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What is Infinite?
It isn’t a number.
While watching Roger Penrose’s Mind-Bending Theory of Reality, on YouTube, I realized how even savvy people have difficulty with infinity. The program features an interviewer who is not an expert in some of Penrose’s specialties (which is one of the things that makes this program so interesting) but she is smart and perceptive in her detailed questioning. When she gets something wrong, Penrose takes the opportunity to explain in ways that only Penrose can. I highly recommend this video.
Nonetheless, at one point, the interviewer made a common mistake regarding the comparison of two infinities. She said that Georg Cantor wrote that there are an infinite number of positive integers:
1, 2, 3, 4…
Since that sequence never stops, it is infinite in size. She then said that surely there were only half as many positive even integers:
2, 4, 6, 8…
This caused Penrose to pause and explain, without going into detail, that it wasn’t that simple. Cantor himself was clear on this point; but, the concept was so puzzling that it led Cantor into a downward spiral of depression which ended in his death in a sanatorium in 1918. Surprisingly, he explained that there are also exactly as many powers of ten:
