fom
Posts:
1,968
Registered:
12/4/12


Re: Ordinals describable by a finite string of symbols
Posted:
Jun 30, 2013 10:44 AM


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?"
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.

