On Sunday, June 30, 2013 12:19:01 PM UTC+2, apoorv wrote: > An ordinal, like any object of our discourse can be described by a string of > > Symbols. Suppose we consider the set S of all ordinals that can be described > > By a finite string of symbols. Now S must be an ordinal. Because if it were > > Not so, then its members must not form an unbroken chain. so, there is an ordinal > > X which is not in S , while the successor of X or some bigger ordinal is in S. > > But if X is not describable by a finite string, then the successor of X also > > Cannot be so, nor any bigger ordinal. > > Now S, being an ordinal cannot be in itself. > > So S, finitely described as ' The set of all ordinals that can be described > > By a finite string of symbols' Cannot be ' a set describable by a finite string of > > Symbols'. > > The set S must then not exist. Then the Set S must be the set of all ordinals, > > As that is the only set whose members form a chain, that does not exist. > > Thus the set S = set of all ordinals. > > Whence, all ordinals must be describable by a finite string of symbols. > > But then, the set of all ordinals is countable. > > From which, we get that there is some countable limit ordinal that does not > > Exist. > > So where is the flaw in the above reasoning ? >
If you consider the set of all ordinals definable in the first-order language of set theory, it doesn't have to be an ordinal. There must be a least ordinal not definable in the first-order language of set theory, and this must be a countable ordinal. It is definable in the first-order language of set theory with a truth predicate. There is no end to the hierarchy of languages in which you can define ordinals.