Further results and examples for formal mathematical systems with structural induction
aa r X i v : . [ m a t h . L O ] S e p FURTHER RESULTS AND EXAMPLES FORFORMAL MATHEMATICAL SYSTEMSWITH STRUCTURAL INDUCTION
MATTHIAS KUNIK
Abstract.
In the former article “Formal mathematical systemsincluding a structural induction principle” we have presented aunified theory for formal mathematical systems including recursivesystems closely related to formal grammars, including the predi-cate calculus as well as a formal induction principle. In this paperwe present some further results and examples in order to illustratehow this theory works.
Keywords:
Formal mathematical systems, elementary prooftheory, languages and formal grammars, structural induction.Mathematics Subject Classification: 03F03, 03B70, 03D03, 03D051.
Introduction
In this article I refer to my former work [2], which is inspired by Hof-stadter’s book [1] as well as by Smullyan’s “Theory of formal systems”in [3].The recursive systems introduced in [2, Section 1] may be regardedas variants of formal grammars, but they are better adapted for usein mathematical logic and enable us to generate in a simple way therecursively enumerable relations between lists of terms over a finitealphabet, using the R-axioms and the R-rules of inference. The R-axioms of a recursive system are special quantifier-free positive hornformulas. In addition, the recursive system contains R-axioms for the
Date : September 15, 2020. use of equations. Three R-rules of inference provide the use of R-axioms, the Modus Ponens Rule and a simple substitution mechanismin order to obtain conclusions from the given R-axioms.In Section 2 of the paper on hand we present the first example of arecursive system which represents the natural numbers in two differentways. The recursive system generates a specific relation between thedual representation of any natural number n and its representation asa tally a n = a . . . a of length n with the single symbol a .In [2, Section 3] a general recursive system S is embedded into aformal mathematical system [ M ; L ] based on the predicate calculusand a formal induction principle. The set L of restricted argumentlists contains the variables and is closed with respect to substitutions.The embedding is consistent in the sense that the R-axioms of S withargument lists in L will become special axioms of [ M ; L ] and that theR-rules of inference with substitutions of lists restricted to L will bespecial rules of inference in [ M ; L ]. The formal structural induction ina mathematical system is performed with respect to the axioms of theunderlying recursive system S , and the formal induction principle forthe natural numbers is a special case.The three examples in Section 3.1 make use of the axioms and rulesfor managing formulas with quantifiers in the formal mathematical sys-tems, and namely the first example is needed in Section 3.6.In Section 3.2 we will present some technical results concerning thesubstitution of variables in formulas, because substitutions in formulaswith quantifiers need special care.The first example with formal induction involving equations will bepresented in Section 3.3. The underlying recursive system of [ M ; L ]is simple and similar to that in Section 2, but its R-axioms containequations, and the formulas which we will deduce in [ M ; L ] need moreeffort than it seems at a first glance.In Section 3.4 we present a simple procedure in order to eliminatecertain prime formulas from formal proofs which do not occur with agiven arity in the basis axioms of the mathematical system. In [2, Section 5] we have stated Conjecture (5.4) and especially haveproved that it implies the consistency of PA. Conjecture (5.4) charac-terizes the provability of variable-free prime formulas in special axioma-tized mathematical systems [ M ; L ] whose basis-axioms coincide withthe basis R-axioms of their underlying recursive systems. In Section3.5 of the paper on hand we present a proof of this conjecture. At leastunder a natural interpretation of the formulas we will show that theaxioms and rules of inference including the Induction Rule (e) from [2,(3.13)(e)] correspond to correct methods of deduction.In Section 3.6 we will come back to the recursive system S fromSection 2 and will present another instructive example for the use ofthe Induction Rule (e).2. Recursive systems
For the preparation of this section we need [2, (1.1)-(1.12)]. Thererecursively enumerable relations are defined. These are special relationsbetween lists of symbols, and they are generated in a very simple wayby three rules of inference, namely Rules (a), (b) and (c) given in (1.11).We start with the following example:2.1.
Dual representation of natural numbers.
We consider the recursive system S = [ A ; P ; B ] with A = [ a ; 0; 1], P = [ D ], with distinct variables x, y ∈ X and with B consisting of thefollowing six basis R-axioms:( α ) D β ) → D x D x γ ) → D x D x δ ) D , a ( ε ) → D x, y D x , yy ( ζ ) → D x, y D x , yya The 1-ary predicate
D x represents natural numbers x in dual form.The 2-ary predicate D x, y gives the dual representation x of a naturalnumber y = a n , represented as a tally with the single symbol “ a ”. Note MATTHIAS KUNIK that the predicate symbol “ D ” is used 1-ary as well as 2-ary within thesame recursive system S , which has to be mentioned separately in eachcase. There results another recursive system ˜ S if we replace the firstthree R-axioms by a single one → D x, y D x . The elementary primeR-formulas derivable in S and ˜ S are the same. In Section 3.6 we willcome back to the recursive system S and will present an instructiveexample for a mathematical system with formal induction.Now we present an R-derivation of the formula D , aaaaa in therecursive system S . It means that 5 (represented by a = aaaaa ) hasthe dual representation 101:(1) D , a Rule (a) and ( δ ).(2) → D x, y D x , yy Rule (a) and ( ε ).(3) → D x, y D x , yya Rule (a) and ( ζ ).(4) → D , y D , yy Rule (c), (2) with x = 1.(5) → D , a D , aa Rule (c), (4) with y = a .(6) D , aa Rule (b), (1) and (5).(7) → D , y D , yya Rule (c), (3) with x = 10.(8) → D , aa D , aaaaa Rule (c), (7) with y = aa .(9) D , aaaaa Rule (b), (6) and (8).3.
Formal mathematical systems
For the preparation of this section we need [2, (3.1)-(3.15)]. In [2, Sec-tion 3] a recursive system S is embedded into a formal mathematicalsystem M . This embedding is consistent in the sense that the R-axiomsof S will become special axioms of M and that the R-rules of inferencewill be special rules of inference in M . In [2, (3.13)] we use five rulesof inference, namely Rules (a)-(e). Rule (e) enables formal inductionwith respect to the recursively enumerable relations generated by theunderlying recursive system S .In [2, (3.15)] formal mathematical systems [ M ; L ] with restrictions inthe argument lists of the formulas are introduced. The set of restrictedargument lists L contains the variables and is closed with respect tosubstitutions. Generally valid formulas with quantifiers.Example 1:
This is needed in Section 3.6. Let F be a formula of amathematical system [ M ; L ] and let x ∈ X be a variable. Then weobtain the following proof of the generally valid formula → F ∃ x F in[ M ; L ], using the rules in [2, (3.13)].(1) → ∀ x ¬ F ¬ F Rule (a), quantifier axiom (3.11)(a).(2) → → ∀ x ¬ F ¬ F → F ¬ ∀ x ¬ F Rule (a) with the identically true propositional function → → ξ ¬ ξ → ξ ¬ ξ .(3) → F ¬ ∀ x ¬ F Rule (b), (1), (2).(4) ↔ ¬ ∀ x ¬ F ∃ x F Rule (a), quantifier axiom (3.11)(c).(5) → → F ¬∀ x ¬ F → ↔ ¬ ∀ x ¬ F ∃ x F → F ∃ x F Rule (a) with the identically true propositional function → → ξ ξ → ↔ ξ ξ → ξ ξ .(6) → ↔ ¬ ∀ x ¬ F ∃ x F → F ∃ x F Rule (b), (3), (5).(7) → F ∃ x F Rule (b), (4), (6).
Example 2:
This example plays a crucial role in the proof of [2, (3.17)Theorem]. Let
F, G be formulas of a mathematical system [ M ; L ] andlet x ∈ X be a variable. Then we obtain the following proof of thegenerally valid formula → ∀ x → F G → ∀ xF ∀ xG in [ M ; L ], usingthe rules in [2, (3.13)].(1) → ∀ xF F Rule (a), quantifier axiom (3.11)(a).(2) → ∀ x → F G → F G
Rule (a), quantifier axiom (3.11)(a).(3) → → ∀ xF F → → ∀ x → F G → F G → ∀ x → F G → ∀ xF G
Rule (a) with the identically true propositional function → → ξ ξ → → ξ → ξ ξ → ξ → ξ ξ MATTHIAS KUNIK and ξ = ∀ xF ; ξ = F ; ξ = ∀ x → F G ; ξ = G .(4) → → ∀ x → F G → F G → ∀ x → F G → ∀ xF G
Rule (b), (1), (3).(5) → ∀ x → F G → ∀ xF G
Rule (b), (2), (4).(6) ∀ x → ∀ x → F G → ∀ xF G
Rule (d), (5).(7) Next we apply Rule (a), quantifier axiom (3.11)(b): → ∀ x → ∀ x → F G → ∀ xF G → ∀ x → F G ∀ x → ∀ x F G (8) → ∀ x → F G ∀ x → ∀ x F G Rule (b), (6), (7).(9) → ∀ x → ∀ x F G → ∀ x F ∀ x G Rule (a), quantifier axiom (3.11)(b).(10) → → ∀ x → F G ∀ x → ∀ x F G → → ∀ x → ∀ x F G → ∀ x F ∀ x G → ∀ x → F G → ∀ x F ∀ x G Rule (a) with the identically true propositional function → → ξ ξ → → ξ → ξ ξ → ξ → ξ ξ and ξ = ∀ x → F G ; ξ = ∀ x → ∀ x F G ; ξ = ∀ x F ; ξ = ∀ x G .(11) → → ∀ x → ∀ x F G → ∀ x F ∀ x G → ∀ x → F G → ∀ x F ∀ x G Rule (b), (8), (10).(12) → ∀ x → F G → ∀ x F ∀ x G Rule (b), (9), (11).The formulas in Example 1 and Example 2 are generally valid be-cause we have only used non-basis axioms and the rules of inference.
Example 3:
We show that x free( F ) is an essential restriction forthe quantifier axiom → ∀ x → F G → F ∀ xG in [2, (3.11)(b)]. Let a ∈ A be a constant and put F = G = ∼ x, a with x ∈ free( F ). Ignor-ing the condition x free( F ) would give the invalid “proof”(1) → ∀ x → ∼ x, a ∼ x, a → ∼ x, a ∀ x ∼ x, a incorrect use of [2, (3.11)(b)].(2) → ∼ x, a ∼ x, a Rule (a), since → ξ ξ is identically true.(3) ∀ x → ∼ x, a ∼ x, a Rule (d), (2).(4) → ∼ x, a ∀ x ∼ x, a Rule (b), (1), (3).(5) → ∼ a, a ∀ x ∼ x, a Rule (c), (4) with x = a .(6) ∼ x, x Rule (a), axiom of equality.(7) ∼ a, a Rule (c), (6) with x = a .(8) ∀ x ∼ x, a Rule (b), (5), (7).In general the result ∀ x ∼ x, a is false.3.2. Collision-free substitutions in formulas.
Let [ M ; L ] be a mathematical system with restricted argument listsin L , let F , G be formulas in [ M ; L ] and x, y, z ∈ X . We makeespecially use of [2, (3.1)-(3.7)] and want to present the whole proof of[2, Lemma (3.16)(a)] with technical details. This is necessary becausesubstitutions in formulas with quantifiers need special care. The lemmastates that we have for all x, z ∈ X with z / ∈ var( F ):(i) CF( F ; z ; x ) and(ii) CF( F zx ; x ; z ) and(iii) F zx xz = F . Proof.
We say that a formula F in [ M ; L ] satisfies the condition ( ⋆ ) ifwe have for all x ∈ X \ free( F ) and for all µ ∈ L :CF( F ; µ ; x ) and F µx = F .
We first use induction on well-formed formulas to show that condition( ⋆ ) is satisfied for all formulas F in [ M ; L ]. Afterwards we prove [2,Lemma (3.16)(a)]. MATTHIAS KUNIK (a) From [2, (3.7)(a)] and [2, (3.5)], [2, (3.6)(a)] we see that condi-tion ( ⋆ ) is satisfied for all prime formulas F .(b) If ( ⋆ ) is satisfied for a formula F , then also for the formula ¬ F due to [2, (3.7)(b)] and [2, (3.6)(b)].(c) Next we assume that F and G both satisfy condition ( ⋆ ) (induc-tion hypothesis) and put H = J F G for J ∈ {→ ; ↔ ; & ; ∨} .Let x ∈ X \ free( H ), µ ∈ L . From free( H ) = free( F ) ∪ free( G )we obtain that x ∈ X \ free( F ) as well as x ∈ X \ free( G ).We conclude from our induction hypothesis and [2, (3.7)(c)]that CF( F ; µ ; x ), CF( G ; µ ; x ) and CF( H ; µ ; x ). Next we obtainfrom our induction hypothesis and [2, (3.6)(c)] thatSbF( H ; µ ; x ) = SbF( J F G ; µ ; x )= J SbF( F ; µ ; x ) SbF( G ; µ ; x )= J F G = H . (d) Assume that F satisfies condition ( ⋆ ) (induction hypothesis)and put H = Qy F for Q ∈ {∀ ; ∃} . Let x ∈ X \ free( H ), µ ∈ L and note that free( H ) = free( F ) \ { y } . We have CF( H ; µ ; x ) immediately from [2, (3.7)(d)] and ob-tain that x ∈ X \ free( F ) or x = y . For the substitution wedistinguish two cases according to [2, (3.6)(d)]:Case 1: x = y . ThenSbF( H ; µ ; x ) = SbF( Qy F ; µ ; x ) = Qy F = H .
Case 2: x = y and x ∈ X \ free( F ). Then we obtain from ourinduction hypothesis thatSbF( H ; µ ; x ) = SbF( Qy F ; µ ; x ) = Qy SbF( F ; µ ; x ) = Qy F = H .
We have shown that condition ( ⋆ ) is valid for all formulas F in [ M ; L ].We say that a formula F in [ M ; L ] satisfies the condition ( ⋆⋆ ) if itsatisfies (i), (ii) and (iii) for all x, z ∈ X with z / ∈ var( F ). For the proofof [2, Lemma (3.16)(a)] we use induction on well-formed formulas toshow that condition ( ⋆⋆ ) is satisfied for all formulas F in [ M ; L ]. (a) From [2, (3.7)(a)] and [2, (3.5)], [2, (3.6)(a)] we see that condi-tion ( ⋆⋆ ) is satisfied for all prime formulas F .(b) If ( ⋆⋆ ) is satisfied for a formula F , then also for the formula ¬ F due to [2, (3.7)(b)] and [2, (3.6)(b)].(c) Assume that F and G both satisfy condition ( ⋆⋆ ) (inductionhypothesis) and put H = J F G for J ∈ {→ ; ↔ ; & ; ∨} . Let x, z ∈ X with z / ∈ var( H ) = var( F ) ∪ var( G ). We have z / ∈ var( F ) , z / ∈ var( G )and conclude from our induction hypothesis and [2, (3.7)(c)],[2, (3.6)(c)] that(i) ′ CF( F ; z ; x ), CF( G ; z ; x ) and hence CF( H ; z ; x ) ,(ii) ′ CF( F zx ; x ; z ), CF( G zx ; x ; z ) and hence CF( H zx ; x ; z ) ,(iii) ′ H zx xz = J SbF( F zx ; x ; z ) SbF( G zx ; x ; z ) = J F G = H . (d) Assume that F satisfies condition ( ⋆⋆ ) (induction hypothesis)and put H = Qy F for Q ∈ {∀ ; ∃} . Let x, z ∈ X , z / ∈ var( H )and note that z / ∈ var( F ) ∪ { y } , especially z = y .Case 1: We suppose that x / ∈ free( H ) . Since H satisfies theformer condition ( ⋆ ) we obtain that CF( H ; z ; x ), H zx = H , andfrom z / ∈ var( H ) that CF( H zx ; x ; z ) as well as H zx xz = H xz = H .Hence H satisfies condition ( ⋆⋆ ) in case 1.Case 2: We suppose that x ∈ free( H ) = free( F ) \ { y } . We haveCF( F ; z ; x ) from our induction hypothesis, recall that y = z and conclude CF( H ; z ; x ) from [2, (3.7)(d)ii)]. Next we usethat y = x and have from the induction hypothesis H zx = Qy F zx and CF(
F zx ; x ; z ) . We obtain CF( H zx ; x ; z ) from [2, (3.7)(d)ii)] and seeSbF( H zx ; x ; z ) = Qy SbF(
F zx ; x ; z ) = Qy F = H from z = y and the induction hypothesis. Hence H satisfiescondition ( ⋆⋆ ) in case 2.We have shown that condition ( ⋆⋆ ) is valid for all formulas F in [ M ; L ]. (cid:3) An example with formal induction and equations.
With A S := [ a ; b ; f ] and P S := [ W ] we define a recursive system S = [ A S ; P S ; B S ] by the following list B S of basis R-axioms, where x, y, s, t, u, v ∈ X are distinct variables:(1) W a (2)
W b (3) → W x → W y W xy (4) ∼ f ( a ) , a (5) ∼ f ( b ) , b (6) → W x → W y ∼ f ( xy ) , f ( y ) f ( x ) .The strings consisting of the symbols a and b are generated by theR-axioms (1)-(3). They are indicated by the predicate symbol W ,which is used only 1-ary here, whereas f denotes the operation whichreverses the order of such a string. For example, ∼ f ( abaab ) , baaba isR-derivable, and equations like ∼ f ( abaab ) , f ( aab ) ba and R-formulaslike W f ( aab ) ba are also R-derivable. The R-formula( ⋆ ) → W x ∼ f ( f ( x )) , x is not R-derivable in S . But we will show that the latter formula isprovable in the mathematical system [ M ; L ] with M = [ S ; A S ; P S ; B S ]and the set L generated by the following rules:(i) x ∈ L for all x ∈ X ,(ii) a ∈ L and b ∈ L ,(iii) If λ, µ ∈ L then λµ ∈ L ,(iv) If λ ∈ L then f ( λ ) ∈ L .The R-axioms (1)-(6) also form a proof in the mathematical system[ M ; L ] which is extended to the following proof in [ M ; L ]:(7) ∼ x, x Rule (a), axiom of equality.(8) ∼ f ( s ) , f ( s ) Rule (c), (7) with x = f ( s ).(9) → ∼ f ( s ) , f ( s ) → ∼ s, t ∼ f ( s ) , f ( t )Rule (a), axiom of equality.(10) → ∼ s, t ∼ f ( s ) , f ( t ) Rule (b), (8), (9).(11) → ∼ f ( a ) , t ∼ f ( f ( a )) , f ( t ) Rule (c), (10) with s = f ( a ).(12) → ∼ f ( a ) , a ∼ f ( f ( a )) , f ( a ) Rule (c), (11) with t = a . (13) ∼ f ( f ( a )) , f ( a ) Rule (b), (4), (12).(14) → ∼ s, t → ∼ t, u ∼ s, u Rule (a), axiom of equality.(15) → ∼ f ( f ( a )) , t → ∼ t, u ∼ f ( f ( a )) , u Rule (c), (14) with s = f ( f ( a )).(16) → ∼ f ( f ( a )) , f ( a ) → ∼ f ( a ) , u ∼ f ( f ( a )) , u Rule (c), (15) with t = f ( a ).(17) → ∼ f ( f ( a )) , f ( a ) → ∼ f ( a ) , a ∼ f ( f ( a )) , a Rule (c), (16) with u = a .(18) → ∼ f ( a ) , a ∼ f ( f ( a )) , a Rule (b), (13), (17).(19) ∼ f ( f ( a )) , a Rule (b), (4), (18).(20) → ∼ f ( b ) , t ∼ f ( f ( b )) , f ( t ) Rule (c), (10) with s = f ( b ).(21) → ∼ f ( b ) , b ∼ f ( f ( b )) , f ( b ) Rule (c), (20) with t = b .(22) ∼ f ( f ( b )) , f ( b ) Rule (b), (5), (21).(23) → ∼ f ( f ( b )) , t → ∼ t, u ∼ f ( f ( b )) , u Rule (c), (14) with s = f ( f ( b )).(24) → ∼ f ( f ( b )) , f ( b ) → ∼ f ( b ) , u ∼ f ( f ( b )) , u Rule (c), (23) with t = f ( b ).(25) → ∼ f ( f ( b )) , f ( b ) → ∼ f ( b ) , b ∼ f ( f ( b )) , b Rule (c), (24) with u = b .(26) → ∼ f ( b ) , b ∼ f ( f ( b )) , b Rule (b), (22), (25).(27) ∼ f ( f ( b )) , b Rule (b), (5), (26).(28) → ∼ s, s → ∼ s, t ∼ t, s Rule (a), axiom of equality.(29) ∼ s, s Rule (a), axiom of equality.(30) → ∼ s, t ∼ t, s Rule (b), (28), (29).(31) → ∼ f ( a ) , t ∼ t, f ( a ) Rule (c), (30) with s = f ( a ).(32) → ∼ f ( a ) , a ∼ a, f ( a ) Rule (c), (31) with t = a .(33) ∼ a, f ( a ) Rule (b), (4), (32).(34) → ∼ f ( b ) , t ∼ t, f ( b ) Rule (c), (30) with s = f ( b ).(35) → ∼ f ( b ) , b ∼ b, f ( b ) Rule (c), (34) with t = b .(36) ∼ b, f ( b ) Rule (b), (5), (35).(37) → ∼ s, t → W s W t
Rule (a), axiom of equality.(38) → ∼ a, t → W a W t
Rule (c), (37) with s = a .(39) → ∼ a, f ( a ) → W a W f ( a ) Rule (c), (38) with t = f ( a ). (40) → W a W f ( a ) Rule (b), (33), (39).(41) W f ( a ) Rule (b), (1), (40).(42) → ∼ b, t → W b W t
Rule (c), (37) with s = b .(43) → ∼ b, f ( b ) → W b W f ( b ) Rule (c), (42) with t = f ( b ).(44) → W b W f ( b ) Rule (b), (36), (43).(45) W f ( b ) Rule (b), (2), (44).(46) → W a → W f ( a ) → ∼ f ( f ( a )) , a & & W a W f ( a ) ∼ f ( f ( a )) , a Rule (a), axiom of the propositional calculus.(47) → W f ( a ) → ∼ f ( f ( a )) , a & & W a W f ( a ) ∼ f ( f ( a )) , a Rule (b), (1), (46).(48) → ∼ f ( f ( a )) , a & & W a W f ( a ) ∼ f ( f ( a )) , a Rule (b), (41), (47).(49) & &
W a W f ( a ) ∼ f ( f ( a )) , a Rule (b), (19), (48).(50) → W b → W f ( b ) → ∼ f ( f ( b )) , b & & W b W f ( b ) ∼ f ( f ( b )) , b Rule (a), axiom of the propositional calculus.(51) → W f ( b ) → ∼ f ( f ( b )) , b & & W b W f ( b ) ∼ f ( f ( b )) , b Rule (b), (2), (50).(52) → ∼ f ( f ( b )) , b & & W b W f ( b ) ∼ f ( f ( b )) , b Rule (b), (45), (51).(53) & &
W b W f ( b ) ∼ f ( f ( b )) , b Rule (b), (27), (52).At this place we stop the proof in the mathematical system [ M ; L ],introduce two different and new constant symbols c , d not occurringin [ M ; L ] and define the extension A := [ a ; b ; f ; c ; d ] of the alphabet A S . With M A := [ S ; A ; P S ; B S ] and L ′ := { λ t x ... t m x m : λ ∈ L , x , . . . , x m ∈ X, t , . . . , t m ∈ { c, d } , m ≥ } there results the mathematical system [ M A ; L ′ ] due to [2, Definition(4.2)(d)] and [2, Corollary (4.9)(a)]. Next we make use of the abbrevi-ation G ( λ ) := & & W λ W f ( λ ) ∼ f ( f ( λ )) , λ with λ ∈ L ′ and adjoin to [ M A ; L ′ ] the two statements( ⋆⋆ ) ϕ := G ( c ) , ϕ := G ( d ) . There results the extended mathematical system [ M ′ ; L ′ ] with M ′ := M A ( { ϕ , ϕ } ) = [ S ; A ; P S ; B S ∪ { ϕ , ϕ } ]due to [2, Definition (4.2)(b)]. Note that the abbreviations G ( λ ), ϕ , ϕ are not part of the formal system. We keep in mind that any proofin [ M ; L ] is also a proof in [ M ′ ; L ′ ] and that the mathematical systems[ M ; L ], [ M A ; L ′ ] and [ M ′ ; L ′ ] all have the same underlying recursivesystem S . Hence (1)-(53) also constitutes a proof in [ M ′ ; L ′ ], and weextend it to the following proof of the formula G ( cd ) in [ M ′ ; L ′ ]:(54) G ( c ) Rule (a) with axiom ϕ = G ( c ).(55) G ( d ) Rule (a) with axiom ϕ = G ( d ).(56) → G ( c ) W c
Rule (a), axiom of the propositional calculus.(57) → G ( c ) W f ( c )Rule (a), axiom of the propositional calculus.(58) → G ( c ) ∼ f ( f ( c )) , c Rule (a), axiom of the propositional calculus.(59) → G ( d ) W d
Rule (a), axiom of the propositional calculus.(60) → G ( d ) W f ( d )Rule (a), axiom of the propositional calculus.(61) → G ( d ) ∼ f ( f ( d )) , d Rule (a), axiom of the propositional calculus.(62)
W c
Rule (b), (54), (56).(63)
W d
Rule (b), (55), (59).(64)
W f ( c ) Rule (b), (54), (57).(65) W f ( d ) Rule (b), (55), (60).(66) ∼ f ( f ( c )) , c Rule (b), (54), (58).(67) ∼ f ( f ( d )) , d Rule (b), (55), (61).(68) → W c → W y W cy
Rule (c), (3) with x = c .(69) → W c → W d W cd
Rule (c), (68) with y = d .(70) → W d W cd
Rule (b), (62), (69).(71)
W cd
Rule (b), (63), (70).(72) → W f ( d ) → W y W f ( d ) y Rule (c), (3) with x = f ( d ). (73) → W f ( d ) → W f ( c ) W f ( d ) f ( c )Rule (c), (72) with y = f ( c ).(74) → W f ( c ) W f ( d ) f ( c ) Rule (b), (65), (73).(75) W f ( d ) f ( c ) Rule (b), (64), (74).(76) → W c → W y ∼ f ( cy ) , f ( y ) f ( c )Rule (c), (6) with x = c .(77) → W c → W d ∼ f ( cd ) , f ( d ) f ( c )Rule (c), (76) with y = d .(78) → W d ∼ f ( cd ) , f ( d ) f ( c ) Rule (b), (62), (77).(79) ∼ f ( cd ) , f ( d ) f ( c ) Rule (b), (63), (78).(80) → ∼ f ( cd ) , t ∼ t, f ( cd )Rule (c), (30) with s = f ( cd ).(81) → ∼ f ( cd ) , f ( d ) f ( c ) ∼ f ( d ) f ( c ) , f ( cd )Rule (c), (80) with t = f ( d ) f ( c ).(82) ∼ f ( d ) f ( c ) , f ( cd ) Rule (b), (79), (81).(83) → ∼ f ( d ) f ( c ) , t → W f ( d ) f ( c ) W t
Rule (c), (37) with s = f ( d ) f ( c ).(84) → ∼ f ( d ) f ( c ) , f ( cd ) → W f ( d ) f ( c ) W f ( cd )Rule (c), (83) with t = f ( cd ).(85) → W f ( d ) f ( c ) W f ( cd ) Rule (b), (82), (84).(86) W f ( cd ) Rule (b), (75), (85).(87) → ∼ st, st → ∼ t, v ∼ st, sv Rule (a), axiom of equality.(88) ∼ st, st Rule (c), (7) with x = st .(89) → ∼ t, v ∼ st, sv Rule (b), (87), (88).(90) → ∼ st, sv → ∼ s, u ∼ st, uv Rule (a), axiom of equality.(91) → → ∼ t, v ∼ st, sv → → ∼ st, sv → ∼ s, u ∼ st, uv → ∼ s, u → ∼ t, v ∼ st, uv Rule (a) with the identically true propositional function → → ξ ξ → → ξ → ξ ξ → ξ → ξ ξ and ξ = ∼ t, v ; ξ = ∼ st, sv ; ξ = ∼ s, u ; ξ = ∼ st, uv .(92) → → ∼ st, sv → ∼ s, u ∼ st, uv → ∼ s, u → ∼ t, v ∼ st, uv Rule (b), (89), (91).(93) → ∼ s, u → ∼ t, v ∼ st, uv Rule (b), (90), (92).(94) → W f ( d ) → W y ∼ f ( f ( d ) y ) , f ( y ) f ( f ( d ))Rule (c), (6) with x = f ( d ).(95) → W f ( d ) → W f ( c ) ∼ f ( f ( d ) f ( c )) , f ( f ( c )) f ( f ( d ))Rule (c), (94) with y = f ( c ).(96) → W f ( c ) ∼ f ( f ( d ) f ( c )) , f ( f ( c )) f ( f ( d ))Rule (b), (65), (95).(97) ∼ f ( f ( d ) f ( c )) , f ( f ( c )) f ( f ( d ))Rule (b), (64), (96).(98) → ∼ f ( f ( c )) , u → ∼ t, v ∼ f ( f ( c )) t, uv Rule (c), (93) with s = f ( f ( c )).(99) → ∼ f ( f ( c )) , c → ∼ t, v ∼ f ( f ( c )) t, cv Rule (c), (98) with u = c .(100) → ∼ f ( f ( c )) , c → ∼ f ( f ( d )) , v ∼ f ( f ( c )) f ( f ( d )) , cv Rule (c), (99) with t = f ( f ( d )).(101) → ∼ f ( f ( c )) , c → ∼ f ( f ( d )) , d ∼ f ( f ( c )) f ( f ( d )) , cd Rule (c), (100) with v = d .(102) → ∼ f ( f ( d )) , d ∼ f ( f ( c )) f ( f ( d )) , cd Rule (b), (66), (101).(103) ∼ f ( f ( c )) f ( f ( d )) , cd Rule (b), (67), (102).(104) → ∼ f ( f ( d ) f ( c )) , t → ∼ t, u ∼ f ( f ( d ) f ( c )) , u Rule (c), (14) with s = f ( f ( d ) f ( c )).(105) → ∼ f ( f ( d ) f ( c )) , f ( f ( c )) f ( f ( d )) → ∼ f ( f ( c )) f ( f ( d )) , u ∼ f ( f ( d ) f ( c )) , u Rule (c), (104) with t = f ( f ( c )) f ( f ( d )).(106) → ∼ f ( f ( d ) f ( c )) , f ( f ( c )) f ( f ( d )) → ∼ f ( f ( c )) f ( f ( d )) , cd ∼ f ( f ( d ) f ( c )) , cd Rule (c), (105) with u = cd .(107) → ∼ f ( f ( c )) f ( f ( d )) , cd ∼ f ( f ( d ) f ( c )) , cd Rule (b), (97), (106). (108) ∼ f ( f ( d ) f ( c )) , cd Rule (b),(103), (107).(109) → ∼ f ( cd ) , t ∼ f ( f ( cd )) , f ( t )Rule (c), (10) with s = f ( cd ).(110) → ∼ f ( cd ) , f ( d ) f ( c ) ∼ f ( f ( cd )) , f ( f ( d ) f ( c ))Rule (c), (109) with t = f ( d ) f ( c ).(111) ∼ f ( f ( cd )) , f ( f ( d ) f ( c )) Rule (b), (79), (110).(112) → ∼ f ( f ( cd )) , t → ∼ t, u ∼ f ( f ( cd )) , u Rule (c), (14) with s = f ( f ( cd )).(113) → ∼ f ( f ( cd )) , f ( f ( d ) f ( c )) → ∼ f ( f ( d ) f ( c )) , u ∼ f ( f ( cd )) , u Rule (c), (112) with t = f ( f ( d ) f ( c )).(114) → ∼ f ( f ( cd )) , f ( f ( d ) f ( c )) → ∼ f ( f ( d ) f ( c )) , cd ∼ f ( f ( cd )) , cd Rule (c), (113) with u = cd .(115) → ∼ f ( f ( d ) f ( c )) , cd ∼ f ( f ( cd )) , cd Rule (b), (111), (114).(116) ∼ f ( f ( cd )) , cd Rule (b), (108), (115).(117) → W cd → W f ( cd ) → ∼ f ( f ( cd )) , cd G ( cd )Rule (a), axiom of the propositional calculus.(118) → W f ( cd ) → ∼ f ( f ( cd )) , cd G ( cd ) Rule (b), (71), (117).(119) → ∼ f ( f ( cd )) , cd G ( cd )Rule (b), (86), (118).(120) G ( cd ) Rule (b), (116), (119).We have deduced G ( cd ) in [ M ′ ; L ′ ]. It follows from the DeductionTheorem [2, (4.5)] that the formula → G ( c ) → G ( d ) G ( cd ) is prov-able in [ M A ; L ′ ]. From the generalization of the constant symbols c , d according to [2, Corollary (4.9)(b)] we see that → G ( x ) → G ( y ) G ( xy )is provable in the original mathematical system [ M ; L ]. Moreover, theformulas G ( a ) in (49) and G ( b ) in (53) are also provable in [ M ; L ]. Weapply Rule (e) in [ M ; L ] on the last three formulas and finally concludethat the formulas → W u G ( u ) and hence → W x W f ( x ) and → W x ∼ f ( f ( x )) , x are provable in [ M ; L ]. On prime formulas not occurring in the basis axioms.
In this note we determine a simple procedure in order to eliminateprime formulas from formal proofs which do not occur with a givenarity in the basis axioms of a mathematical system.Let [ M ; L ] with M = [ S ; A M ; P M ; B M ] be a mathematical systemwith restricted argument lists in L . Assume that q ∈ P M does not occur j -ary in the basis axioms B M , where j ≥ F ; . . . ; F l ] be a proof in [ M ; L ] with the steps F ; . . . ; F l .For a variable z ∈ X not involved in B S we put as abbreviation thecontradiction C = & ∀ z ∼ z, z ¬∀ z ∼ z, z . If replace in each formula F of [ M ; L ] all subformulas of the form qλ , . . . , λ j with λ , . . . , λ j ∈ L by the contradiction C then we obtainthe formula C ( F ) with argument lists in L . We will show that[ C (Λ)] = [ C ( F ); . . . ; C ( F l )]is again a proof in [ M ; L ], where q does not occur j -ary in [ C (Λ)] .We can subsequently apply this procedure and obtain the followingresult: Apart from the equations we can replace all prime formulas inthe original proof [Λ] by C which do not appear as subformulas witha given arity in the basis axioms of [ M ; L ]. All other prime formulaswhich occur as subformulas in the steps of [Λ] are not affected by thisprocedure.In the sequel we fix the quantities q ∈ P M and j ∈ N in the definitionof C and C ( · ). Lemma:
Let F be a formula in [ M ; L ] Then for every list µ ∈ L andfor all variables x ∈ X with CF ( F ; µ ; x ) there holds the conditionCF( C ( F ); µ ; x ) and the equation C (SbF( F ; µ ; x )) = SbF( C ( F ); µ ; x ) . Proof.
We use induction with respect to the rules for generating for-mulas in [ M ; L ] and fix a variable x ∈ X as well as a list µ ∈ L .We say that a formula F in [ M ; L ] satisfies Condition ( ∗ ) if the condi-tion CF( F ; µ ; x ) implies the condition CF( C ( F ); µ ; x ) and the equa-tion C (SbF( F ; µ ; x )) = SbF( C ( F ); µ ; x ) . We prove that Condition ( ∗ ) is satisfied for all formulas F in [ M ; L ].We use the definitions [2, (3.6) and (3.7)] and the notations occurringthere by treating the corresponding cases (a)-(d) in these definitions.(a) If F is a prime formula in [ M ; L ] of the form qλ , . . . , λ j then C ( F ) = C with CF( F ; µ ; x ) and CF( C ; µ ; x ), and we have C (SbF( F ; µ ; x )) = C = SbF( C ( F ); µ ; x )Otherwise F is a prime formula in [ M ; L ] different from qλ , . . . , λ j with C ( F ) = F . In both cases we have confirmed Condition ( ∗ )for the prime formulas.(b) We assume that Condition ( ∗ ) is satisfied for a formula F in[ M ; L ] and that CF( ¬ F ; µ ; x ). Then there holds the condi-tion CF( F ; µ ; x ), and we have C ( ¬ F ) = ¬ C ( F ). Since F satisfies Condition ( ∗ ), we conclude that CF( C ( F ); µ ; x ) andCF( C ( ¬ F ); µ ; x ) are valid and that the equations C (SbF( ¬ F ; µ ; x )) = ¬ C (SbF( F ; µ ; x )) = SbF( C ( ¬ F ); µ ; x )are satisfied. Thus we have confirmed Condition ( ∗ ) for ¬ F .(c) We assume that Condition ( ∗ ) is satisfied for the [ M ; L ]-formulas F, G and that CF(
J F G ; µ ; x ) holds. We obtain CF( F ; µ ; x ) andCF( G ; µ ; x ). Since F and G satisfy Condition ( ∗ ), we concludethat CF( C ( F ); µ ; x ) and CF( C ( G ); µ ; x ) are both valid. There-fore CF( J C ( F ) C ( G ); µ ; x ) and CF( C ( J F G ); µ ; x ) are satisfied.Since F and G satisfy Condition ( ∗ ), we obtain C (SbF( J F G ; µ ; x )) = C ( J F µx G µx ) = J C ( F µx ) C ( G µx )= J C ( F ) µx C ( G ) µx = SbF( C ( J F G ); µ ; x ) , i.e. Condition ( ∗ ) is satisfied for J F G .(d) We assume that Condition ( ∗ ) is satisfied for a formula F in[ M ; L ] and that there holds CF( Q y F ; µ ; x ). We further keep inmind that free( C ( F )) ⊆ free( F ) and that C ( Q y F ) =
Q y C ( F ).In the case x / ∈ free( F ) \ { y } we have x / ∈ free( C ( F )) \ { y } andconclude that CF( C ( Q y F ); µ ; x ) as well as C (SbF( Q y F ; µ ; x )) = C ( Q y F ) = SbF( C ( Q y F ); µ ; x ) . Otherwise we use that CF(
Q y F ; µ ; x ) is satisfied with x = y and conclude that y / ∈ var( µ ) and CF( F ; µ ; x ). Recall that F satisfies the Condition ( ∗ ) which implies CF( C ( F ); µ ; x ). From y / ∈ var( µ ) and CF( C ( F ); µ ; x ) we conclude CF( Q y C ( F ); µ ; x ),i.e. CF( C ( Q y F ); µ ; x ) is again satisfied. Since F satisfies theCondition ( ∗ ), we finally conclude due to x = y that C (SbF( Q y F ; µ ; x )) = Q y C (SbF( F ; µ ; x ))= Q y
SbF( C ( F ); µ ; x )= SbF( C ( Q y F ); µ ; x ) , i.e. Condition ( ∗ ) is satisfied for Q y F .Thus we have proved the lemma. (cid:3)
Theorem:
With the assumptions of this subsection we obtain that[ C (Λ)] = [ C ( F ); . . . ; C ( F l )]is again a proof in [ M ; L ] . Proof.
We employ induction with respect to the rules of inference. Firstwe note that for the ”initial proof” [Λ] = [ ] we can choose [ C (Λ)] = [ ].In the sequel we assume that [Λ] as well as [ C (Λ)] = [ C ( F ); ... ; C ( F l )]are both proofs in [ M ; L ].(a) Let H be an axiom in [ M ; L ]. Then [Λ ∗ ] = [Λ ; H ] is also aproof in [ M ; L ] due to Rule (a). We show that C ( H ) is again anaxiom. Then [ C (Λ ∗ )] = [ C (Λ) ; C ( H ) ] is a proof in [ M ; L ] dueto Rule (a). For this purpose we distinguish four cases.1.) Let α = α ( ξ , ..., ξ m ) be an identically true propositionalfunction of the distinct propositional variables ξ , ..., ξ m , m ≥ m propositionalvariables occur in α . If H ,..., H m are formulas in [ M ; L ] with H = α ( H , ..., H m ), then C ( H ) = α ( C ( H ) , ..., C ( H m )) is againan axiom of the propositional calculus in [ M ; L ].2.) If H is an axiom of equality [2, (3.10)(a),(b)] in [ M ; L ]then C ( H ) = H . For [2, (3.10)(c)], p = q or n = j we haveagain C ( H ) = H . For [2, (3.10)(c)], p = q and n = j we obtainthat C ( H ) is an axiom of the propositional calculus. H is a quantifier axiom [2, (3.11)] then C ( H ) is again aquantifier axiom. For the quantifier axioms (3.11)(b) we furtherhave to note that x / ∈ free( F ) implies x / ∈ free( C ( F )).4.) For H ∈ B M we obtain again C ( H ) = H .(b) Let F , G be two formulas in [ M ; L ] and F , → F G both steps ofthe proof [Λ]. Then [Λ ∗ ] = [Λ ; G ] is also a proof in [ M ; L ] dueto Rule (b). It follows that C ( F ) and C ( → F G ) = → C ( F ) C ( G )are both steps of the proof [ C (Λ)] due to our assumptions, anddue to Rule (b) we put [ C (Λ ∗ )] = [ C (Λ) ; C ( G ) ] for the requiredproof in [ M ; L ].(c) Let F ∈ [Λ], x ∈ X and λ ∈ L . Suppose that there holds thecondition CF( F ; λ ; x ). Then [Λ ∗ ] = [Λ ; F λx ] is also a proof in[ M ; L ] due to Rule (c). We obtain from the previous lemmathat there holds the condition CF( C ( F ); λ ; x ) and the equa-tion C ( F λx ) = C ( F ) λx . Since C ( F ) ∈ [ C (Λ)] we conclude that[ C (Λ ∗ )] = [ C (Λ) ; C ( F λx ) ] is a proof in [ M ; L ] due to Rule (c).(d) Let F ∈ [Λ] and x ∈ X . Then [Λ ∗ ] = [Λ ; ∀ x F ] is also a proofin [ M ; L ] due to Rule (d). Since F ∈ [Λ] implies C ( F ) ∈ [ C (Λ)]and since C ( ∀ x F ) = ∀ x C ( F ), we can apply Rule (d) on [ C (Λ)], C ( F ) in order to conclude that [ C (Λ ∗ )] = [ C (Λ) ; C ( ∀ x F ) ] is aproof in [ M ; L ].(e) In the following we fix a predicate symbol p ∈ P S , a list x , ..., x i of i ≥ G in [ M ; L ]. Wesuppose that x , ..., x i and the variables of G are not involvedin B S .Then to every R-formula F of B S there corresponds exactlyone formula F ′ of the mathematical system, which is obtainedif we replace in F each i − ary subformula p λ , ..., λ i , where λ , ..., λ i are lists, by the formula G λ x ... λ i x i . Note that in thiscase λ , ..., λ i ∈ L .Suppose that F ′ is a step of [Λ] for all R-formulas F ∈ B S for which p occurs i − ary in the R-conclusion of F . Then[Λ ∗ ] = [Λ; → p x , ..., x i G ]is also a proof in [ M ; L ] due to Rule (e). We distinguish two cases: In the first case we assume that p = q and i = j . Then we can apply Rule (a) on the formula C ( → p x , ..., x i G ) = → C C ( G ) , which is an axiom of the propositional calculus , and concludethat [ C (Λ ∗ )] = [ C (Λ) ; C ( → p x , ..., x i G ) ]is a proof in [ M ; L ]. See also [2, (3.14), Example 2].In the second case we assume that p = q or i = j . For everyR-formula F of B S we define the formula F ′′ of [ M ; L ] whichis obtained if we replace in F each i − ary subformula p λ , ..., λ i with λ , ..., λ i ∈ L by the formula C ( G ) λ x ... λ i x i and note thatthe variables in C ( G ) are not involved in B S , because we haveassumed that the bound variable z in the contradiction C doesnot occur in B S . We have assumed that q does not occur j -aryin B M , hence in the formula F ′ the symbol q can only occur j -ary within the subformulas G λ x ... λ i x i . From the previouslemma we obtain C ( G ) λ x ... λ i x i = C ( G λ x ... λ i x i ) . We see that F ′′ = C ( F ′ ) and recall that [Λ] as well as [ C (Λ)] =[ C ( F ); ... ; C ( F l )] are both proofs in [ M ; L ]. Hence F ′′ is a stepof the proof [ C (Λ)] for all R-formulas F ∈ B S for which p occurs i − ary in the R-conclusion of F . Therefore we can apply Rule(e) on [ C (Λ)] and conclude that[ C (Λ ∗ )] = [ C (Λ) ; C ( → p x , ..., x i G ) ]with C ( → p x , ..., x i G ) = → p x , ..., x i C ( G ) is a proof in[ M ; L ].Thus we have proved the theorem. (cid:3) The lemma and theorem of this subsection have a strong resemblanceto [2, (4.7) Lemma, (4.8) Theorem], and the proofs are very similar.It arises the question whether there is a more general result which isrelevant in elementary proof theory.
Proof of a general conjecture and its application to PA.
We consider a mathematical system M = [ S ; A M ; P M ; B M ] with anunderlying recursive system S = [ A S ; P S ; B S ] such that A M = A S , P M = P S and B M = B S and assume that [ M ; L ] is a mathematicalsystem with restricted argument lists in L . We suppose that L isenumerable. We will consider this mathematical system [ M ; L ] untilwe discuss its application to PA. Definition 3.1.
An R-derivation [Λ] in [ S ; L ] is defined as an R-derivation in S with the following restrictions: The R-formulas in [Λ]and the R-formulas F , G in [2, (1.11)] have only argument lists in L , and the use of the Substitution Rule [2, (1.11)(c)] is restricted to λ ∈ L . Then the R-formulas in [Λ] are called R-derivable in [ S ; L ]. ByΠ R ( S ; L ) we denote the set of all R-derivable R-formulas in [ S ; L ] .From [2, Section 3, Example 2] we know that the formula ¬ q x , . . . , x j with x , . . . , x j ∈ X is provable in [ M ; L ] whenever q ∈ P S does notoccur j -ary in B S . On the other hand we have shown the consistencyof [ M ; L ] in [2, (5.1) Proposition].We will first simplify the syntax of the formulas F in [ M ; L ] by re-moving the quantifier ∃ and the symbols ∨ , & and ↔ . By Form( M ; L )we denote the set of all formulas in [ M ; L ]. Let be F the set of allformulas in Form( M ; L ) without the symbols ∃ , ∨ , & and ↔ and definethe mapping Θ : F orm ( M ; L )
7→ F as follows:1. Θ( F ) = F if F is a prime-formula in [ M ; L ].2. Θ( ¬ F ) = ¬ Θ( F ) for all formulas F in [ M ; L ].3. For all F, G ∈ Form( M ; L ) we havei. Θ( → F G ) = → Θ( F )Θ( G ).ii. Θ( ∨ F G ) = → ¬ Θ( F )Θ( G ).iii. Θ(& F G ) = ¬ → Θ( F ) ¬ Θ( G ).iv. Θ( ↔ F G ) = ¬ → → Θ( F )Θ( G ) ¬ → Θ( G )Θ( F ).4. i. Θ( ∀ xF ) = ∀ x Θ( F ) for all x ∈ X and F ∈ Form( M ; L ).ii. Θ( ∃ xF ) = ¬∀ x ¬ Θ( F ) for all x ∈ X , F ∈ Form( M ; L ). Theorem 3.2.
Let [Λ] = [ F ; . . . ; F l ] be a proof in [ M ; L ] with the steps F ; . . . ; F l . Then [Θ(Λ)] = [Θ( F ); . . . ; Θ( F l )] is again a proof in [ M ; L ] . For all k = 1 , . . . , l the formula Θ( F k ) can be derived with the same rule of inference that was used for thederivation of F k in the proof [Λ] .Proof. We employ induction with respect to the rules of inference. Firstwe note that for the ”initial proof” [Λ] = [ ] we can choose [Θ(Λ)] = [ ].In the sequel we assume that [Λ] as well as [Θ(Λ)] = [Θ( F ); ... ; Θ( F l )]are both proofs in [ M ; L ].(a) The basis axioms in [ M ; L ] are exactly the basis R-axioms of theunderlying recursive system, hence Θ( F ) = F for all formulas F ∈ B M = B S . If F is an axiom of equality then we have againΘ( F ) = F . If F is an axiom of the propositional calculus, thenalso Θ( F ). If F is an quantifier axiom (3.11)(a),(b), then alsoΘ( F ). If F is an quantifier axiom (3.11)(c), then Θ( F ) is anaxiom of the propositional calculus. We conclude that Θ mapsaxioms into axioms.(b) Let F , G be two formulas in [ M ; L ] and F , → F G both stepsof the proof [Λ]. Then [Λ ∗ ] = [Λ ; G ] is also a proof in [ M ; L ]due to Rule (b). It follows thatΘ( F ) and Θ( → F G ) = → Θ( F ) Θ( G )are both steps of the proof [Θ(Λ)] due to our assumptions, anddue to Rule (b) we put [Θ(Λ ∗ )] = [Θ(Λ) ; Θ( G ) ] for the requiredproof in [ M ; L ].(c) Let F ∈ [Λ], x ∈ X and λ ∈ L . Suppose that there holds thecondition CF( F ; λ ; x ). Then [Λ ∗ ] = [Λ ; F λx ] is also a proof in[ M ; L ] due to Rule (c). We obtain that there holds the condi-tion CF(Θ( F ); λ ; x ) and the equation Θ( F λx ) = Θ( F ) λx . SinceΘ( F ) ∈ [Θ(Λ)] we conclude that [Θ(Λ ∗ )] = [Θ(Λ) ; Θ( F λx ) ] is aproof in [ M ; L ] due to Rule (c).(d) Let F ∈ [Λ] and x ∈ X . Then [Λ ∗ ] = [Λ ; ∀ x F ] is alsoa proof in [ M ; L ] due to Rule (d). Since F ∈ [Λ] implies Θ( F ) ∈ [Θ(Λ)] and since Θ( ∀ x F ) = ∀ x Θ( F ), we can applyRule (d) on [Θ(Λ)], Θ( F ) in order to conclude that [Θ(Λ ∗ )] =[Θ(Λ) ; Θ( ∀ x F ) ] is a proof in [ M ; L ].(e) In the following we fix a predicate symbol p ∈ P S , a list x , ..., x i of i ≥ G in [ M ; L ]. Wesuppose that x , ..., x i and the variables of G are not involvedin B S .Then to every R-formula F of B S there corresponds exactlyone formula F ′ of the mathematical system, which is obtainedif we replace in F each i − ary subformula p λ , ..., λ i , where λ , ..., λ i are lists, by the formula G λ x ... λ i x i . Note that in thiscase λ , ..., λ i ∈ L .Suppose that F ′ is a step of [Λ] for all R-formulas F ∈ B S for which p occurs i − ary in the R-conclusion of F . Then[Λ ∗ ] = [Λ; → p x , ..., x i G ]is also a proof in [ M ; L ] due to Rule (e).For every R-formula F of B S we define the formula F ′′ of[ M ; L ] which is obtained if we replace in F each i − ary subfor-mula p λ , ..., λ i with λ , ..., λ i ∈ L by the formula Θ( G ) λ x ... λ i x i and note that the variables in Θ( G ) are not involved in B S , be-cause Θ( G ) and G both have the same variables. We see that F ′′ = Θ( F ′ ) and recall that [Λ] and [Θ(Λ)] = [Θ( F ); ... ; Θ( F l )]are both proofs in [ M ; L ]. Hence F ′′ is a step of the proof [Θ(Λ)]for all R-formulas F ∈ B S for which p occurs i − ary in the R-conclusion of F . Therefore we can apply Rule (e) on [Θ(Λ)] andconclude that[Θ(Λ ∗ )] = [Θ(Λ) ; Θ( → p x , ..., x i G ) ]with Θ( → p x , ..., x i G ) = → p x , ..., x i Θ( G ) is a proof in[ M ; L ].Thus we have proved the theorem. (cid:3) Now we will roughly divide the formulas in F into equivalence classes.This is used in order to present a well defined interpretation of theformulas F ∈ F in the mathematical system. Definition 3.3.
Equivalence classes h F i of formulas F ∈ F .1. By P we denote the set of all prime formulas in [ M ; L ]. For anyprime formula F ∈ P we have F ∈ F and put h F i = P .2. For F ∈ F we also have ¬ F ∈ F and put h¬ F i = ¬h F i = {¬ F ′ : F ′ ∈ h F i } .
3. For
F, G ∈ F we also have → F G ∈ F and put h→ F G i = → h F ih G i = {→ F ′ G ′ : F ′ ∈ h F i , G ′ ∈ h G i } .
4. For x ∈ X , F ∈ F we also have ∀ xF ∈ F and put h∀ x F i = ∀h F i = {∀ x ′ F ′ : x ′ ∈ X, F ′ ∈ h F i } . The sets h F i with F ∈ F give a well-defined partition of F , two for-mulas F and F ′ in F are equivalent if and only if h F i = h F ′ i for theirequivalence classes h F i and h F ′ i , respectively. The construction of eachclass starts with P and terminates in a finite number of steps. It ispurely syntactic, for example ¬¬∀ → PP and ∀ → PP are disjoint.By F ∗ we denote the set of all formulas in F without free variables, alsocalled statements . Let L ∗ be the set of all lists in L without variables.We suppose that L ∗ is not empty. Now we will give an interpretationof all statements F ∈ F ∗ in the mathematical system [ M ; L ].Using the verum ⊤ , the empty set ∅ and formulas F, G ∈ F we definethe following function V : F ∗
7→ {∅ , {⊤}} :1. If λ, µ ∈ L ∗ thenV( ∼ λ, µ ) = ( {⊤} if ∼ λ, µ ∈ Π R ( S ; L ) , ∅ otherwise . Let p ∈ P S and λ , ..., λ i ∈ L ∗ for i ≥ A S -listsin L ∗ . Then we evaluateV( p λ , ..., λ i ) = ( {⊤} if p λ , ..., λ i ∈ Π R ( S ; L ) , ∅ otherwise .
2. For ¬ F ∈ F ∗ we also have F ∈ F ∗ and requireV( ¬ F ) = {⊤} \ V( F ) .
3. For → F G ∈ F ∗ we also have F, G ∈ F ∗ and requireV( → F G ) = ( {⊤} \ V( F )) ∪ V( G ) .
4. For x ∈ X , ∀ x F ∈ F ∗ and λ ∈ L ∗ we have F λx ∈ F ∗ , recall L ∗ = ∅ and requireV( ∀ x F ) = \ λ ∈L ∗ V (cid:18) F λx (cid:19) . We say that a formula F ∈ F ∗ is true if and only if ⊤ ∈ V( F ). The sets h F i ∗ = h F i ∩ F ∗ with F ∈ F ∗ form a partition of F ∗ , and induction onthe equivalence classes h F i ∗ in F ∗ shows that the function V is well-defined. Of course, in general the evaluation of V ( F ) must be highlynon-constructive. Definition 3.4.
Let F be a formula in [ M ; L ]. Let x = x j , . . . , x m = x j m be the list of all free variables in F , ordered with increasingindizes j < . . . < j m of the variables. We defineFree( F ) = [ x ; . . . ; x m ] , Gen( F ) = ∀ x . . . ∀ x m F , namely the list Free( F ) of free variables in F and the generalizationof the formula F , respectively. Especially for statements F we haveFree( F ) = [ ] and Gen( F ) = F .Now we make use of Theorem 3.2 which guarantees that proofs withformulas in F are not a real restriction and present our main result,namely Theorem 3.5.
Let F ∈ F be a formula which is provable in [ M ; L ] .Suppose that the set L ∗ of all lists in L without variables is not emptyand that L is enumerable. Then ⊤ ∈ V ( Gen ( F )) .Proof. Let F ∈ F be a formula in [ M ; L ] with Free( F ) = [ x ; . . . ; x m ].Then ⊤ ∈ V (Gen( F )) iff ⊤ ∈ V ( F λ x . . . λ m x m ) for all λ , . . . λ m ∈ L ∗ .This will be used throughout the whole proof.We want to show for each proof [Λ] in [ M ; L ] with steps only in F that ⊤ ∈ V (Gen( F )) for all F ∈ [Λ]. We employ induction with respectto the rules of inference. First we note that the statement is true forthe ”initial proof” [Λ] = [ ]. Let [Λ] = [ F ; . . . ; F l ] be a proof in [ M ; L ] with the steps F ; . . . ; F l ∈ F .Our induction hypothesis is ⊤ ∈ V (Gen( F k )) for all k = 1 , . . . , l .(a) Here we show that ⊤ ∈ V (Gen( F )) for all axioms F ∈ F . Thenthe extended proof [Λ ∗ ] = [Λ; F ] will also satisfy the statement. • The basis axioms and the axioms of equality in [ M ; L ] areR-axioms of the underlying recursive system. Assume that F is such an axiom with Free( F ) = [ x ; . . . ; x m ] and that λ , . . . λ m ∈ L ∗ . Then ˜ F = F λ x . . . λ m x m is an elementaryR-formula in F . We have ⊤ ∈ V ( ˜ F ) iff there is an R-premise in ˜ F which is not R-derivable in [ S ; L ] or if theR-conclusion of ˜ F is R-derivable in [ S ; L ]. But due to theModus Ponens Rule the R-conclusion of ˜ F is R-derivablein [ S ; L ] if all R-premises in ˜ F are R-derivable in [ S ; L ].We see ⊤ ∈ V ( ˜ F ) and hence ⊤ ∈ V (Gen( F )) in the casethat F is a basis axiom or an axiom of equality in themathematical system [ M ; L ]. • Suppose that α = α ( ξ , ..., ξ j ) is an identically true propo-sitional function defined in [2, (3.8)] which is only con-structed with the negation symbol “ ¬ ” and the implicationarrow “ → ” and that F ,..., F j ∈ F are formulas in [ M ; L ].Then the formula F = α ( F , ..., F j ) ∈ F is an axiom of thepropositional calculus. Prescribe λ , . . . λ m ∈ L ∗ and put˜ F = F λ x . . . λ m x m , ˜ F k = F k λ x . . . λ m x m for k = 1 , . . . , j and Free( F ) = [ x ; . . . ; x m ] . For any twoformulas F ′ , F ′′ ∈ F ∗ we have ⊤ ∈ V ( ¬ F ′ ) iff ⊤ / ∈ V ( F ′ )and ⊤ ∈ V ( → F ′ F ′′ ) iff ⊤ ∈ V ( F ′ ) implies ⊤ ∈ V ( F ′′ ),respectively. We see that ˜ F = α ( ˜ F , . . . , ˜ F j ) ∈ F ∗ is anaxiom of the propositional calculus with ⊤ ∈ V ( ˜ F ). • Suppose that x ∈ X , that F ∈ F andFree( ∀ xF ) = [ x ; . . . ; x m ] . We put H = → ∀ xF F and note that x / ∈ [ x ; . . . ; x m ]. Wesee ⊤ ∈ V (Gen( H )) iff ⊤ ∈ V (SbF( ˜ H ; µ ; x )) = V ( → ∀ x ˜ F SbF( ˜ F ; µ ; x ))for all µ, λ , . . . λ m ∈ L ∗ , using ˜ F = F λ x . . . λ m x m and˜ H = H λ x . . . λ m x m = → ∀ x ˜ F ˜ F as abbreviations.Now ⊤ ∈ V ( ∀ x ˜ F ) implies indeed ⊤ ∈ V (SbF( ˜ F ; µ ; x ))for all µ, λ , . . . λ m ∈ L ∗ , independent of x ∈ Free( F ) or x / ∈ Free( F ). • Suppose that x ∈ X , that F, G ∈ F and that x / ∈ Free( F ),Free( ∀ x → F G ) = Free( → F ∀ xG ) = [ x ; . . . ; x m ]. We put H = → ∀ x → F G → F ∀ xG , fix arbitrary lists λ , . . . λ m ∈ L ∗ and make use of the ab-breviations ˜ F = F λ x . . . λ m x m and ˜ G = G λ x . . . λ m x m . We have˜ H = H λ x . . . λ m x m = → ∀ x → ˜ F ˜ G → ˜ F ∀ x ˜ G with ˜ H ∈ F ∗ . In order to show ⊤ ∈ V ( ˜ H ) we assume ⊤ ∈ V ( ∀ x → ˜ F ˜ G ) and note that x / ∈ Free( ˜ F ). Then ⊤ ∈ V ( ∀ x → ˜ F ˜ G ) iff ⊤ ∈ V ( → ˜ F SbF( ˜ G ; λ ; x ))for all λ ∈ L ∗ . Hence ⊤ ∈ V ( ˜ F ) implies ⊤ ∈ V (SbF( ˜ G ; λ ; x ))for all λ ∈ L ∗ , i.e. ⊤ ∈ V ( ˜ F ) implies ⊤ ∈ V ( ∀ x ˜ G ), andwe have shown ⊤ ∈ V ( → ˜ F ∀ x ˜ G ) and ⊤ ∈ V ( ˜ H ). • Recall that the quantifier axiom (3.11)(c) is replaced by anaxiom of the propositional calculus due to Theorem 3.2.(b) Suppose that F and H = → F G are both steps of the proof[Λ] = [ F ; . . . ; F l ] with Free( → F G ) = [ x ; . . . ; x m ] . Then ⊤ ∈ V (Gen( F )) and ⊤ ∈ V (Gen( H )) from our inductionhypothesis. Fix λ , . . . λ m ∈ L ∗ and put ˜ F = F λ x . . . λ m x m ,˜ G = G λ x . . . λ m x m . For ˜ H = H λ x . . . λ m x m = → ˜ F ˜ G we have ˜ F , ˜ G, ˜ H ∈ F ∗ , ⊤ ∈ V ( ˜ F ), ⊤ ∈ V ( ˜ H ) and ⊤ ∈ V ( ˜ G ).Note that substitutions of variables in [ x ; . . . ; x m ] not occurringin F or G are allowed, because they do not have any effect. Weobtain that the extended proof [Λ ∗ ] = [Λ; G ] also satisfies ourstatement.(c) Let x ∈ X and suppose that F ∈ F is a step of the proof[Λ] = [ F ; . . . ; F l ]. Let λ ∈ L and suppose that there holdsthe condition CF( F ; λ ; x ). Note that ⊤ ∈ V (Gen( F )) from ourinduction hypothesis.Without loss of generality we may assume that x ∈ free( F ),where we use the set free( F ) = { x, x , . . . , x m } (instead of or-dered lists) with distinct variables x, x , . . . , x m ∈ X , and putΦ( F ) = { F λ x λ x . . . λ m x m : λ , λ , . . . , λ m ∈ L ∗ } . We write var( λ ) = { y , . . . , y k } and λ = λ ( y , . . . , y k ) . From x ∈ free( F ) and CF( F ; λ ; x ) we see thatvar( λ ) ⊆ free (cid:18) F λx (cid:19) . Hence we can write free( F λx ) = { y , . . . , y n } with n ≥ k distinctvariables y , . . . , y n ∈ X and define the new setΦ( F ; λ ; x ) = { F λx µ y . . . µ n y n : µ , . . . , µ n ∈ L ∗ } . Again from CF( F ; λ ; x ) we conclude thatΦ( F ; λ ; x ) = (cid:26) F λ ( µ , . . . , µ k ) x µ y . . . µ n y n : µ , . . . , µ n ∈ L ∗ (cid:27) , hence Φ( F ; λ ; x ) ⊆ Φ( F ) and V (Gen( F )) = \ G ∈ Φ( F ) V ( G ) ⊆ \ G ∈ Φ( F ; λ ; x ) V ( G ) = V (cid:18) Gen (cid:18)
F λx (cid:19)(cid:19) . We obtain from our induction hypothesis ⊤ ∈ V (Gen( F )) that ⊤ ∈ V (Gen( F λx )). Now the extended proof [Λ ∗ ] = [Λ; F λx ]satisfies our statement.(d) Let F be a step of the proof [Λ] = [ F ; . . . ; F l ]. If x ∈ X is nota free variable of F then F λx = F for all λ ∈ L ∗ and V (Gen( ∀ x F )) = V (Gen( F )) . Then ⊤ ∈ V (Gen( ∀ x F )) from our induction hypothesis. Nowwe suppose that x ∈ free( F ) = { x , . . . , x m } with distinct vari-ables x , . . . , x m . In this case we see ⊤ ∈ V (Gen( ∀ x F )) iff forall λ , . . . , λ m ∈ L ∗ ⊤ ∈ V (cid:18) F λ x . . . λ m x m (cid:19) , i.e. V (Gen( ∀ x F )) = V (Gen( F )), and obtain ⊤ ∈ V (Gen( ∀ x F ))again from our induction hypothesis. In any case the extendedproof [Λ ∗ ] = [Λ; ∀ x F ] satisfies our statement.(e) In the following we fix a predicate symbol p ∈ P S , a list x , ..., x i of i ≥ G ∈ F . We supposethat x , ..., x i and the variables of G are not involved in B S .Then to every R-formula F of B S there corresponds exactlyone formula F ′ ∈ F of the mathematical system, which is ob-tained if we replace in F each i − ary subformula p λ , ..., λ i ,where λ , ..., λ i are lists, by the formula G λ x ... λ i x i . We sup-pose that F ′ is a step of a proof [Λ] for all R-formulas F ∈ B S for which p occurs i − ary in the R-conclusion of F . Now[Λ ∗ ] = [Λ; → p x , ..., x i G ] is also a proof in [ M ; L ] withformulas in F . To finish the proof of the main theorem it re-mains to show that ⊤ ∈ V (Gen( → p x , ..., x i G )) . We maywrite free( → p x , ..., x i G ) = { x , ..., x m } with m ≥ i distinctvariables x , ..., x m . For m > i we choose ˜ λ i +1 , . . . , ˜ λ m ∈ L ∗ arbitrary but fixed and put˜ G = G ˜ λ i +1 x i +1 . . . ˜ λ m x m , and otherwise we put ˜ G = G . It is sufficient to show that ⊤ ∈ V (Gen( → p x , ..., x i ˜ G )) with the formula ˜ G ∈ F satisfyingfree( ˜ G ) ⊆ { x , ..., x i } . Note that the variables of ˜ G are notinvolved in B S . For λ , . . . , λ i ∈ L we can also write˜ G ( λ , . . . , λ i ) = ˜ G λ x . . . λ i x i , provided that x , ..., x i and the variables of G are not involvedin λ , . . . , λ i . Especially for i = 0 we put ˜ G ( λ , . . . , λ i ) = ˜ G .We have to show that pλ , . . . , λ i ∈ Π R ( S ; L ) ⇒ ⊤ ∈ V ( ˜ G ( λ , . . . , λ i ))for all λ , . . . , λ i ∈ L ∗ , see Definition 3.1.We will show that ˜ G ( λ , . . . , λ i ) can be derived in [ M ; L ] fromthe given proof [Λ] = [ F ; . . . ; F l ] by using only Rules (a)-(d)whenever pλ , . . . , λ i ∈ Π R ( S ; L ) for λ , . . . , λ i ∈ L ∗ . Then wecan first apply Theorem 3.2 in order to obtain an extension ofthe proof [Λ] which consists only on formulas in F and whichcontains the formula ˜ G ( λ , . . . , λ i ) as a final step. This willconclude the proof of the theorem because [Λ] satisfies the in-duction hypothesis and Rules (a)-(d) applied step by step onthe extensions of [Λ] with formulas in F can only produce fur-ther new formulas F satisfying ⊤ ∈ V (Gen( F )).Following our strategy we can construct an algorithm A withthe following properties: • A generates an infinite sequence R ; R ; R ; . . . of R-formu-las such that each finite part [ R ; . . . ; R n ] with n ∈ N is anR-derivation in [ S ; L ]. Note that A makes only use of therules of inference (1.11)(a),(b),(c) in [2]. • All elementary prime R-formulas in Π R ( S ; L ) occur at leastone time in the sequence R ; R ; R ; . . . . • We suppose that x , ..., x i and the variables of G are notinvolved in R ; R ; R ; . . . , which is not a real restriction.Depending on A we define a second algorithm B with the fol-lowing properties: • B generates an infinite sequence of formulas F ; F ; F ; . . . in [ M ; L ]. Each finite part [ F ; . . . ; F n ] with n ∈ N is aproof in [ M ; L ], and for all n > l algorithm B makes onlyuse of the rules of inference (3.13)(a)-(d) in [2] in order toderive F n . • First of all we start with algorithm B and prescribe theformulas F ; F ; . . . ; F l in the proof [Λ]. Next we extend [Λ]to a proof [Λ ] by applying only Rule (c) a finite numberof times in order to substitute all variables x i +1 , . . . , x m by˜ λ i +1 , . . . , ˜ λ m in the formulas F ′ ∈ [Λ] for all R-formulas F ∈ B S for which p occurs i − ary in the R-conclusion of F .After the construction of [Λ ] we pause B and start A . • Each time when A has generated a prime R-formula R κ (in-cluding equations and with or without variables) we pausealgorithm A and activate algorithm B to generate R κ aswell in the sequence of formulas F ; F ; F ; . . . . Moreover,if R κ = p λ , ..., λ i with lists λ , ..., λ i ∈ L , then B willalso generate the formula ˜ G ( λ , ..., λ i ) in a finite numberof steps. Afterwards we pause algorithm B and activatealgorithm A again, and so on.It is clear that any R-derivation in [ S ; L ] can also be performedin [ M ; L ]. To prove that algorithm B is well defined we have toshow that it is able to generate the formulas ˜ G ( λ , ..., λ i ) oncealgorithm A has produced the next prime formula of the form p λ , ..., λ i . This will be explained now.Let F be any R-formula in [ S ; L ] and suppose that x , . . . , x i and the variables of G are not involved in F . To F therecorresponds exactly one formula ˜ F ∈ F of [ M ; L ] which re-sults if we replace in F each i − ary subformula p λ , ..., λ i with λ , ..., λ i ∈ L by the formula ˜ G ( λ , ..., λ i ). We have assumedthat the variables of G are not involved in B S . Then we obtainthat ˜ F is a step of the extended proof [Λ ] for all R-formulas F ∈ B S for which p occurs i − ary in the R-conclusion of F . We have ˜ F = F if p does not occur i -ary in F . Beside the axioms F in B S for which p occurs i − ary in the R-conclusion of F algorithm A can also make use of the R-axiomsof equality (1.9)(c) with n = i in order to deduce prime R-formulas p λ , ..., λ i in [ S ; L ] from equations and these R-axioms.Let F = → ∼ y , y ′ . . . → ∼ y i , y ′ i → p y , . . . , y i p y ′ , . . . , y ′ i be such an R-axiom of equality with variables y k , y ′ k ∈ X . Wesuppose that x , ..., x i and the variables of G are not involvedin F . Then we infer in [ M ; L ] the formula˜ F = → ∼ y , y ′ . . . → ∼ y i , y ′ i → ˜ G ( y , . . . , y i ) ˜ G ( y ′ , . . . , y ′ i ) . That this is possible can be seen by using [2, (4.9) Corollary]combined with the Deduction Theorem [2, (4.5)], by using theaxioms of equality and the Equivalence Theorem [2, Theorem(3.17)(a)]. This will only require the use of the Rules (a)-(d).Let R λ be any R-axiom generated by algorithm A and assumethat p occurs i -ary in the R-conclusion of R λ . Then we sum-marize and keep in mind that we can derive the correspondingformula ˜ R λ in [ M ; L ] by using only Rules (a)-(d).For α ≥ R ; . . . ; R α of the R-formulasfrom Π R ( S ; L ) generated by the algorithm A , and we assumethat R α is a prime R-formula. Then we activate algorithm B and proceed with the further expansion of the list of formulas F ; . . . ; F β from Π( M ; L ) until we have derived R α and F β = ˜ R α .This can be achieved if B mimics the R-derivation R ; . . . ; R α in the following way: • For any R-axiom R λ with λ ≤ α algorithm B generates theformula R λ as well. If p occurs i -ary in the R-conclusionof R λ then B generates ˜ R λ . • Suppose that the R-formula R λ was derived from the primeR-formula R κ and the R-formula → R κ R λ with Rule (b), κ < λ ≤ α . Then algorithm B derives the formula R λ aswell. Suppose that ˜ R κ and → ˜ R κ ˜ R λ were already derivedby B , but not ˜ R λ . Then B generates ˜ R λ from Rule (b). • Assume that the R-formula R λ = R κ νx with x ∈ X and ν ∈ L was derived from the R-formula R κ with Rule (c), κ < λ ≤ α . Then algorithm B derives the formula R λ aswell. Suppose that ˜ R κ was already derived by B , but not˜ R λ . Then B generates ˜ R λ = ˜ R κ νx from Rule (c). (cid:3) Under the mild additional condition L ∗ = ∅ we have proved a slightlymore general version of [2, (5.4) Conjecture], namely the following the-orem, which makes use of Definition 3.1: Theorem 3.6.
Let M = [ S ; A M ; P M ; B M ] be a mathematical systemwith an underlying recursive system S = [ A S ; P S ; B S ] such that A M = A S , P M = P S , B M = B S . Suppose that [ M ; L ] is a mathematicalsystem with restricted argument lists in L and that L is enumerable .Let L ∗ = ∅ be the set of all A S -lists in L without variables. i. Let be λ, µ ∈ L ∗ . Then ∼ λ, µ ∈ Π( M ; L ) ⇔ ∼ λ, µ ∈ Π R ( S ; L ) . ii. Let p ∈ P S and λ , ..., λ i ∈ L ∗ for i ≥ be elementary A S -lists.Then p λ , ..., λ i ∈ Π( M ; L ) ⇔ p λ , ..., λ i ∈ Π R ( S ; L ) . Remark 3.7.
We have assumed that L ∗ = ∅ in order to avoid troublewith the definition of the interpretation V of the formulas F ∈ F ∗ .From [2, Section 5.3] and Theorem 3.6 we obtain a consistency prooffor the following Peano arithmetic PA: Let ˜ S be the recursive sys-tem ˜ S = [ ˜ A ; ˜ P ; ˜ B ] where ˜ A , ˜ P and ˜ B are empty, and introduce thealphabets A P A = [ 0; s ; + ; ∗ ], P P A = [ ]. We define the set L ofnumeral terms by the recursive definition(i) 0 and x are numeral terms for any x ∈ X .(ii) If ϑ is a numeral term, then also s ( ϑ ).(iii) If ϑ , ϑ are numeral terms, then also +( ϑ ϑ ) and ∗ ( ϑ ϑ ).We define the mathematical system M ′ = [ ˜ S ; A P A ; P P A ; B P A ] bygiving the following basis axioms for B P A ( α ) ∀ x ∼ +(0 x ) , x ( β ) ∀ x ∀ y ∼ +( s ( x ) y ) , s (+( xy ))( γ ) ∀ x ∼ ∗ (0 x ) , δ ) ∀ x ∀ y ∼ ∗ ( s ( x ) y ) , +( ∗ ( xy ) y )( ε ) ∀ x ∀ y → ∼ s ( x ) , s ( y ) ∼ x, y ( ζ ) ∀ x ¬ ∼ s ( x ) , . Moreover, for all formulas F (with respect to A P A and P P A ) whichhave only numeral argument lists, the following formulas belong to B P A according to the Induction Scheme(IS) → ∀ x & SbF( F ; 0 ; x ) → F SbF( F ; s ( x ) ; x ) ∀ x F .The system PA of Peano arithmetic is given by PA = [ M ′ ; L ], i.e. theargument lists of PA are restricted to the set L of numerals. TheInduction Rule (3.13)(e) is not used in PA since ˜ A , ˜ P and ˜ B are emptyhere and since we are using the Induction Scheme (IS). Let us define arecursive system S = [ A S ; P S ; B S ] as follows:We choose A S = A P A = [ 0; s ; + ; ∗ ], P S = [ N ] and B S consisting ofthe basis R-axioms(1) N → N x N s ( x )(3) → N x ∼ +(0 x ) , x (4) → N x → N y ∼ +( s ( x ) y ) , s (+( xy ))(5) → N x ∼ ∗ (0 x ) , → N x → N y ∼ ∗ ( s ( x ) y ) , +( ∗ ( xy ) y )(7) → N x → N y → ∼ s ( x ) , s ( y ) ∼ x, y .Using the results in [2, Chapter (5.3)] and Theorem 3.6 the inconsis-tency of PA would imply that there is an elementary numeral term λ , i.e. a numeral term without variables, such that N λ as well as ∼ s ( λ ) , R -derivable in S , which is impossible.3.6. A further example with formal induction.
Finally we go back to the recursive system S = [ A ; P ; B ] introduced inSection 2.1 with A = [ a ; 0; 1], P = [ D ] and the set B consisting of thesix basis R-axioms ( α )-( ζ ). Let L be the set generated by the rules(i) x ∈ L for all x ∈ X ,(ii) 0 ∈ L , 1 ∈ L and a ∈ L ,(iii) If λ, µ ∈ L then λµ ∈ L . We define the mathematical system [ M ; L ] with M = [ S ; A ; P ; B ]and will show that the formula ∀ x ↔ D x ∃ y D x, y is provable in[ M ; L ]. We will present a short semi-formal proof. Due to Rule (d)it is sufficient to show that ↔ D x ∃ y D x, y is provable in [ M ; L ].For this purpose we will apply the Induction Rule (e) twice to deduce → ∃ y D x, y D x as well → D x ∃ y D x, y in [ M ; L ].Let x, y, u, v ∈ X be distinct variables. Due to Rule (a) the R -axiomsin B are provable in [ M ; L ]:1. D → D x D x → D x D x D , a → D x, y D x , yy → D x, y D x , yya For the first application of Rule (e) we put p = D , i = 2, x = u , x = v and G = D u . In axioms 4.-6. we replace the prime subformulas
D λ , λ by D λ and obtain axioms 1.-3. for the 1-ary predicate “ D ”.Due to Rule (e)7. → D u, v D u is provable in [ M ; L ], and also the formulas8. → ¬ D x ¬ D x, y ∀ y → ¬ D x ¬ D x, y from 8. & Rule (d)10. → ¬
D x ∀ y ¬ D x, y with 9. & quantifier axiom (3.11)(b)11. → ¬ ∀ y ¬ D x, y D x → ∃ y D x, y D x with 11. & quantifier axiom (3.11)(c)This is the first implication.For the second one we deduce the following formulas in [ M ; L ]:13. → D x, y ∃ y D x, y from example 1 in Section 3.114. → D , a ∃ y D , y from 13. and two times Rule (c)15. ∃ y D , y with 4. and 14.16. → D x , y ∃ y D x , y from example 1 in Section 3.1 → D x , yy ∃ y D x , y from 16. and Rule (c)18. → D x, y ∃ y D x , y with 5. and 17.19. → ¬ ∃ y D x , y ¬ D x, y ∀ y → ¬ ∃ y D x , y ¬ D x, y from 19. & Rule (d)21. → ¬ ∃ y D x , y ∀ y ¬ D x, y with 20. & quantifier axiom (3.11)(b)22. → ¬ ∀ y ¬ D x, y ∃ y D x , y → ∃ y D x, y ∃ y D x , y with 22. & quantifier axiom (3.11)(c)24. → D x , y ∃ y D x , y from example 1 in Section 3.125. → D x , yya ∃ y D x , y from 24. and Rule (c)26. → D x, y ∃ y D x , y with 6. and 25.27. → ¬ ∃ y D x , y ¬ D x, y ∀ y → ¬ ∃ y D x , y ¬ D x, y from 27. & Rule (d)29. → ¬ ∃ y D x , y ∀ y ¬ D x, y with 28. & quantifier axiom (3.11)(b)30. → ¬ ∀ y ¬ D x, y ∃ y D x , y → ∃ y D x, y ∃ y D x , y with 30. & quantifier axiom (3.11)(c)From formulas 15. 23., 31. and [2, Theorem (3.17)(b)] we obtain thatthe formulas32. ∃ v D , v → ∃ v D x, v ∃ v D x , v → ∃ v D x, v ∃ v D x , v are provable in [ M ; L ].For the second application of Rule (e) we put p = D , i = 1, x = u and G = ∃ v D u, v . We replace the prime subformulas D λ in axioms1.-3. by ∃ v D λ , v and obtain formulas 32.-34., respectively.Due to Rule (e) we see that → D u ∃ v D u, v and hence → D x ∃ y D x, y are both provable in [ M ; L ].4. Conclusions and outlook
We have presented contributions to elementary proof theory. Espe-cially in Section 3.4 we have determined a simple procedure in order toeliminate prime formulas from formal proofs which do not occur with a given arity in the basis axioms of a mathematical system. We alsohope to develop a method in order to eliminate equations from formalproofs if there are no equations in the basis axioms.Our most important result is the proof of [2, (5.4) Conjecture], seeTheorem 3.6 in Section 3.5, which is a general result of mathematicallogic concerning formal induction. Though it implies the consistency ofthe Peano arithmetic PA, its meaning seems to go beyond this specialapplication.It would be very interesting to create a computer program which is ableto check semiformal proofs like in Section 3.6. First a machine shouldbe able to check fully formalized proofs with certain restrictions. Forexample, the number of propositional variables in the axioms [2, (3.9)]must be small enough for an efficient calculation. In a next step the pro-gram should be extended to analyze the use of the axioms and rules inorder to develop further composed rules of inference, especially for thepropositional calculus and for the treatment of equations. An advancedprogram should also make use of [2, (3.17),(4.5),(4.8),(4.9),(4.10)].
References [1] Hofstadter, D.R. “G¨odel, Escher, Bach. Ein Endloses Geflochtenes Band”,¨Ubersetzung: Philipp Wolff-Windegg und Hermann Feuersee. Neuausgabe:Klett-Cotta Verlag, Stuttgart (2006).[2] Kunik, M., “Formal mathematical systems including a structural inductionprinciple”, Third revised version of the Preprint Nr. 31/2002, Fakult¨at f¨urMathematik, Otto-von-Guericke-Universit¨at Magdeburg. Available online, seearXiv:2005.04951 (2020).[3] Smullyan, R.M., “Theory of formal systems”, Annals of Math. Stud. No. ,Princeton Univ. Press (1961). Universit¨at Magdeburg, IAN, Geb¨aude 02, Universit¨atsplatz 2, D-39106 Magdeburg, Germany
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