aha! thank you for replying. i'll say more when
 OOO  O O       i get a moment, but let me first clean-up my
   O OO O       sloppy language:
 OO   O O       
    O O            > ??? - There is no way of assigning equal
 OO O O O          > nonvanishing probability to infinitely
  O O O O          > many mathematical structures, each being
    O O O          > represented by a finite set of axioms.
 OO O O O       
        O       okay - strictly speaking, you are correct. but a
 OOOOOOO        common trick is to compute equal-probabilities
        O       on finite subsets of the infinite set. and then
O OOOOO O       you can take the limit as those subsets "grow"
O O     O       to the size of the infinite set.
   OOO  O       
 O    OO        the "growing" here is important - very often the
 O OO   O       order in which you add members to the set change
 O   OOOO       how the series converges. but for the case of
 O OOO          expected complexity, it does not.
   O O OO       
 O      O       
 O OOOO O       
O               -k

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