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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
Re: Ordinals describable by a finite string of symbols
Posted: Jun 30, 2013 1:24 PM
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On Sunday, June 30, 2013 10:31:35 PM UTC+5:30, dull...@sprynet.com wrote:
> On Sun, 30 Jun 2013 03:19:01 -0700 (PDT), apoorv <ski..@gmail.com>
>
> 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 ?
>
>
>
> The condition "can be described in a finite set of symbols" is
>
> too vague to be part of an actual mathematical proof.
>
> Described in what sense? In what language?
>
>
>
> See, this matters. Suppose you give a precise
>
> formal definition of "can be described in a finite
>
> set of symbols" - some formal language, etc.
>
>
>
> Doesn't matter what the formalism is. Let's
>
> say a set is a D set if it can be described in
>
> finitely many symbols, _in_ the sense specified
>
> by whatever definition you chose. Now
>
> consider the string
>
>
>
> (*) "S is the union of all the ordinals which are D-sets."
>
>
>
> That _is_ a finite string of symbols. But it doesn't
>
> show that S is a D-set, because we haven't shown
>
> that the string (*) _is_ one of the "descriptions"
>
> allowed in whatever our definition of "D-set" was!

If we take 'describable' as describable (definable?) by any
String of symbols, then S is indeed a D -set.
Like all theories in a countable language, ZFC has a countable model.
In this model, the set of all ordinals, which does not exist, must be countable.
In view thereof,there is one countable limit ordinal , which cannot be assumed
To exist.?
-Apoorv
>
>
> One possibility is that (*) is simply _not_ one of
>
> the "descriptions" allowed by the definition of
>
> D-set. Another possibility is that it _is_ a valid
>
> D-set description. In the second case you haven't
>
> obtained a real contradiction, you've just shown
>
> that the formal system you used to define D-sets
>
> is inconsistent.
>
>
>
> (Assuming ZFC is consistent, the first possibility
>
> arises if you try to give a definition of D-set in
>
> the context of ZFC. The second possibility arises
>
> if you do this in various other versions of set theory,
>
> which _are_ inconsistent for example by Russell's
>
> paradox.)
>
>
>

> >
>
> >- Apoorv




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