Re: Computability and logic


george wrote:
Tom wrote:
It is just that little
part I mentioned that has continued to make me bounce off for several
months now.

In this context (as opposed to the Post context), substitution is
for supplying arguments to functions, or for applying functions
defined with argument-VARIABLES, TO argument-TERMS.
You have a function definition like F(x,y,z)=df=
'stuff'x'otherstuff'y'morestuff'z
and you want to know what F(1,2,3) is; you want to substitute 1 for x,
2 for y,
and 3 for z to get F(1,2,3)='stuff'1'otherstuff'2'morestuff'3.

I have shown the thing on the right, the result, as a string, composed
of other concatenated strings, but in this paradigm, that is NOT the
default composition of things in general. Things in general are TERMS.
They look like p(a,b,c) or s(s(s(0)); they the SAME kind of TREE
structure
that function-applications and predicate-applications have.

These trees do not have to be binary but they can all be re-represented
as binary.

My point is that there is a vast amount of literature in logic and
term-
rewriting involving TERM-substitutions (for variables) as opposed to
substring
substitutions that will accomplish the same objective.

Yes, George, I understand. Thank you. I was just thinking I am really
trying to establish what Goedel's Bew(x,y) really is, you know. I mean,
I am trying to reach a single string which incorporates both the
starting sequences and the rules of inference (Bew) from the inside,
which is impossible. But, again, if it is impossible how we do that at
all? I mean, given a TM description and a tape, how do you apply the
description to the tape to obtain another tape. Don't you need a meta,
and meta-meta, and meta-meta-meta ... machine to do that? You do.
Still, we manage to do the thing. How?

Thank you very much for writing.
Tom

.

Relevant Pages

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  • Re: Computability and logic
    ... In this context (as opposed to the Post context), substitution is ... default composition of things in general. ... They look like por s); they the SAME kind of TREE ...
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  • Re: Computability and logic
    ... In this context (as opposed to the Post context), substitution is ... default composition of things in general. ... They look like por s); they the SAME kind of TREE ...
    (sci.logic)