[FOM] Platonism and Formalism

Sandy Hodges SandyHodges at alamedanet.net
Fri Sep 12 14:34:54 EDT 2003

Gaifman writes: "This is essentially the position Abraham Robinson took

Thanks for the reference.

Gaifman writes:
It is a coherent description of a "formalist's behavior". But perhaps we
should distinguished between describing one's *behavior* and giving an
*account* of a certain view. For example, we believe that the system we
employ is consistent. Can a formalist give grounds for this belief, over
and above the brute fact that so far no contradiction has been

I proceed merrily along, making assertions in my U-language, such as 2 <
3, and 0 < 1.    There may be some system S, such that all the
assertions I make happen to be either axioms of S, or follow from
earlier assertions that I made, by rules of S.   Perhaps one such system
S is the very one proposed in FOM by Harvey Friedman a few months ago,
the one with axioms and rules for first-order logic, sets, pairs, and
real numbers.    However, were I to say: "0 < 1, and that's an axiom."
then I would have made an assertion that is neither an axiom of S nor
follows by a rule of S from prior assertions, since Friedman's system
does not contain a predicate that means "is an axiom."

A system that would both allow me to assert that 0 < 1, and to claim
that my assertion (or claim, sentence, token, proposition, formula,
pointer, etc.) "0 < 1" is an axiom, would be a system that acts as its
own meta-langauge, and such systems are as you know somewhat rare.

If I am making assertions in my U-language, and the systems to which my
assertions happen to conform are of the sort which do not act as their
own meta-language, then I can't give "grounds" for asserting "0 < 1."
I can only assert it.     Not only can I not give grounds for claiming
that the system I am using is consistent, I can't refer to the system I
am using at all.

I find the whole business, which seems to be taken for granted, of
jumping from assertions made in one's U-language, to assertions in a
meta-langauge of that U-language, somewhat mystifying.   Perhaps you can
give an account of how this is done, in a more formal way than is
usually seen?    It seems to me that any such a formal account will need
to have two classes of formulas, the U-language ones and the
meta-language ones.    The result will be a system that acts as its own
meta-language, in that it has formulas that refer to its own formulas.
It will merely be a weak system, compared with other systems that can
act as their own meta-langauges.

So it seems to me that any question that requires jumping to a
meta-language, can only be understood, if we insist on a rigorous
account, in the context of a system that can act as its own
meta-language.  There are hard choices to be made in selecting any such

Of course, speaking in my U-language, and without recourse to
meta-language, I can define a set P of numbers; and it may happen that
there is a one-to-one correspondence between the numbers in P and the
theorems of some system S, to which my assertions happen to conform.
The closest I could come to saying that my own system is consistent
(without using meta-language) would be to assert that some number is not
in P.    But if I say this, I have not actually said that the system I
am using is consistent, since I am unable to assert, without recourse to
meta-language, that the one-to-one correspondence exists.    And in any
case, I can't assert that some number is not in P, while still
conforming to S.

Thus I claim that we can't give a formally rigorous account of a
Platonist, or anyone else, giving grounds for her belief that the system
she employs is consistent, without considering systems that can act as
their own meta-language.

Systems that act as their own meta-language are the focus of my interest
in logic, and good references would be much appreciated.

- Sandy

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Sandy Hodges / Alameda,  California,   USA
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