WM

2020-11-20 12:24:12 UTC

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PermalinkHow did the belief emerge that there were actually infinitely many natural numbers following upon every natural number, all of them were definable, but between all these natural numbers and ω nothing was existing?

First, it is clear that in potential infinity infinitely many natural numbers are following upon every defined natural number. Alas there is no ω. But set theorists are eagerly concerned with blurring the difference between potential and actual infinity. Many even deny that there were any difference.

On the other hand, many students of mathematics are attracted by counter intuitive statements and may be even quite a bit proud of being able to "understand" that stuff. Further the gateway drug is not that strong: For the moment they are taught harmlessly that every natural number is followed by a larger natural number, but that no natural number is larger than all others. And with the plausible explanation that everybody has a mother, but there is no mother of everybody, the argument appears not problematic.

If in the gapless sequence of ordinal numbers, on the "ordinal axis", one ordinal is missing before ω, then this gap is not standing out. It occurs in the infinite and its position cannot be verified anyway.

After this little gap has been swallowed, the dose can be increased. Then ℵo gaps of missing numbers will soon be accepted. And the recognition that in any case there are more numbers missing between n and ω than are existing before n enters an immunized brain that considers every sceptic simply as incapable of receiving the higher ordinations of infinity.

Yet the understanding of the potentially infinite class of definable numbers should appear counter intuitive enough. This class can be arbitrarily increased without end whereas the class of dark numbers can be decreased infinitely but without losing its actually infinite size.

A lecture about this topic will be given on Saturday, 21 Nov, 6 PM c.t. here:

https://hs-augsburg.zoom.us/j/92096767073?pwd=aG1DUzJOdnFqTzNTYk1FU0RFQUZ0QT09

Regards, WM