
Re: Uncountability of the Real Numbers Without Decimals
Posted:
Dec 3, 2013 1:56 PM


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, ... ".
> > > 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.
> > 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.
> "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.
> 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.".
> > However, your gripes do Not point out any flaw in the proof. > > 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. 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.
> 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".
This result is CounterIntuitive, but Not Contradictory.
> Regards, WM
ZG

