
Re: Uncountability of the Real Numbers Without Decimals
Posted:
Dec 5, 2013 6:41 AM


Am Dienstag, 3. Dezember 2013 19:56:05 UTC+1 schrieb Zeit Geist: > On Tuesday, December 3, 2013 4:42:54 AM UTC7, WM wrote: > > > Am Montag, 2. Dezember 2013 21:41:33 UTC+1 schrieb Zeit Geist: > > > > > > > > The set of positive rational numbers that is less than the natural number n and has not been enumerated by the first n natural numbers grows with n. It is impossible eneumerate all rational numbers, i.e., to remove all rationals from the state of being not enumerated to the state of being enumerated. > > > > > > > Impossible? How about a proof. > > > > > > How can we prove anything about the limit of an infinite sequence other than by considering the finite formula? In the finite formula we have the set of rationals less than n but not enumerated by naturals less than n: > > > > > > M_(n) = (M_(n1) U (n1, n]) \ q_n > > > with M_0 = { } > > > > > > This sequence grows without end. > > > > So does the Set of Natural Numbers, but I can still say, "For All n e N, ... ".
chuckle. Of course you can say so. But that does not change any mathematical facts. > > > > > > > Here the intentional confusion of "for every n in N" and "for all n in N" has been used. It has been neglected that beyond every n there are infinitely many following, such that never all can have been used. > > > > Suppose for some Property phi, that For Every n e N, phi(n) is the case. > > Hence, there is NO Natural Number, m, such that ~phi(m) is the case. > > Therefore, For All n e N, phi(n) is the case. > > > > Unless, of course, you reject the Law of the Excluded Middle. > I do not. You can state that every natural number is either even or od. But you cannot "take" most of the natural numbers for any purpose. > > > > > You are welcome to work your own Mathematical System sans the AoI. > > > > > > That is not a question about AoI. It simply shows that your kind of matheology blinds its adherents with respect to the fact that > > > > > > > But it is. The AoI allows us to form the Set of Natural Numbers, all of them.
> From this we can form function from N to other Sets. This, we can prove that the Set of Rational Numbers are Not Countable. > Have you a definition of "set". Who told you that you can take all natural numbers for your function? And if you don't believe me, then try it. Try to take one of the natural numbers that do not belong to the set of the first 0 %. Why can't you simply convince yourself?
> > > > "every rational number can be enumerated if and only if beyond that rational number infinitely many rationals are following." > > > > No, there could be last Rational in my ordering, say q_l. However, the Set Q \ {q_l} must contain a SubOrder isomorphic to omega, the Order Type of the Natural Numbers.
First answer whether you have been able to take a natural number that does not belong to the first 0 %. Then you will no longer consider nonsense like infinite Order Types. > > > > > There is no way to circumvent this simple fact. Closing eyes or stamping feet or shouting axioms will not help. > > > > You're the one who seems to be stamp his feet and shouting, "Infinity Can Not Be Completed.".
No. I know simply that all you boast to use belongs to a 0%set.
> > > Of course there is the flaw I mentioned above. Further there is the flaw that the limit has been defined from the scratch. So it belongs to the set of definable numbers. It is irrelevant how that defining happens if only every definition is assumed to be finite. And that is the case  up to now. Perhaps matheologians will also drop that fact. > > > > You mention no flaw, just an illfounded belief.
Have you looked into Cantor's work? Have you ssen how he denotes the uncounted number? a^(oo). Like this the diagonal is not constructed from only indexed digits d_n. It requires the digit d_oo. > > The sequence, { a_n  n e N }, has no specific values. It consists of arbitrary real Numbers chosen Arbitrarily from a NonEmpty Set of Real Numbers. The Axioms of the Real Numbers are then used to show that this Sequence results in a Contradiction.
If you assume defined real numbers in the proof, you get a defined real number in the result. Otherwise you get nothing. > > > > > So you may continue with your matheology, but you know anyhow that it is completely useless. Nothing can be accomplished, even mathematics is contradicted by the fact that never undefinable numbers can result from mathematical operations. The reason is simple: Matheology is selfcontradictory as shown above and in many further instances. > > > > The Powerset Operation can, but Not necessarily does, produce Real Numbers that must Exist but may have a precise "description".
Every accessible set has an accessible powerset. Since 100 % of the set N is not accessible also its powerset is not. > > > > This result is CounterIntuitive, but Not Contradictory.
It is not counter intuitive. Try to find a natural number that is not an element of the 0%set. That is a mathematically disprovable action. Afterwards you can decide about the quality of your intuition.
Regards, WM

