?? so... ?? harvey friedman says statement something like "every sequence of rational numbers has a subsequence converging as fast as s" (for some reasoanble particular s, apparently) implies consistency of peano arithmetic, if i understood correctly ... ?? apparently inherently a "second-order" statement, thus escaping becoming celebrated as intuitively clear and difficult-to-doubt first-order statement presumably unprovable in peano arithmetic ... ??? .... ?? ...

??idea that ... ??? maybe there's a sort of demonstrating of usefulness of hilbert's idea of "formalism" here ?? ... "usefulness of reasoning about ideal objects that you might be ambivalent about actually believing in" ... ??? ....

?? negation of first-order, though ????? .... ???? well, that's obviously not the right way to say it... what was i trying to say then??... ..hmm, i may have lost the thought here, or it may not have made any sense ... ???vaguely thinking about ... turing machine ... witness ... statement which if true can be proved, but not necessarily for its negation ... ?? ...

i still have a fair amount of confusion about some stuff here ... which would be interesting to try to clear up some day, probably ... can't articulate (not surprisingly perhaps) my confusions too well at the moment ... ?? interaction between "ground floor equivalence of statements" and "talking about itself" interpretation and "referential opacity" .... ??? ... ???? ....

??well, but what about very clever (in bishop's sense...) sequence ... very cagey about making you think that certain conspicuous subsequences of it might be converging ... ???? .... ???....