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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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David C. Ullrich

Posts: 3,238
Registered: 12/13/04
Re: Ordinals describable by a finite string of symbols
Posted: Jul 1, 2013 11:02 AM
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On Sun, 30 Jun 2013 10:24:27 -0700 (PDT), apoorv <skjshr@gmail.com>
wrote:

>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.


Sigh. Taking 'describable' to mean 'describable (definable?)
by any String of symbols' makes no sense! Symbols don't
mean anything - it's impossible to use a string of symbols
to describe anything.

A string of symbols _in_ a well-defined system where
symbols have been _assigned_ "meanings" can be
used to describe or define something.

>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.?


You're confused. Look up "Lowenheim-Skolem paradox" somewhere...

>-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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