A Step by Step Guide to
Gödel’s Incompleteness Proof:
3: Axioms and rules of the formal system
3: Axioms and rules of the formal system
Page last updated 28 Dec 2022
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.
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.
- 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.
- 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:
|⊃ ‘implies’||a ⊃ b is equivalent to (¬a) ∨ b|
|∃ ‘there exists’||(∃u)b is equivalent to ¬(u∀(¬b))|
|≡ ‘equivalence’||a ≡ b is equivalent to (a ⊃ b) ∧ (b ⊃ a)|
|= ‘equals’||x1 = y1 is defined as x2∀ (x2(x1) ⊃ x2(y1)) (Footnote: For all type 2 classes x2, if x1 is a member of that class, then y1 is a member of that class. Since this applies to all type 2 classes, then if both x1 and y1 are a member of a class that has only one member, then it must be the case that x1 is identical to y1. Similarly for higher type classes. )|
|∧ ‘and’||c ∧ d is equivalent to ¬((¬c) ∨ (¬d))|
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.
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.
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, ‘p 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)’
In these Axiom Schemas,
a is any formula,
v is any variable,
b is a formula and
c is a string;
b and c are subject to certain conditions.
“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 string 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 string of the form f…fu where u is either 0 or a variable of type 1.”
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).
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
For every formula, there is a corresponding class whose members satisfy that formula.
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.
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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