# [FOM] 18 Word Proof of the Godel, Rosser and Smullyan Incompleteness Theorems

Harvey Friedman friedman at math.ohio-state.edu
Fri Jul 16 00:32:34 EDT 2010

```On Jul 11, 2010, at 9:10 PM, Charlie V wrote:

> Godel’s 1931 First Incompleteness Theorem is equivalent to the
> assertion that truth and provability do not coincide.

This is incorrect. You are ignoring hidden assumptions, which is an
important consideration for 18 word proofs.

The usual statements are

1. There is a true sentence that is not provable.
2. There is a sentence that is neither provable nor refutable.

To get 1, your argument requires that a) every provable sentence is
true, b) the set of provable sentences is recursively enumerable.
To get 2, your argument also requires that a) every provable sentence
is true, b) the set of provable sentences is recursively enumerable.

But then you have the problem of not having said enough about truth
and provability for b) to make sense. Then once you explain what b)
means, other issues arise concerning a), and you can't even begin to

So in addition to this not being satisfactory definitionally, it is
not sensitive to the weaker assumptions used by Goedel and Rosser,
which in turn lead to stronger theorems involving general classes of
formal systems:

Goedel's proof of 1 for PA requires less - that no sentence and its
negation are provable.

Goedel's proof of 2 for PA requires less - that every provable Sigma01
sentence is true.

Rosser proved 2 for PA using even less - that no sentence and its
negation are provable.

Furthermore, R. Robinson identified a very weak finite fragment Q of
PA such that what Rosser proved above holds with PA replaced by any
extension of Q.

> Rosser’s 1936 extension is equivalent to the assertion that
> provability and unrefutability do not coincide.  Smullyan’s 1961
> Dual Form Theorem is equivalent to the assertion that truth and
> unrefutability do not coincide.
>
> We can prove that these three sets, the true, provable and
> unrefutable sentences, are three different sets by referring to the
> property of being recursively enumerable, as stated in theorems of
> the Theory of Computation:
>
> The sets of true, provable and unrefutable sentences pairwise differ
> w.r.t. whether they or their complement is r.e.

We can resurrect what you are trying to say if we focus only on
systems like PA - and take the realist viewpoint.

Of course, the relevant recursion theory was not around when Goedel
proved his incompleteness theorems.

Once you try to make what you say mathematically rigorous, the usual
complexities enter in as to the proper formulations.

Harvey Friedman

```