On Apr 22, 2009, at 10:27 PM, Bart Gajderowicz wrote:
> JK> How does a cyclic definition introduce incompleteness?
>
> When sentences contain cyclic sentences, they become inconsistent,
> think of the halting problem (undecidability) with turing machines.
> If we were to introduce recursion to our interpretations, it would be
> limited to partial recursion. The task, then, would be to create an
> interpretation of the sentence which is acyclic, and our sentence
> consistent. The interpretation would be decidable, and our recursive
> function total. (01)
Bart, you keep using terms in ways I don't understand. (02)
1. Did I miss the definition of a "cyclic" sentence? Could you
provide a reasonably rigorous definition and then demonstrate how (as
it seems you are saying) any sentence containing a cyclic sentence
"becomes inconsistent"? (03)
2. How could the halting problem demonstrate anything about cyclic
sentences (no matter how they are defined)? The halting problem is
the problem of whether a certain welldefined function is computable
by a Turing machine. On the face of it, it has nothing whatever to do
with sentences, cyclic or otherwise. (04)
3. What does it mean to "introduce recursion in our interpretations"?
Recursion is a way of defining one function in terms of others. An
interpretation might interpret a *theory* that is capable of defining
functions by recursion, for example, but I have no idea what it means
to "introduce recursion in" an interpretation; it seems like a
category mistake. (05)
4. Interpretations are mathematical structures. What do you mean by a
"decidable" interpretation? One whose domain is decidable? (06)
chris (07)
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