FOM: Saying what you mean

[Roger Bishop Jones]

When I said in a previous posting that in second order logic you can "say
what you mean", this is what I meant:

In second order logic, when you write down the sentences of arithmetic, you
express the propositions of arithmetic.

In first order logic, when you write down the same sentences, you don't
quite succeed in expressing the same propositions.

The difference is most conspicuous when the truth value differs, as in
con(PA).
A sentence intended to express the claim that PA is consistent, expresses in
second order logic a proposition which is true.
When expressed in first order logic it expresses a proposition which has no
definite truth value, since it is true in some, and false in others of the
first order models of PA.

As suggested by my posting about "expressiveness", the propositions of
arithmetic cannot be expressed in any complete logic, because in no
consistent logic are all the true propositions of arithmetic provable and in
no complete logic are there any unprovable truths.

Roger Jones

RBJones at RBJones.com