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Good Math by Mark C. Chu-Carroll

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19Transfinite Numbers: Counting and Ordering Infinite Sets

One of the deepest questions you can ask is the very first one that set theory was used to answer: Can you have something infinitely large that is larger than something else that is infinitely large? Once you know the answer to that question and recognize that there are degrees of infinity, that opens up a new question: How can you talk about infinity using numbers? How does arithmetic work with infinity? What do infinitely large numbers mean? We’ll answer those questions in this section by looking at numbers through the lens of sets and at Cantor’s definition of cardinal and ordinal numbers in terms of sets. This will lead us to looking at a new kind of number: Cantor’s transfinite ...

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