Mathematical Cranks and Fallacies

pboily

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What's your favorite mathematical fallacy?

I've just encountered a delightful one that actually got published, in the Transactions of the Wisconsin Academy of Sciences, Arts and Letters, in 1974, by William Dilworth.

In it, Dilworth procedes to show that the the interval [0,1] is countable (that is, that you can enumerate, or list (even if the list is infinite) all the real numbers in the interval [0,1]), as follows:

first, enumerate all the real numbers with no (repeating) digit after the decimal (there are 2 of them):

Code:
0 1

Then, enumerate all the real numbers with exactly one digit after the decimal (there are 11 of them, minus the ones that were already counter above, as 0=0.0 and 1=1.0):

Code:
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Then, enumerate all the real numbers with exactly two digits after the decimal (there are 101 of them, minus the ones that were already counted above):

Code:
0.00 0.01 0.02 0.03 ... 0.55 0.56 0.57 ... 0.97 0.98 0.99 1.00

...

Then, enumerate all the real numbers with exactly n digits after the decimal (there are 10^n+1 of them, minus the ones that were already counted above)

Code:
0.000......000 0.000......001 0.000......002 ... 0.999......998 0.999......999 1.000......000

...

since for all n, the list of digits is countable, the union of all lists must also be countable (being a countable union of countable sets, an easy theorem of set theory).

He concludes: R is countable. QED (or is it?)
==================================

Did you spot the mistake? Do you know of any other such "proofs"? And how could this get into TWSLA? Is it not a refereed journal?
 
I don't understand the premise! "only no digit after the decimal" means what?

Uhh, I guess my favourite would be the one where you prove 1=2, but you do it by dividing by zero, or something.
 
Mr. Do said:
I don't understand the premise! "only no digit after the decimal" means what?

that should read "no digit after the decimal". I did a bad copy and paste from my computer.
 
since for all n, the list of digits is countable, the union of all lists must also be countable (being a countable union of countable sets, an easy theorem of set theory).

Could the fallacy be that the number, n, of lists, is not countable?
 
pboily said:
that should read "no digit after the decimal". I did a bad copy and paste from my computer.

Yeah, I still don't get it :blush: Why does 1 have no digit after the decimal but 2 does? Do I need to go read a maths textbook to understand?
 
I forgot to define what countable means:

a set X is countable if there is a bijection from the N to X, i.e.,

that X can be written as {x_1,x_2,x_3,....}, without missing any of the elements of the set.

For instance, the set of positive integers is countable, since it can be written as N={1,2,3,....}.

Similarly, the set of integers is countable as it can be written

Z={0,1,-1,2,-2,3,-3,...,n,-n,...}
 
Mr. Do said:
Yeah, I still don't get it :blush: Why does 1 have no digit after the decimal but 2 does? Do I need to go read a maths textbook to understand?

No, no. 0 has no (repeating) digit following the decimal, but 0.112 has three digits following the decimal.
 
Is it that the interval he has proven countable is not the same as R, ie in the Set R there exist transindental etc. numbers for which n can be proven infinite? (or something like that)
 
Re: the spoilered hint-

That was my first thought, but if n continues to infinity without a problem, repeating decimals will be included eventually.
 
I hate math its my worst subject. Too bad everything in the computer field requires lots of math. :(
 
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