Yes, infinity is commonly accepted in mathematics, primarily as a concept within set theory rather than a tangible number.
While "actual infinity" as a specific term is largely obsolete, the concept of infinite sets is foundational in modern mathematics. This shift, spearheaded by mathematicians like Bolzano and Cantor in the 19th century, revolutionized our understanding of quantity and sets the stage for a more rigorous and abstract approach to mathematical concepts.
The Evolution of Infinity
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Early Conceptions: Historically, infinity was often viewed with suspicion, perceived as an incomplete or potential entity, rather than a completed, actual thing. This "potential infinity" represented the idea of endless continuation, but not an actual, finalized infinite quantity.
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Bolzano and Cantor's Revolution: Bernard Bolzano and Georg Cantor challenged this view. Cantor, in particular, developed set theory, demonstrating that infinite sets can have different "sizes." For example, the set of all real numbers is "larger" than the set of all natural numbers, a counterintuitive but mathematically sound conclusion. This was a crucial step in accepting the existence of completed, infinite sets.
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Infinite Sets Today: Today, mathematicians routinely work with infinite sets without necessarily grappling with the philosophical implications of "actual infinity." The focus is on the properties and relationships between these sets, not whether infinity "exists" in a physical or metaphysical sense.
Key Concepts
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Infinite Sets: A set is infinite if it can be put into a one-to-one correspondence with a proper subset of itself. For example, the set of natural numbers {1, 2, 3,...} is infinite because it can be mapped to the set of even numbers {2, 4, 6,...}.
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Cardinality: Cardinality is a measure of the "size" of a set. Infinite sets have infinite cardinalities. Cantor proved that there are different infinite cardinalities, showing that some infinities are "larger" than others.
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Ordinality: Ordinality describes the order of elements within a set. Infinite sets can also have infinite ordinality, indicating an endless sequence of elements.
Implications and Applications
The acceptance of infinity as a mathematical concept has profound implications:
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Calculus: The concept of limits, essential to calculus, relies on the notion of approaching a value infinitely closely.
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Set Theory: Set theory, which deals directly with infinite sets, provides a foundation for much of modern mathematics.
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Computer Science: Some models of computation, such as Turing machines, involve potentially infinite processes.
Conclusion
In summary, while the term "actual infinity" is less commonly used now, the concept of infinity, particularly in the form of infinite sets, is a well-established and essential part of modern mathematics. It's a fundamental tool for understanding complex mathematical structures and has applications in various fields.
