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 ? >
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".