# [isabelle] The presentation of Goedel's First Incompleteness Theorem by Lawrence C. Paulson (and others)

```Dear Gottfried Barrow,

There has been a misunderstanding. I claimed neither that Isabelle is
inconsistent nor the contrary.

However, a proof of Paulson's claimed theorem 'proved_iff_proved_PfP' is not
available in axiomatic (Hilbert-style) deductive systems, because in
Hilbert-style systems this claimed theorem cannot even be expressed, since it
does not contain well-formed formulae only. Therefore this claimed theorem is
not a mathematical theorem or metatheorem. For now, please allow me to focus on
this single point.

This can be demonstrated easily by looking at the structure of the claimed
theorem 'proved_iff_proved_PfP' available at
http://www.cl.cam.ac.uk/~lp15/papers/Formath/Goedel-ar.pdf (p. 21)
http://www.owlofminerva.net/files/Goedel_I_Extended.pdf (p. 19)

which can be written
{} > a  <->  {} > PfP "a"

if we use '>' for the deduction symbol (turnstile) and '"' for the Goedel
encoding quotes, and simplified without change of meaning to
> a  <->  > PfP "a"

expressing that 'a' is a theorem if and only if there is a proof of 'a'.

Now, recalling the quotes by Alonzo Church and Peter B. Andrews available at
https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2015-December/msg00018.html
https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2015-December/msg00042.html

in an axiomatic (Hilbert-style) deductive system, the claimed theorem
'proved_iff_proved_PfP' could be either a theorem (a theorem of the object
language) or a metatheorem (a theorem of the mathematical meta-language).

Case 1 (theorem of the object language, language level 1):

As a trivial example for a theorem of the object language, we shall use
(T & T) = T

usually written
> (T & T) = T

as presented in [Andrews, 2002, p. 220 (5211)], with the preceding deduction
symbol (turnstile) in order to express that '(T & T) = T' is a theorem.

In this notation it has, like all theorems of the object language of Q0,
exactly one occurrence of the deduction symbol (turnstile).
Hence, the claimed theorem 'proved_iff_proved_PfP', having two occurrences,
cannot be a theorem of the object language.

Case 2 (theorem of the meta-language, language level 2):

As a trivial example for a theorem of the meta-language, we shall use
If H > A and H > A => B, then H > B.

presented in [Andrews, 2002, p. 224 (5224 = Modus Ponens)], expressing that if
there is a proof of A (from the set of hypotheses H) and a proof of A => B
(from H), then there is a proof of B (from H).

Note that this example shows some of the typical formal criteria of a
metatheorem:
1. Multiple occurrences of the deduction symbol (turnstile).
2. Use of syntactical variables (denoted by bold letters in the works of both
Church and Andrews).
3. Use of the informal words "If" and "then" instead of logical symbols in the
meta-language (according to Church's proposal).

It should be emphasized that metatheorems in proofs can always be replaced by
the proof of the concrete theorems (the syntactical or schematic variable
instantiated) when carrying out the proof, such that metatheorems are actually
not necessary (but reveal properties of the object language that help finding
proofs).

In the notation of Isabelle (natural deduction) this metatheorem would be
expressed by
[H > A; H > A => B]  -->  H > B

and, if we would add subscripts for the language levels, by
[H >1 A; H >1 A => B]  -->2  H >1 B

So metatheorems infer from theorems of the object language (language level 1)
to another theorem of the object language, and this relation between theorems
of the object language is expressed at a higher level: the meta-language
(language level 2).

But the claimed theorem 'proved_iff_proved_PfP'
> a  <->  > PfP "a"

cannot be a metatheorem either, since both ways of dealing with it, either
semantically (subcase a) or syntactically (subcase b), fail.

Case 2 Subcase a (semantically):

In the claimed theorem 'proved_iff_proved_PfP'
> a  <->  > PfP "a"

the right-hand side (PfP "a"), expressing the provability of theorem 'a', is,
by its meaning, itself a metatheorem, not a theorem of the object language, and
we would have some kind of meta-metatheorem like
>1 a  <->3  >2 PfP "a"

If we ignore the fact that his meta-metatheorem violates the language level
restrictions and nevertheless proceed further, then from a theorem of the
object language a theorem of the meta-language could be inferred and vice
versa, which would again violate language level restrictions, as for example a
metatheorem would be added to the list of theorems of the object language and
treated as such, leading to a confusion of language levels.

This is, in principle, the construction of the proofs of Andrews and Rautenberg
[cf. Kubota, 2013], in which 'proved_iff_proved_PfP' is used as an implicit
rule, not as a proven theorem/metatheorem. Of course, they both fail also
simply by not providing a proof using syntactical means only.

Case 2 Subcase b (syntactically):

In Paulson's claimed theorem 'proved_iff_proved_PfP'
> a  <->  > PfP "a"

the right-hand side (PfP "a") needs to be a well-formed formula.

But the Goedel encoding in Paulson's proof is implemented by the use of means
which are not available in the object language (i.e., in mathematics).

According to Paulson at
http://www.cl.cam.ac.uk/~lp15/papers/Formath/Goedel-ar.pdf (p. 16)

"[i]t is essential to remember that GÃdel encodings are terms (having type tm),
not sets or numbers. [...] First, we must define codes for de Bruijn terms and
formulas.

function quot_dbtm :: "dbtm -> tm"
where
"quot_dbtm DBZero = Zero"
[...]"

Paulson's definition goes beyond the use of purely mathematical means. After
the introduction of the definition of quot_dbtm, it is used as follows:

"We finally obtain facts such as the following:
lemma quot_Zero: "'Zero' = Zero"
[...]"

But with its purely syntactical means the object language cannot explicitly
reason about its own properties directly.
Propositions in Andrews' logic Q0 have type 'o' (Boolean truth values), and one
could define a function 'foo': o -> o, o -> i, or o -> nat (with nat = (o(oi))
= (i -> o) -> o; natural numbers as "equivalence classes of sets of
individuals" [Andrews, 2002, p. 260]), etc.
But since a type "tm" (for term) does not exist in Q0 [cf. Andrews, 2002, p.
210] or R0, one cannot define a mathematical function 'quot_dbtm': dbtm -> tm.
Of course, there are rules for construing well-formed formulae (wffs), but
these (in R0 hardcoded) rules are used implicitly for construing wffs and are
not part (are not theorems) of the object language itself.
Explicit meta-reasoning as with lemma 'quot_Zero' might extend (and, hence,
violate) the object language, as it actually introduces new rules of inference
to the object language, which again may be considered as a confusion of
language levels.

Type "tm" (for term) is a notion of the (technical) meta-language, but not a
mathematical type. Therefore the function 'quot_dbtm' (type: dbtm -> tm) is not
a mathematical well-formed formula (wff), subsequently the Goedel encoding
function ('" "') and the Goedel encoding of proposition 'a' ('"a"') are not
either, and hence the right-hand side (PfP "a") is not a wff and therefore not
a proposition. Finally,
PfP "a"
or
> PfP "a"
cannot be a theorem, and for this reason the claimed theorem
'proved_iff_proved_PfP'
> a  <->  > PfP "a"
cannot be a metatheorem.

Obviously the (technical) meta-language and the object language in Isabelle are
not strictly separated, since the type "tm" (for term) is treated as a
mathematical type in the construction of wffs of the object language, which is
not mathematically safe. Mathematically, a proposition has only type 'o'
(Boolean truth values), but not a type "tm" (for term).

All definitions of Q0 are only shorthands for established wffs. In my R0
implementation, a definition label added to a wff is used for input (parsing)
and output (display) only, and remains irrelevant for syntactical inference and
can be removed or replaced at any time. This means that a definition label for
Goedel encodings (in this case the quotation marks) must represent a
mathematical well-formed formula (wff) when used in 'proved_iff_proved_PfP',
which may be a function with a mathematical type as input (domain) type such as
the type of truth values (type: o -> *), but not with types of the
meta-language as input (domain) type (type: dbtm -> *, or tm -> *), as this
violates the rules for construing mathematical wffs [cf. Andrews, 2002, p. 211].

Of course, one could introduce Goedel numbering in order to arithmetize the
object language and reason about the Goedel numbers. But the reasoning would
then be restricted to these Goedel numbers, and there would be no possibility
to relate these Goedel numbers directly to theorems of the object language as
done in the claimed theorem 'proved_iff_proved_PfP'
> a  <->  > PfP "a"

since the Goedel encodings in Paulson's proof (requiring a type "tm") are not
definable with purely mathematical means of the object language (e.g., in R0).
Since the proposition 'a' has only type 'o' (Boolean truth values), the logical
arithmetic is not stronger than propositional calculus, ruling out Goedel
encodings requiring a type "tm".

The concept of the Goedel encoding function generally violates the type
restrictions for construing mathematical wffs, as with the type of truth values
as input (domain) type there would be only two different Goedel numbers.

As in other claimed proofs, non-mathematical means are used in order to
establish a relation between the object language (proposition 'a') and the
meta-language (its provability: PfP "a") as the translation mechanism between
both language levels necessary for the self-reference part of the antinomy,
since Goedel's antinomy is construed across language levels. Note that the
antinomy seems to cause inconsistency in axiomatic (Hilbert-style) deductive
systems, but not necessarily in natural deduction [cf. Kubota, 2015, p. 14].

References

Andrews, Peter B. (2002), An Introduction to Mathematical Logic and Type
Theory: To Truth Through Proof. Second edition. Dordrecht / Boston / London:
Kluwer Academic Publishers. ISBN 1-4020-0763-9. DOI: 10.1007/978-94-015-9934-4.

Kubota, Ken (2013), On Some Doubts Concerning the Formal Correctness of GÃdel's
Incompleteness Theorem. Berlin: Owl of Minerva Press. ISBN 978-3-943334-04-3.
DOI: 10.4444/100.101. See: http://dx.doi.org/10.4444/100.101

Kubota, Ken (2015), GÃdel Revisited. Some More Doubts Concerning the Formal
Correctness of GÃdel's Incompleteness Theorem. Berlin: Owl of Minerva Press.
ISBN 978-3-943334-06-7. DOI: 10.4444/100.102. See:
http://dx.doi.org/10.4444/100.102

____________________

Ken Kubota
doi: 10.4444/100
http://dx.doi.org/10.4444/100

```

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