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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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fom

Posts: 1,968
Registered: 12/4/12
Re: Ordinals describable by a finite string of symbols
Posted: Jun 30, 2013 10:44 AM
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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.
>>
>>> 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 ?
>>
>>>
>>
>>
>>
>> Pretty good.
>>
>>
>>
>> This is why one must postulate the existence of
>>
>> a limit ordinal (different from the null-class
>>
>> which can be thought of as satisfying the definition
>>
>> 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?"

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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