Date: Jun 30, 2013 6:19 AM
Author: apoorv
Subject: Ordinals describable by a finite string of symbols
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 ?

- Apoorv