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Topic: Ordinals describable by a finite string of symbols
Replies: 16   Last Post: Jul 9, 2013 7:59 AM

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apoorv

Posts: 53
Registered: 4/11/13
Ordinals describable by a finite string of symbols
Posted: Jun 30, 2013 6:19 AM
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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




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