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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 12:57 PM
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On Sunday, June 30, 2013 8:14:25 PM UTC+5:30, fom wrote:
> 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.
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> >>
>
> >>> But if X is not describable by a finite string, then the successor of X also
>
> >>
>
> >>> Cannot be so, nor any bigger ordinal.
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> >>
>
> >>> Now S, being an ordinal cannot be in itself.
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> >>
>
> >>> So S, finitely described as ' The set of all ordinals that can be described
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> >>
>
> >>> By a finite string of symbols' Cannot be ' a set describable by a finite string of
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> >>
>
> >>> Symbols'.
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> >>
>
> >>> The set S must then not exist. Then the Set S must be the set of all ordinals,
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> >>
>
> >>> As that is the only set whose members form a chain, that does not exist.
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> >>
>
> >>> Thus the set S = set of all ordinals.
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> >>
>
> >>> Whence, all ordinals must be describable by a finite string of symbols.
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> >>
>
> >>> But then, the set of all ordinals is countable.
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> >>
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> >>> From which, we get that there is some countable limit ordinal that does not
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> >>
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> >>> Exist.
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> >>
>
> >>> So where is the flaw in the above reasoning ?
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> >>
>
> >>>
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> >>
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> >>
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> >>
>
> >> Pretty good.
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> >>
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> >>
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> >>
>
> >> This is why one must postulate the existence of
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> >>
>
> >> a limit ordinal (different from the null-class
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> >>
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> >> which can be thought of as satisfying the definition
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> >>
>
> >> 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?"

Any string of symbols (language of set theory suitably augmented?) that can unequivocally tell us which ordinal we are referring to.
0 is Ax~x e 0 and Ex x=0
w is 0 e w and Ax x e w-->Sx e --> w and Ex x = w.
Etc.
-Apoorv

>
>
> 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.





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