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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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Rupert

Posts: 3,806
Registered: 12/6/04
Re: Ordinals describable by a finite string of symbols
Posted: Jul 9, 2013 7:59 AM
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On Sunday, June 30, 2013 12:19:01 PM UTC+2, 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 ?
>


If you consider the set of all ordinals definable in the first-order language of set theory, it doesn't have to be an ordinal. There must be a least ordinal not definable in the first-order language of set theory, and this must be a countable ordinal. It is definable in the first-order language of set theory with a truth predicate. There is no end to the hierarchy of languages in which you can define ordinals.



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