Re: My investigations into Godels Incompleteness Theorem

John Jones wrote:
MoeBlee wrote:
For formal mathematical languages, a string is a certain kind of
function. And a sentence is a certain kind of string.

No, a sentence isn't a certain sort of string. If it was, it would be a
string.

The usual definition of 'sentence' in mathematcial logic is that a
sentence in a formal language is a certain kind of string (where
'certain kind of' is filled in by a certain recursive definition). In
this regard, all sentences are strings but not all strings are
sentences. This definition of 'sentence' is a stipulative definition.
If you're inclined to say that certain stipulative definitions are
incorrect qua definitions, then there's no basis of a rational
discussion with you.

The meaning of a
sentence depends upon a structure for the language.

No it doesn't. You are talking about translation.

No, I'm talking about an interpretation of the language. What this
involves is specified in any textbook on mathematical logic.
Interpretation of a language is different from translation between
languages.

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?)

Sentences don't have meanings. Again, you are talking about
interpretation and not language.

I am talking about interpretations of languages. 'meaning' is an
informal notion, but certain important aspects of that notion are
captured by scuh formal semantics as I am talking about.

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.


That's mauling the definition of sentence. There isn't anything
sentence-like about a sentence except expression. There is no 'it' in
'its expression'.

The definition of 'sentence' is a stipulative definition that I have
not "mauled" by the above remarks.

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.

Not at all. Analysis of language is after the fact of language - it
merely works upon the dead letters. Sorry about all that.

I said that I am talking about formal languages (such as a first order
language for a mathematical theory). Your remarks are, as usual,
completely aside the point even when, at best, they are coherent.

MoeBlee

.

Relevant Pages

  • Re: Operator overloading in C
    ... All development of C as an independent language has ... making any changes or improvements to the standard ... The lack of a counted string data structure, ... Pointers can't be used for arg1 or arg2. ...
    (comp.std.c)
  • Re: syntax...
    ... B&D on the part of the language designer. ... probably handle concatenation of string literals by itself, ... bitwise XOR, or if not that, then exponentiation.) ...
    (comp.lang.misc)
  • Why C Is Not My Favourite Programming Language
    ... C has no string type. ... compiler take care of the rest. ... Why does any normal language ... the programmer fail. ...
    (comp.lang.c)
  • Re: Controlling Javascript from server side
    ... but five different language implementations here. ... 'true' means that the request must be handled asynchronously. ... There is exactly *no* reason for such a thing here. ... | percent-endoded string). ...
    (comp.lang.javascript)
  • Re: Theodore Adorno, a prophet of data systems design
    ... C may not be a forgiving language, ... >> attention get burned with the memory leaks, and buffer overruns, ... by a small, compact team of brilliant programmers, or it may be ... so you agree that it is not a limit on the length of the string, ...
    (comp.programming)
Quantcast