Thanks for your response.
Quote:
Originally Posted by TravisT
condition II can be rewritten as for some i

I thought of this, however consider, for example, a function that satisfies
. Then if
,
for
. Informally, a loop might occur, but it might not loop right back to the beginning.
Quote:
Originally Posted by TravisT
are you sure you mean that ? if I have then that's a busted function.

I certainly do mean
. The other possibility
would not include all of the relevant functions, for example,
, if
then
will still always be a natural number.
Your example is a busted function, however a rather trivial one. It would perhaps be possible to define
, but then
is not well defined. Basically the functions of any real interest are smooth functions, like polynomials, exponentations, etc..
Quote:
Originally Posted by TravisT
what does bustible mean? Either a function is busted or it is not, bustible sounds like we can do something to the function to make sure it is busted.

Yes I shouldn't have used the term 'bustible'. This is a human word which I used to mean "it is possible to prove that this function is busted." In fact lets make it a definition.
Definition: Let
be a function. Then if there exists a proof that
is busted, we say
is
bustible.