apoorv
Posts:
53
Registered:
4/11/13


Re: Ordinals describable by a finite string of symbols
Posted:
Jun 30, 2013 12:57 PM


On Sunday, June 30, 2013 8:14:25 PM UTC+5:30, fom wrote: > On 6/30/2013 8:34 AM, apoorv wrote: > > > On Sunday, June 30, 2013 5:38:07 PM UTC+5:30, fom wrote: > > >> On 6/30/2013 5:19 AM, 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 ? > > >> > > >>> > > >> > > >> > > >> > > >> Pretty good. > > >> > > >> > > >> > > >> This is why one must postulate the existence of > > >> > > >> a limit ordinal (different from the nullclass > > >> > > >> which can be thought of as satisfying the definition > > >> > > >> a limit ordinal in a trivial sense). > > > The axiom specifying the limit ordinal, such as AoI, is a finite description of that limit ordinal. > > > The argument is that the set of all finitely describable ordinals, described so finitely,should also not be finitely describable. > > > Whence it does not exist. So it is the same as the set of all ordinals, both having elements > > > That are finitely describable., and hence countable. Whence one of the countable limit ordinals > > > Cannot beast ulster to exist. > > > Apoorv > > > > > > If this is to be your argument, then you have > > a problem. > > > > You are using natural language terms in decidedly > > informal ways. > > > > Let us begin with "What precisely constitutes a > > description?" Any string of symbols (language of set theory suitably augmented?) that can unequivocally tell us which ordinal we are referring to. 0 is Ax~x e 0 and Ex x=0 w is 0 e w and Ax x e w>Sx e > w and Ex x = w. Etc. Apoorv
> > > Indeed, it is one thing to scrawl chicken scratch > > consecutively. It is an entirely different thing > > to invoke its pragmatic meaning to explain the > > semantics of its own representation.

