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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 8:08 AM
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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."


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".

You win of course.

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