Roads to Infinity Revisited

After half a dozen years, try again: John Stillwell's Roads to Infinity, which I read and partly understood in 2010, still stretches beyond the mind's elastic limit. But perhaps, among the silly symbols, some parts may make slight sense. In another decade, more? Meanwhile, powerful principles emerge:

So there's not one "infinity", or even an "infinity of infinities" — mathematicians know how to play that game too well! — but there are sets of rules within which one can speak consistently, can define and discern objects, and can deduce implications. Sometimes the transfinite casts shadows onto the (seemingly) finite. Take, for instance, Goodstein's Theorem: pick a starting number and write it as sums of powers of two. Following Stillwell's example (in Section 2.7), 87 = 64 + 16 + 4 + 2 + 1 = 26 + 24 + 22 + 21 + 1. Then if an exponent in that representation is larger than 2, write it as sums of powers of two as well; so the 6th power in the first term of 87's expansion is written as 22 + 2, etc. OK, now replace all those 2's by 3's, and subtract 1. Rearrange to make the resulting number a proper sum of powers of three. OK, now replace all those 3's by 4's, and subtract 1. Rearrange, and repeat ("... and so on..."). After a long, long, long, LONG time, Goodstein's Theorem says, you end up at 0.

Far from obvious, and not provable without tiptoeing toward infinity, or maybe beyond. And the variations on "infinity", starting with ordinals and cardinals, lead to extraordinarily strange birds indeed, like the elusive Inaccessible cardinals.

And so on ...

^z - 2016-06-16