In article <7a3e3a5a-2202-4cbb-9556-e3766ee5a745@k39g2000yqd.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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 proper set theory, one must be able to determine of an object whether or not it is a member of a set, but since those vague "meanings" of a string of symbols are so indeterminate, one cannot determine whether a "meaning" does belong to a "symbol" so WM's "sets" are not well enough defined to be sets in any proper set theory. > > In mathematics we can calculate or estimate things even if not all can > be named.
But set membership must be less ambiguous than WM's. > > > > > Therefore the list contains only countably many finite definitions.
Except that it is not at all clear what has been listed. > > > > 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.
Only for a while. > > > 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.
We cannot possibly know that. > > 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.
But what everyone else sees is that ZFC is right unless there is a natural between 0 and 1.
