Re: My investigations into Godels Incompleteness Theorem

John Jones wrote:
I say that the properties of a
string are not conferred on the sentence that is constructed out of it.
Any more than the properties of a letter are conferred on the meaning
of a word.
When you make a string you lose the sense of the of the sentence and
merely address individual signs. What you are claiming is that the
meaning of a sentence is exhibited in its string, which is simply not
true.

I'm not referencing the specifics of your discussion with Daryl
McCullough, but I'd like to add my own notion, which is a common one:

For formal mathematical languages, a string is a certain kind of
function. And a sentence is a certain kind of string. The meaning of a
sentence depends upon a structure for the language. A structure for a
language is yet another certain kind of function. A sentence has
different meanings depending on different structures for the language.
(Or maybe (I don't know) some people say that the single overall
meaning of a sentence is its class of structure-specific meanings?)

And we may distinguish a sentence from the proposition that the
sentence expresses. The sentence is just a syntactical object - a
string. The proposition expressed by the sentence is something else,
which is harder to pin down. In Church's introductory chapter of his
textbook, he says that a proposition is that which only synonymous
sentences (even sentences not in the same language) have in common.

But whatever vagaries there are in the notion of a propostion, I think
that in mathematics we can get by without the notion of a propostion if
we take sentences to be syntactical and take the semantical aspects to
be given by the method of structures for a language.

MoeBlee

.

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