pboily
fingerlickinmathematickin
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):
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):
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):
...
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)
...
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'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?