On 6/30/2013 7:08 AM, fom wrote: > 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." > > Leibniz > > > > 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". > Whoops... injective; well-defintion is the other direction.