Mathematicians Measure Infinities, Find They’re Equal
I find Cantor&#x27;s diagonal argument unconvincing.<p>The claim is that there are more real numbers in the range from zero to one than there are natural numbers. To see that this is false simply realize that you don&#x27;t actually have to write a decimal point to specify the real numbers in this range. Without the decimal point these real numbers just become natural numbers. Can a rational person believe that there are infinite sequences of digits in the form of real numbers but not infinite sequences of digits in the form of natural numbers? The natural numbers are just an infinite sequence of finite numbers. If you believe n is a natural number then you must also believe that n*10 is a natural number. One more digit! There is always one more digit (that is what infinity implies). If there really are an infinite number of natural numbers then some of them must be of a transfinite number of digits or else you would be including numbers in the list more than once.<p>The problem with Cantor&#x27;s argument comes down to the fact that the procedure he uses to find a number not in the set is essentially the same as the procedure he uses for creating the infinite set in the first place. The only difference is our understanding of randomness. His procedure for finding a number not in the set may not seem very random but it might be as random as any other. A truly random coin could theoretically come up heads every time. The important part of his argument is that the infinite list of real numbers has no repeats. The diagonalization procedure similarly ensures that there are no repeats. On the one hand he claims the infinite set of real numbers exists. On the other hand he argues that the diagonalization that yields a number not in the set has not already been done. He takes away infinity and then gives it back!<p>There is only one infinity. It means &quot;repeat&quot;. It is simply the interplay of finite state with process. You can think of it as an &quot;infinite loop&quot; in programming. To say that one infinity is smaller than another is to deny that the smaller is infinite. Infinite means without bound.