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Mathematicians Measure Infinities, Find They’re Equal | Quanta Magazine

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At the time, mathematicians knew that “The real numbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size, or is there a size in between?” said Maryanthe Malliaris of the University of Chicago, co-author of the new work along with Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University.

What about the set of just the even numbers, or just the prime numbers? Each of these sets would at first seem to be a smaller subset of the natural numbers. Over any finite stretch of the number line, there are about half as many even numbers as natural numbers, and still fewer primes. Mathematicians call sets of this size “Countable,” because you can assign one counting number to each element in each set.

Some problems remained including a question from the 1940s about whether p is equal to t. Both p and t are orders of infinity that quantify the minimum size of collections of subsets of the natural numbers in precise ways.

What’s more important is that mathematicians quickly figured out two things about the sizes of p and t. First, both sets are larger than the natural numbers.

Second, p is always less than or equal to t. Therefore, if p is less than t, then p would be an intermediate infinity – something between the size of the natural numbers and the size of the real numbers.


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