
Re: Ordinals describable by a finite string of symbols
Posted:
Jul 1, 2013 11:02 AM


On Sun, 30 Jun 2013 10:24:27 0700 (PDT), apoorv <skjshr@gmail.com> wrote:
>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.
Sigh. Taking 'describable' to mean 'describable (definable?) by any String of symbols' makes no sense! Symbols don't mean anything  it's impossible to use a string of symbols to describe anything.
A string of symbols _in_ a welldefined system where symbols have been _assigned_ "meanings" can be used to describe or define something.
>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.?
You're confused. Look up "LowenheimSkolem paradox" somewhere...
>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

