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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: 3,145
Registered: 12/13/04
Re: Ordinals describable by a finite string of symbols
Posted: Jun 30, 2013 1:01 PM
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On Sun, 30 Jun 2013 03:19:01 -0700 (PDT), apoorv <skjshr@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!

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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