Selmer Bringsjord
Question 1: Is it possible that the machine can print all true sentences? No.
Here is a true sentence that the machine cannot print:
: 
Question 2: How can it be proved that is true but can't
be printed?
Proof. is true iff the mirror of 
cannot be printed.
So either is true and not printable, or false and printable. The
second possibility violates the assumption that the machine 
never prints false sentences. Thus is true but unprintable. QED
Questions 3 & 4: What about 
-- can
it be printed? Why?
This sentence is false since its negation -- -- is true. Since
it's false, it cannot, by hypothesis, be printed by .
Now, replace our alphabet with
and identify natural numbers with their correlates in binary notation.
To each expression we assign the Gödel number of the expression. We do this according to the following scheme:
The mirror of an expression 
is the expression followed
by its Gödel number.
A sentence is an
expression having one of the following four forms (where 
is any
number): 
, 
, 
, 
.
Naturally, is true iff is the Gödel number of a
printable expression, is true iff is the Gödel number
of an expression whose mirror is printable, and so on.
Question 1: Given once again that our machine never prints a false sentence, is there a true sentence the machine cannot print? If so, what is it?