real numbers than natural numbers. This discovery revolutionized set theory and mathematics as a whole.
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Projective Geometry: In projective geometry, we often add a "point at infinity" to a line or a plane. This allows us to deal with parallel lines intersecting at infinity, which simplifies many geometrical theorems and proofs.
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Why "Actual Infinity" Is Outdated: The term "actual infinity" was used historically to distinguish between the potential infinity of a process that can go on forever (like counting) and a completed, actual infinite collection (like the set of all natural numbers). Modern set theory provides a rigorous framework for handling infinite sets directly, rendering the distinction somewhat obsolete.
In summary, while you cannot find a specific "infinity" number, the concept of infinity is fundamental and well-defined in various branches of mathematics, especially in set theory, calculus, and geometry. It's not about proving infinity "exists" in a physical sense, but rather understanding and manipulating it within a logical and consistent mathematical framework. The idea of "actual infinity" as a philosophical debate has largely been replaced by the more precise and useful concept of infinite sets.
