[FOM] Concerning definition of formulas

[Thomas Forster]

I have been following this discussion with great interest, and copying the
various contributions away for future reference:  i am writing a book on
the axioms of set theory and over the years a number of my supervisees
have come up with some version or another of this circularity problem . I
have been trying for some years to find the most illuminating brief
observation to make.

  I have felt for some time now that the most helpful thing i can say to
people who lose sleep over it that what is causing their unease is their
taking too seriously the foundational claims of Set Theory.  It is
certainly true that once one has set up set theory one can be very formal
indeed about defining recursive datatypes of formulae - in particular the
language of set theory and then.. and so on. All of this is very cute.  
However that's not the way the mathematics develops and one shouldn't
expect it to.  Foundationalism can be a terrible distraction.

  I'm not claiming that this answer banishes all metaphysical concerns.
What it does do is give my fourth years a good reason for not worrying
about the circularity.  God knows they have enough other things to worry
about as it is.

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