7/25/2019 On Formally Undecidable Propositions of TNT and Related Systems

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On Formally Undecidable Propositions of TNT and Related Systems

By Jos Emmanuel Sainz Jaime

The preceding discussion got us to the point where we saw how TNT can "introspect" on the notion

of TNT -theoremhood. This is the essence of the first part of the proof. We now wish to press on to

the second major idea of the proof, by developing a notion which allows the concentration of this

introspection into a single formula. To do this, we need to look at what happens to the Godel

number of a formula when you modify the formula structurally in a simple way. In fact, we shall

consider this specific modification: replacement of all free variables by a specific numeral.

We now have reached the crucial point where we can combine all of our disassembled parts into

one meaningful whole. We want to use the machinery of the TNT-PROOF-PAlR and SUB formulas in

some way to construct a single sentence of TNT whose interpretation is: "This very string of TNT is

not a TNT -theorem." How do we do it? Even at this point, with all the necessary machinery in front

of us, the answer is not easy to find.

A curious and perhaps frivolous-seeming notion is that of substituting a formula's own G6del

number into itself. This is quite parallel to that other curious, and perhaps frivolous-seeming, notion

of "quining" in theAir on C's String. Yet quining turned out to have a funny kind of importance, in

that it showed a new way of making a self-referential sentence. Selfreference of the Quine varietysneaks up on you from behind the first time you see it-but once you understand the principle, you

appreciate that it is quite simple and lovely. The arithmetical version of quining-let's call it

arithmoquining-will allow us to make a TNT-sentence which is "about itself".

Now if you look back in the Air on G's String, you will see that the ultimate trick necessary for

achieving self-reference in Quine's way is to quine a sentence which itself talks about the concept

of quining. It's not enough just to quine-you must quine a quine-mentioning sentence! All right,

thenthe parallel trick in our case must be to arithmoquine some formula which itself is talking about

the notion of arithmoquining.

Since G's interpretation is true, the interpretation of its negation -G is false. And we know that no

false statements are derivable in TNT. Then us neither G nor its negation -G can be a theorem of

TNT. We have found a "hole" in our system-an undecidable proposition. This has a number oframifications.

Here is one curious fact which follows from G's undecidability: although neither G nor -G is a

theorem, the formula is a theorem, since the rules of the Propositional Calculus ensure that

all well-formed formulas of the form are theorems.

This is one simple example where an assertion inside the system and an assertion about the system

seem at odds with each other. It makes one wonder if the system really reflects itself accurately.

Does the "reflected

metamathematics" which exists inside TNT correspond well to the metamathematics which we do?

This was one of the questions which intrigued Godel when he wrote his paper. In particular, he was

interested in whether it was possible, in the "reflected metamathematics", to prove TNT's

consistency. Recall that this was a great philosophical dilemma of the day: how to prove a system

consistent. Godel 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.