On 16 Jun., 17:55, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > WM <mueck...@rz.fh-augsburg.de> writes: > > It is very probable that every line has many different meaning, but no > > line has uncounatbly many meanings. > > Every line has an indefinite and indeterminate number of possible > meanings. It makes no sense to speak of the cardinality of the totality > of all possible meanings of a string of symbols.
It is fact that all possible meanings must be defined by finite definitions. Therefore the meanings are countable. Therefore it makes sense to call the set of all meanings countable.
In mathematics we can calculate or estimate things even if not all can be named. > > > Therefore the list contains only countably many finite definitions. > > The list contains just random, meaningless strings. Whenever we instill, > with mathematical precision, these strings with some definite meaning, > so that they become definitions of reals, we find there are definitions > not included in the list. There is an absolute notion of computability > in logic.
If this notion yields numbers only that are in trichotomy with each other, then the notion is acceptable. If this notion yields numbers like you gave examples for (IIRC) like n = (1 if Obama gets a second term and 0 otherwise) or so, then this notion together with your logic should be put into the trash can.
> There is no absolute notion of definability.
We need not an absolute notion if we know that all possible definitions of the notion of definability belong to a countable set.
It is enough to prove by estimation that set theory is wrong. Compare the famous irrationality proofs and transcendence proofs of number theory. We need not calculate its deviation from truth to the fifth digit. It is enough to see that ZFC is wrong unless there is a natural between 0 and 1.
Wissenschaft bedingt Zweifel am Glaubhaften. Matheologie erfordert Glauben an das Zweifelhafte.