C:\Documents and Settings\dave\Desktop\Godel, Escher, Bach ...

the day: how to prove a system consistent. Gödel found a simple way to express the
statement "TNT is consistent" in a TNT formula; and then he showed that this formula
(and all others which express the same idea) are only theorems of TNT under one
condition: that TNT is inconsistent. This perverse result was a severe blow to optimists
who expected that one could find a rigorous proof that mathematics is contradiction-free.
How do you express the statement "TNT is consistent" inside TNT It hinges on
this simple fact: that inconsistency means that two formulas, x and x, one the negation of
the other, are both theorems. But if both x and -- x are theorems, then according to the
Propositional Calculus, all well-formed formulas are theorems. Thus, to show TNT's
consistency, it would suffice to exhibit one single sentence of TNT which can be proven
to be a nontheorem. Therefore, one way to express "TNT is consistent" is to say "The
formula -0=0 is not a theorem of TNT". This was already proposed as an exercise a few
pages back. The translation is:
---3a:TNT-PROOF- PAIR{a,SSSSS
SSSSSOIa'}
223,666,111,666 S's
It can be shown, by lengthy but fairly straightforward reasoning, that-as long as TNT is
consistent-this oath-of-consistency by TNT is not a theorem of TNT. So TNT's powers
of introspection are great when it comes to expressing things, but fairly weak when it
comes to proving them. This is quite a provocative result, if one applies it metaphorically
to the human problem of self-knowledge.
TNT Is ω-Incomplete
Now what variety of incompleteness does TNT "enjoy? We shall see that TNT's
incompleteness is of the "omega" variety-defined in Chapter VIII. This means that there
is some infinite pyramidal family of strings all of which are theorems, but whose
associated "summarizing string" is a nontheorem. It is easy to exhibit the summarizing
string which is a nontheorem:
u S's
Va: 3a':
To understand why this string is a nontheorem, notice that it is extremely similar to G
itself-in fact, G can be made from it in one step (viz., according to TNT's Rule of
Interchange). Therefore, if it were a theorem, so would G be. But since G isn't a theorem,
neither can this be.
Now we want to show that all of the strings in the related pyramidal family are
theorems. We can write them own easily enough:
On Formally Undecidable Propositions 444