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The Infinity Inside By Georg Cantor

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Introduction

While perusing certain articles on the news-aggregator app Flipboard, I came across a blog post from the Huffington Post entitled ‘The Infinity Inside.’ The compelling title gave way to an even more intriguing body, which referred to infinity not in mathematical terms but as an abstract principle that governs the universe. This encounter instigated a profound curiosity in me, spurring me to investigate the concept of infinity further; after searching “how large is infinity” on the Internet, I discovered a TedEd video exploring the central question ‘How Big is Infinity?’ This video introduced me to the idea that there exist multiple forms of infinity, many of which are actually larger than each other. This prompted …show more content…

To define infinity, I will first explain the underlying logical framework of Cantor’s Infinite Set Theory. To prove the existence of multiple, larger infinities, I will work through Cantor’s Theorem of power sets. Through listing, I will also explore the different types of possible infinities and test whether or not it is true that all mathematical objects can be arranged in sets.

Basic Concepts of Set Theory

Set theory is considered a fundamental basis of modern mathematics whose notions can be built upon to form more complex areas of study. A set is simply a collection of objects . These objects are called members or elements of the set and are governed by well-defined membership criteria. In order to know whether or not an object belongs or does not belong to a set, there must be clear rules that delineate which objects are acceptable for inclusion. For example, if we had a set of all prime numbers, the number 4 could not belong to the set as it has more positive divisors than just 1 and itself -- it violates the specific criteria for membership in the set.

There are two types of sets as defined by Georg Cantor: finite and infinite. A finite set is simply one whose size, or cardinality, can be defined by a natural number. Cardinality describes the number of elements contained in a set. The members of any set (finite or infinite) can be placed in a

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