In article <email@example.com>, firstname.lastname@example.org (Giles Redgrave) writes: > I'm having a problem understanding Cantor's diagonal argument (CDA). > Specifically it's use in proving the uncountability of the reals from > 0 to 1. > > I don't understand why you can't apply CDA to the natural numbers > themselves. If we list the natural numbers padding to the left with > zeros like so: > > ...000 > ...001 > ...002 > ...003 > . > . > . > > and apply CDA by adding one to the nth digit (from the right) of n and > constructing our new number from these digits. > > We then have a number (consisting of an infinite series of ones) that > is not in the original list because it is different from each n in > it's nth digit. > > But it is clear that this number *is* in the list because it is a > natural number.
Classic issue. The resolution is to realize that this number is not a natural number. All natural numbers have finitely many decimal digits. This number has infinitely many.
Getting a handle on the concept of "arbitrarily large but finite" is sometimes a key to wrapping yourself around this conundrum.
The decimal expansion of any particular natural number might be arbitrarily large. But it will always be finite.
This is different from the decimal expansions of real numbers. Some (most) of those are infinite.