[FOM] Concerning definition of formulas
Richard Heck
rgheck at brown.edu
Sat Sep 29 09:23:12 EDT 2007
Saurav wrote:
> This is seemingly convincing. But as ZF is itself made of formulas, we
> can not adopt this definition.
>
I don't quite understand this. Yes, ZF is a set of formulae. But there's
nothing circular about using set theoretic language to define the notion
of a formula of ZF. You aren't /using/ the notion of a formula in the
definition. The set of formulae is defined in terms of ordinary set
theoretic notions, ultimately in terms of membership.
It's really quite irrelevant whether the background theory is ZF. If
you're going to define this notion, or any other, then you have to use
some background theory. And theories are sets of formulae.
The set theoretic definition is really overkill, though. You can define
the notion of a formula in a very weak arithmetic, using the techniques
pioneered by Goedel.
Richard
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Richard G Heck, Jr
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Brown University
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