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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 10:05 AM
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On Sunday, June 30, 2013 7:04:21 PM UTC+5:30, 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:
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> >
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> > > An ordinal, like any object of our discourse can be described by a string of
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> >
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> > > Symbols. Suppose we consider the set S of all ordinals that can be described
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> >
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> > > By a finite string of symbols. Now S must be an ordinal. Because if it were
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> >
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> > > Not so, then its members must not form an unbroken chain. so, there is an ordinal
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> >
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> > > X which is not in S , while the successor of X or some bigger ordinal is in S.
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> >
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> > > But if X is not describable by a finite string, then the successor of X also
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> >
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> > > Cannot be so, nor any bigger ordinal.
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> >
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> > > Now S, being an ordinal cannot be in itself.
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> >
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> > > So S, finitely described as ' The set of all ordinals that can be described
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> >
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> > > By a finite string of symbols' Cannot be ' a set describable by a finite string of
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> >
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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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> >
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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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> >
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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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> > > Exist.
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> >
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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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> >
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> > a limit ordinal in a trivial sense).
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> 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

The last line got mangled by 'predictive text'. Pl read the last as 'cannot be assumed to exist'.

> >
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> > Limit ordinals are treated differently from successors
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> >
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> > in transfinite recursion as well, are they not?
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> >
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>

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




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