apoorv
Posts:
53
Registered:
4/11/13


Re: Ordinals describable by a finite string of symbols
Posted:
Jun 30, 2013 1:24 PM


On Sunday, June 30, 2013 10:31:35 PM UTC+5:30, dull...@sprynet.com wrote: > On Sun, 30 Jun 2013 03:19:01 0700 (PDT), apoorv <ski..@gmail.com> > > 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 ? > > > > The condition "can be described in a finite set of symbols" is > > too vague to be part of an actual mathematical proof. > > Described in what sense? In what language? > > > > See, this matters. Suppose you give a precise > > formal definition of "can be described in a finite > > set of symbols"  some formal language, etc. > > > > Doesn't matter what the formalism is. Let's > > say a set is a D set if it can be described in > > finitely many symbols, _in_ the sense specified > > by whatever definition you chose. Now > > consider the string > > > > (*) "S is the union of all the ordinals which are Dsets." > > > > That _is_ a finite string of symbols. But it doesn't > > show that S is a Dset, because we haven't shown > > that the string (*) _is_ one of the "descriptions" > > allowed in whatever our definition of "Dset" was! If we take 'describable' as describable (definable?) by any String of symbols, then S is indeed a D set. Like all theories in a countable language, ZFC has a countable model. In this model, the set of all ordinals, which does not exist, must be countable. In view thereof,there is one countable limit ordinal , which cannot be assumed To exist.? Apoorv > > > One possibility is that (*) is simply _not_ one of > > the "descriptions" allowed by the definition of > > Dset. Another possibility is that it _is_ a valid > > Dset description. In the second case you haven't > > obtained a real contradiction, you've just shown > > that the formal system you used to define Dsets > > is inconsistent. > > > > (Assuming ZFC is consistent, the first possibility > > arises if you try to give a definition of Dset in > > the context of ZFC. The second possibility arises > > if you do this in various other versions of set theory, > > which _are_ inconsistent for example by Russell's > > paradox.) > > > > > > > > Apoorv

