Re: Restrictable Primitives?

On Tue, 11 Mar 2008 11:50:45 -0700 (PDT), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:

Just changed your notation and formatting slightly (I hope you don't
mind):


For our first order languages, the definition of /term/ is recursive.
For a language with [...] first order function symbol[s] 'f' ['g', 'h',
...], one of the clauses in the definition of /term/ would be:

If t is a term [and f is a function symbol], then f(t) is a term.

Now that is part of a RECURSIVE definition, so everything going ON
UPWARDS to build more and more terms depends on that clause.

Then for our first order languages, the definition of /satisfies/ is
recursive.

First, though is the recursive definition of what a term denotes:

One of the clauses would be:

If t is a term [and f is a function symbol], the f(t) denotes the
result of the function denoted by f applied to the object denoted
by t. So the function symbol f must map to a function that is total
on the universe of the model, so that we won't be "jammed" when
[we] seek the denotation of f(t) for any term t.

And that is part of a recursive definition, so everything going ON
UPWARDS to determine the denotation of more and more terms depends on
that clause, and then the recursive definition of /satisfies/ depends
on the whole recursive definition of the denotation of a term.

For any fuller explanation than this, since I don't have time to just
pour out an entire chapter of a book in a few posts, you really need
to read a good book on mathematical logic, where all of this is
carried out both formally and with an author's explanation.


Well done.


F.

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