On Sun, 30 Jun 2013 03:19:01 -0700 (PDT), apoorv <email@example.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.)