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A Step by Step Guide to Gödel’s Incompleteness Proof
3: The axioms and rules of the formal system


 

 


 

Note that (provided you have JavaScript enabled) clicking on (hideshow) will reveal further details, while clicking again will hide it. Also, clicking on (hideshow Gödel’s) will reveal relevant parts of Gödel’s text (shown in green), while clicking again will hide it. Please note that older browsers may not display some symbols correctly.

 

(like this)

 

(like this)

 

If there is any difficulty in following any part of the proof, please contact me and I will try to help. And if you have any suggestions as to how this guide might be improved, please contact me. This guide is intended to be read alongside the English translation of Gödel’s original proof which can be viewed online at English translation of Gödel’s original proof or as a PDF file at English translation of Gödel’s original proof, PDF file.

 

The axioms and rules of the formal system

As is the case for the formal system P in general, the precise details of the axioms of this system are not important. It might be wondered whether the system P has sufficient axioms and rules of inference - that perhaps some detail has been omitted which renders the system not powerful enough to be complete. But this would be to miss the point - which is that Gödel’s proof can in principle be applied to any formal system (provided it contains a certain amount of basic arithmetic, such as defining natural numbers and basic operations on those numbers). So that even if there was some deficiency in the system P, the overall thrust of the argument would remain and be applicable to all formal systems which include a basic arithmetical core. That said, you still need to be familiar with the terminology that Gödel uses if you are to follow the principle of Gödel’s argument.

 

Note that, while in most logical systems in common use today, every variable in an axiom is bound by a quantifier, in Gödel’s system P, this is not the case - the axioms can have variables which are not bound by a quantifier. However, every such axiom can be converted to a formula in which all the variables are bound, as will be seen below.

 

The rules of inference of the system P

There are two rules of inference (hideshow Gödel’s) .

  1. If the formula (~(b)) ∨ (c) is an axiom or is a proven formula, and b is an axiom or is proved, then the formula c is proved.
  2. If the formula a is an axiom or is a proven formula, then the formula v∀(a) is proved, where v is any variable.

“A formula c is called an immediate consequence of a and b, if a is the formula (~(b)) ∨(c), and an immediate consequence of a, if c is the formula v∀(a), where v denotes any given variable. The class of provable formulae is defined as the smallest class of formulae which contains the axioms and is closed with respect to the relation ‘immediate consequence of’. ”

 

Hence, given an axiom or proven formula whose variables are not bound by a quantifier, by repeated application of rule 2 we can obtain the same formula where each variable is bound by a quantifier.

 

The axioms of the system P

Gödel divides the axioms into five sections.

 

Note that the axioms II, III, and V are actually Axiom Schemas; this means that each ‘axiom’ actually represents infinitely many axioms, which are given by replacing the appropriate variables in the schemas by specific values. For example, for the Axiom Schema II.1 below (p ∨ p ⊃ p), we could replace p by any formula of the formal system to give a single axiom.

 

Note: the axioms as given include some symbols which are not actually symbols of the formal system P. The reason for this is as follows:

The system P uses a very small set of symbols. There are several symbols in common usage which are effectively abbreviations for long combinations of the basic symbols of the system P. Gödel uses these symbols because the actual representation in the notation of the system P would be very lengthy and difficult to read. The symbols Gödel uses are listed is below, along with the equivalent representation by symbols of the formal system P:

 

 

Note that Gödel also uses the symbol . in the axiom I.3; this is equivalent to the symbol.

 

Note that, for convenience Gödel follows convention in omitting some brackets that would be present in the actual corresponding formula of the formal system. For example, in the following, wherever the symbol (the ‘implies’ symbol) occurs, all of the expression to the left of the symbol implies all of the expression to the right of the symbol, unless that part of the expression containing the symbol is enclosed by brackets.

 

 

Axioms I:

These are elementary axioms about natural numbers.

 

1. ~(fx1 = 0)

No successor of a number can be equal to zero.

 

2. fx1 = fy1 ⊃  x1 = y1

Given two numbers, if we add one to each number, and the resultant numbers are equal, then that implies (the symbol) that the original numbers are both equal.

 

3. x2(0) ∧ x1∀(x2(x1) ⊃ x2(fx1))  ⊃  x1∀(x2(x1))

This is what is usually called an axiom of induction. Given any class of natural numbers, if 0 is a member of that class, and if, for every natural number x1, if x1 being a member of that class implies that x1 + 1 is also a member of that class, then every natural number is a member of that class.

 

Axioms II:

These are Axiom Schemas, based on axioms of classical propositional logic. In the Schemas below p, q and r can be any formula of the formal system.

 

1. p ∨ p  ⊃  p

For any formula p, p or p implies p.

 

2. p  ⊃  p ∨ q

For any formula p, p implies ‘p or any other formula q’.

 

3. p ∨ q  ⊃  q ∨ p

For any formulas p and q, ‘or q’ implies ‘q or p’.

 

4. (p ⊃ q)  ⊃  (r ∨ p  ⊃  r ∨ q)

For any formulas p, q and r, ‘p implies q’ implies that ‘(r or p) implies (r or q)’

 

Axioms III:

In these Axiom Schemas,

a is any formula,

v is any variable,

b is a formula and

c is a sign;

b and c are subject to certain conditions (hideshow Gödel’s).

 

1. v∀(a)  ⊃  Subst a(v|c)

This states that, if for all v, formula a applies, then every formula given by the substitution of a valid value c for v applies.

 

As Gödel notes, the Subst function is a function of the meta-language, not of the formal system P. So while there is no expression of the formal system that corresponds to the above expression when values are simply inserted for a, b, c, and v, there are formulas that correspond to the expressions given when appropriate values are inserted for a, b, c, and v and the value given by the meta-language function Subst a(v|c) (which is a symbol string of the system P) is substituted in place of Subst a(v|c).

 

Note that normally we use the term substitution to refer to the substitution of a variable by a specific value which is a member of the domain of the variable. Here Subst allows variables to be substituted by variables, provided they are of the same type, subject to certain conditions; this is akin to changing the names of the variables in a formula.

 

2. v∀(b ∨ a)  ⊃  b ∨ v∀(a)

 

This states that, ‘if for all v, b or a’ applies, then ‘b or for all v, a’ applies (provided that v is not a free variable in b) .

 

“Every formula derived from the two schemata by making the following substitutions for a, v, b, c (and carrying out in I the operation denoted by “Subst”): for a any given formula, for v any variable, for b any formula in which v does not appear free, for c a sign of the same type as v, provided that c contains no variable which is bound in a at a place where v is free.
c is therefore either a variable or 0 or a sign of the form f…fu where u is either 0 or a variable of type 1.”

 

Axiom IV:

In this Axiom Schema,

v and u are variables, where u is one type higher than v, and

a is formula which does not have u as free variable

 

1. (∃u)(v∀(u(v)≡a))

For every formula, there is a corresponding class whose members satisfy that formula.

 

Axiom V:

In this axiom we have type 1 and type 2 variables.

 

1. x1∀(x2(x1) ≡ y2(x1))  ⊃  x2 = y2

If two classes have precisely the same members, then they are identical.

 

Further axioms are defined by this axiom by applying Type-lift to the basic formula above.

 

 

Footnotes:

 


 

 


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The Lighter Side

 

NEWS

Peter Smith’s ‘Proof’

It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.

 

 

There’s something about Gödel by Francesco Berto

There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.

 

 

Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

 

 

O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

 

 

New page on Chaitin’s Constant

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

 

Previous Blog Posts  

 

16th Mar 2015 Bishops Dancing with Pixies?

 

23rd Feb 2015 Artificial Intelligence

 

31 Mar 2015 Cranks and Crackpots

 

30 Apr 2015 The Chinese Room

 

Links  

 

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

 

– a page with general links:

Gödel Links

 

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

 

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