Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Topic:
listing the mistakes of Cantor and his diagonal argument Chapt28 Summary, Review and Reminders #1210 Correcting Math 3rd ed
Replies:
154
Last Post:
Nov 6, 2012 5:17 PM



netzweltler
Posts:
426
From:
Germany
Registered:
8/6/10


Re: Cantor destroys Probability theory Re: it is a bijection since it is All Possible Digit Arrangements Chapt28 Summary, Review and Reminders #1237 Correcting Math 3rd ed
Posted:
Nov 2, 2012 7:46 PM


On 2 Nov., 20:10, Archimedes Plutonium <plutonium.archime...@gmail.com> wrote: > On Nov 2, 2:04 pm, Archimedes Plutonium > > > > > > <plutonium.archime...@gmail.com> wrote: > > On Nov 1, 3:15 pm, netzweltler <reinhard_fisc...@arcor.de> wrote: > > > > On 1 Nov., 20:40, Archimedes Plutonium > > > > <plutonium.archime...@gmail.com> wrote: > > > > We start with Naturals and we list them and reverse them to obtain the > > > > bijection into the Reals. Since we use the ordering of the Naturals it > > > > may look as though it is not a bijection: > > > > > 1 <> .1 > > > > ?10 <> .01 > > > > 11 <> .11 > > > > 100 <> .001 > > > > 101 <> .101 > > > > 110 <> .011 > > > > etc etc > > > > > Now every Natural has a Real to correspond to it and the reason that > > > > is true is because both sets are All Possible Digit Arrangements and > > > > neither set has a hole in it. > > > > The naturals correspond to the reals which end up in an infinite > > > number of 0's, right? > > > > 1 <> .10000... > > > 10 <> .01000... > > > 11 <> .11000... > > > ... > > > > which naturals do represent the binary complement of the reals above? > > > > ? <> .01111... > > > ? <> .10111... > > > ? <> .00111... > > > ... > > > > which natural does represent .010101(01)...? > > > >  > > > netzweltler > > > Alright, let me correct you. > > > Let me go to decimal rather than binary since everyone uses decimal in > > common life. > > > You do know the Fundamental Counting Principle, although not sure that > > Probability theory was used when Cantor did his diagonal argument. > > > What are all the ways of arranging 10 digits to 2 place value? to 3 > > place value? to 4 place value? Etc etc? > > > The answer is 10^2, 10^3, 10^4, etc etc > > > So we have our Bijection of Naturals onto Reals and Reals onto > > Naturals: > > > All the Possible Digit Arrangements of 10 digits to 2 place value: > > > 00 > > 01 > > 02 > > 03 > > . > > . > > . > > 97 > > 98 > > 99 > > > There are 100 such Naturals > > > How many Reals between 0 and 1 with All Possible Digit Arrangements > > of 10 digits to 2 place value? The answer is 100 and for 3 place value > > the answer is 1000 and for 4 place value the answer is 10^4. > > > So, since All Possible Digit Arrangements of Counting Numbers matches > > All Possible Digit Arrangements of Reals, there is no room in either > > Reals or Counting Numbers for Cantor nonsense. > > > 00 <==> .00 > > 01 <==> .1 > > 02 <==> .2 > > . > > . > > 09 <==> .9 > > 10 <==> .01 > > 11 <==> .11 > > . > > . > > 97 <==> .79 > > 98 <==> .89 > > 99 <==> .99 > > 100 <==> .001 > > > Now I was too lazy to list them all, and listed just a few > > representatives. > > > Anyway the proof is simple enough. If you want to do a Cantor fakery, > > then you make lists which are never All Possible Digit Arrangements > > and you end up with a connartist trick of thinking you have some new > > Real not on the list which you purposely neglected to list all the > > Reals. > > > A true proof would recognize the Reals are All Possible Digit > > Arrangements as well as the Naturals are All Possible Digit > > Arrangements > > > Proof: The Naturals are All Possible Digit Arrangements as well as the > > Reals, since there is no gap or hole in either Reals or Naturals, one > > can never produce a Real when bijecting the Naturals with Reals. There > > are a total of 100 Naturals of 10 digits to 2 place value and there > > are a total of 100 Reals of 10 digits to 2 place value, and there are > > total of 1000 Reals of 10 digits to 3 place value and ditto for the > > Naturals, hence the Reals are 11 Correspondence of Naturals. > > > Now as I noted earlier, if Cantor were correct, and he indeed is > > false, but if he were correct, that would throw Probability theory off > > tremendously, because there are numbers within the Reals, if Cantor > > was correct for which we thought we had a Total Probability Universe > > for, but did not because of what can be called the CantorReals, the > > fake Reals. > > > Now I wonder and do not know, why the mathematicians of Probability > > theory never raised objection to Cantor diagonal, and can only guess > > that they were suppressed by the stupid believers of the ZFC school. > > Sorry I forgot my signature block and added it to the original: > > Archimedes Plutonium ?http://www.iw.net/~a_plutonium?whole entire > Universe is just one big atom ?where dots of the electron dot cloud > are galaxies
If this is the list of all possible digit arrangements of the naturals
1 <> 10000... 2 <> 01000... 3 <> 11000... 4 <> 00100... ...
then the diagonal is 00111..., a digit arrangement ending up in infinitely many 1's. Since the digit arrangements of natural numbers are ending up in infinitely many 0's, this diagonal cannot be on the list of the naturals.
Of course this doesn't work if we do not even agree on the existence of numbers of infinitely many digits.
 netzweltler



