Oh drat. There’s a very silly thinko in Gödel Without Tears. On p. 28, I say that the logical vocabulary of BA, basic arithmetic, comprises just the identity predicate and negation. On p. 29 — yes, the very next page — I’m using the conditional in giving a schema for the axioms of BA; and the conditional features essentially in e.g. a BA derivation on p. 32. But then on p. 33 — yes, the very next page — it is plainly said that the only wffs of BA are equations and their negations, and the conditional has been forgotten about again.
I think I can see that happened. I wanted to slightly simplify the presentation of BA from that in An Introduction to Gödel’s Theorems and in earlier versions of GWT; and I inattentively did so in an internally incoherent way. How annoying.
The current version of GWT with the flawed argument has been downloaded some 4500 times, while over 1400 pointed copies have been sold, and no one has noticed — or at least noticed and thought to tell me — until yesterday. So many thanks then to Ben Selfridge for pointing out the foul-up.
I’ll think about the neatest way of clearing up the mess on the corrections page for the book. But then it will be time, given that there is already a handful of other known typos, for (at least) a corrected reprint for the book. So if you know of any other glitches in the book, now is the time to tell me!
“To make no mistakes is not in the power of man.” Too true, Plutarch, too true. “But from their errors and mistakes the wise and good learn wisdom for the future.” Let’s hope.
Added: the corrections page for GWT has now been updated.