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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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Registered: 12/4/12
Re: Ordinals describable by a finite string of symbols
Posted: Jun 30, 2013 10:39 AM
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On 6/30/2013 7:08 AM, 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).
> Limit ordinals are treated differently from successors
> in transfinite recursion as well, are they not?
> Here it the thing. At the end of the nineteenth
> century, there had been a re-evaluation of Kant and
> progress in these subjects had been influenced by
> the writings of Leibniz. Frege invokes Leibniz in
> formulating his logic (two things are the same if
> one can be substituted for the other without loss
> of truth) and, as far as I know, Cantor also makes
> certain references to Leibniz of a more metaphysical
> nature (I am less clear on Cantor's works).
> Here is one of Leibniz' statements:
> "All existential propositions, though true,
> are not necessary, for they cannot be
> proved unless an infinity of propositions
> is used, i.e., unless an analysis is
> carried to infinity. That is, they can
> be proved only from the complete concept
> of an individual, which involves infinite
> existents. Thus, if I say, "Peter denies",
> understanding this of a certain time, then
> there is presupposed also the nature of
> that time, which also involves all that
> exists at that time. If I say "Peter
> denies" indefinitely, abstracting from
> time, then for this to be true -- whether
> he has denied, or is about to deny --
> it must nevertheless be proved from the
> concept of Peter. But the concept of
> Peter is complete, and so involves infinite
> things; so one can never arrive at a
> perfect proof, but one always approaches
> it more and more, so that the difference
> is less than any given difference."
> Leibniz
> There are certain questions mathematicians generally
> do not ask. One of them is "what is an object?"
> So long as one does not ask what an object is, one
> can count objects in succession.
> But, "objects", "representations of objects", "names
> of objects", etc. all lead to questions involving
> the difference between universal statements and
> existential statements.
> And, in particular, the question of treating real
> numbers as objects for arithmetical exactness leads
> to questions involving completed infinities.
> I am certain a more skeptical reader of your reasoning
> will help you to improve it. But, you are arguing over
> the assumption of an axiom. This is a matter of understanding
> and not reasoning. That Cantor saw these things in
> a metaphysical sense is one thing. But, it is not
> necessary for every mathematician to interpret these
> things metaphysically.
> The directionality needed to support Leibniz' remark
> is in the Peano axioms:
> m+1=n+1 -> m=n
> The identity of the object zero is decided by the
> infinity of identity statements of its successors.
> You, of course, wish to correct me by saying that
> the axiom merely expresses that the successor function
> is "well-defined".

Whoops... injective; well-defintion is the
other direction.

> You win of course.

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