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

Dear Ken,

Please forgive me if my tone may seem condescending. I am not an expert in Goedel's theorems, but I have a lot of experience with Isabelle and I have looked at the incriminated proofs. 

First, please let me assure you of the following, concerning Larryâs proofs: The employed concepts and the proved theorems do not suffer from any consequence of Isabelle/HOL being based on a meta-logic or featuring a natural deduction system. These concepts and theorems are easily reproducible in "pure" systems, lacking a logical framework on top of them â including all the HOL and ZF systems, be they based on natural deduction, sequent calculi or Hilbert systems. Any person having experience with logical frameworks and with Isabelle would give you this assurance after superficially inspecting these proofs. The logical framework merely offers some convenience. 

Now, I say letâs not be dogmatic about how Goedelâs Second has been proved. The fact that Larry and authors before him took some shortcuts, working at an outer level for as much as possible, is perfectly legitimate. It forms the beauty of taming a proof of huge complexity. Again, I ensure you that there is nothing exotic with the employed primitives that led to the stated theorem. In my opinion, the only question worth asking is if the end statement is truthful:  

theorem Goedel II: 
assumes "Â {} â Fls" 
shows "Â {} â Neg (PfP âFlsâ)" 

I hope we agree that the conclusion of the theorem (the "shows" part above) should be "Â {} â Neg phi" where phi is a formula that captures the notion "False is provable in the first-order logic of HF,"  as represented in the language of HF itself. Here, one may argue that taking phi to be "PfP âFlsâ" is not a priori convincing, especially since the definitions leading to PfP are very technical and are spanning two levels. These aspects could be clarified by unfolding the definitions and performing some simplifications that "reify" this term. But I am not sure: Is this what you are arguing?

With best wishes, 

-----Original Message-----
From: cl-isabelle-users-bounces at [mailto:cl-isabelle-users-bounces at] On Behalf Of Ken Kubota
Sent: 22 December 2015 09:03
To: cl-isabelle-users at
Subject: [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 (p. 21) (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

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
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 (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 

function quot_dbtm :: "dbtm -> tm"
	"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"
	> PfP "a"
cannot be a theorem, and for this reason the claimed theorem 
	> 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].


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:

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:


Ken Kubota
doi: 10.4444/100

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