Next Steps (Section 5.8) ∆-definability of sets Terms := { � t � : terms t } = { a ∈ N : a = � t � for some term t } , Formulas := { � ϕ � : formulas ϕ } = { a ∈ N : a = � ϕ � for some formula ϕ } .
∆ -Definition of Terms = { � t � : t is a term } � ¬ α � = � 1 , � α � � � = t 1 t 2 � = � 7 , � t 1 � , � t 2 � � � + t 1 t 2 � = � 13 , � t 1 � , � t 2 � � � <t 1 t 2 � = � 19 , � t 1 � , � t 2 � � � ( α ∨ β ) � = � 3 , � α � , � β � � � 0 � = � 9 � � · t 1 t 2 � = � 15 , � t 1 � , � t 2 � � � v i � = � 2 i � � ( ∀ v i )( α ) � = � 5 , � v i � , � α � � � St � = � 11 , � t � � � Et 1 t 2 � = � 17 , � t 1 � , � t 2 � �
∆ -Definition of Terms = { � t � : t is a term } � ¬ α � = � 1 , � α � � � = t 1 t 2 � = � 7 , � t 1 � , � t 2 � � � + t 1 t 2 � = � 13 , � t 1 � , � t 2 � � � <t 1 t 2 � = � 19 , � t 1 � , � t 2 � � � ( α ∨ β ) � = � 3 , � α � , � β � � � 0 � = � 9 � � · t 1 t 2 � = � 15 , � t 1 � , � t 2 � � � v i � = � 2 i � � ( ∀ v i )( α ) � = � 5 , � v i � , � α � � � St � = � 11 , � t � � � Et 1 t 2 � = � 17 , � t 1 � , � t 2 � � Recall the inductive definition of an L NT -term t : it is either • a variable symbol v i , • St 1 where t 1 is term, • the constant symbol 0, • + t 1 t 2 or · t 1 t 2 or Et 1 t 2 where t 1 , t 2 are terms.
∆ -Definition of Terms = { � t � : t is a term } � ¬ α � = � 1 , � α � � � = t 1 t 2 � = � 7 , � t 1 � , � t 2 � � � + t 1 t 2 � = � 13 , � t 1 � , � t 2 � � � <t 1 t 2 � = � 19 , � t 1 � , � t 2 � � � ( α ∨ β ) � = � 3 , � α � , � β � � � 0 � = � 9 � � · t 1 t 2 � = � 15 , � t 1 � , � t 2 � � � v i � = � 2 i � � ( ∀ v i )( α ) � = � 5 , � v i � , � α � � � St � = � 11 , � t � � � Et 1 t 2 � = � 17 , � t 1 � , � t 2 � � Recall the inductive definition of an L NT -term t : it is either • a variable symbol v i , • St 1 where t 1 is term, • the constant symbol 0, • + t 1 t 2 or · t 1 t 2 or Et 1 t 2 where t 1 , t 2 are terms. Let’s start with ∆-definition of (= { 2 2 i +1 : i = 1 , 2 , . . . } ) . Variables := { � v i � : i = 1 , 2 , . . . } by the formula Variable ( x ) : ≡ ( ∃ y < x )[ Even ( y ) ∧ (0 < y ) ∧ ( x = 2 Sy )] .
∆ -Definition of Terms = { � t � : t is a term } � ¬ α � = � 1 , � α � � � = t 1 t 2 � = � 7 , � t 1 � , � t 2 � � � + t 1 t 2 � = � 13 , � t 1 � , � t 2 � � � <t 1 t 2 � = � 19 , � t 1 � , � t 2 � � � ( α ∨ β ) � = � 3 , � α � , � β � � � 0 � = � 9 � � · t 1 t 2 � = � 15 , � t 1 � , � t 2 � � � v i � = � 2 i � � ( ∀ v i )( α ) � = � 5 , � v i � , � α � � � St � = � 11 , � t � � � Et 1 t 2 � = � 17 , � t 1 � , � t 2 � � Recall the inductive definition of an L NT -term t : it is either • a variable symbol v i , • St 1 where t 1 is term, • the constant symbol 0, • + t 1 t 2 or · t 1 t 2 or Et 1 t 2 where t 1 , t 2 are terms. We would like to write: “ x is � St 1 � for some term t 1 ” “ x is � 0 � ” � �� � � �� � Sy x = 2 10 ∨ ( ∃ y < x )[ Term ( y ) ∧ x = 2 12 · 3 Term ( x ) : ≡ Variable ( x ) ∨ ] � �� � � 11 ,y � ∨ · · · ���� “ x is + t 1 t 2 or · t 1 t 2 or Et 1 t 2 ” However, there is a problem with this “∆-formula”.
∆ -Definition of Terms = { � t � : t is a term } � ¬ α � = � 1 , � α � � � = t 1 t 2 � = � 7 , � t 1 � , � t 2 � � � + t 1 t 2 � = � 13 , � t 1 � , � t 2 � � � <t 1 t 2 � = � 19 , � t 1 � , � t 2 � � � ( α ∨ β ) � = � 3 , � α � , � β � � � 0 � = � 9 � � · t 1 t 2 � = � 15 , � t 1 � , � t 2 � � � v i � = � 2 i � � ( ∀ v i )( α ) � = � 5 , � v i � , � α � � � St � = � 11 , � t � � � Et 1 t 2 � = � 17 , � t 1 � , � t 2 � � Recall the inductive definition of an L NT -term t : it is either • a variable symbol v i , • St 1 where t 1 is term, • the constant symbol 0, • + t 1 t 2 or · t 1 t 2 or Et 1 t 2 where t 1 , t 2 are terms. We would like to write: “ x is � St 1 � for some term t 1 ” “ x is � 0 � ” � �� � � �� � Sy x = 2 10 ∨ ( ∃ y < x )[ Term ( y ) ∧ x = 2 12 · 3 Term ( x ) : ≡ Variable ( x ) ∨ ] � �� � � 11 ,y � ∨ · · · ���� “ x is + t 1 t 2 or · t 1 t 2 or Et 1 t 2 ” This is a not legitimate formula of first-order logic! Note the circular use of the subformula Term ( y ).
∆ -Definition of Terms = { � t � : t is a term } Definition. A term construction sequence for a term t is a finite sequence of terms ( t 1 , . . . , t ℓ ) such that t ℓ : ≡ t and, for each k ∈ { 1 , . . . , ℓ } , the term t k is either • a variable symbol, • the constant symbol 0, • St j for some j < k , or • + t i t j or · t i t j or Et i t j for some i, j < k .
∆ -Definition of Terms = { � t � : t is a term } Definition. A term construction sequence for a term t is a finite sequence of terms ( t 1 , . . . , t ℓ ) such that t ℓ : ≡ t and, for each k ∈ { 1 , . . . , ℓ } , the term t k is either • a variable symbol, • the constant symbol 0, • St j for some j < k , or • + t i t j or · t i t j or Et i t j for some i, j < k . Example. (0 , v 1 , Sv 1 , +0 Sv 1 ) is term construction sequence for the +0 Sv 1 .
∆ -Definition of Terms = { � t � : t is a term } Definition. A term construction sequence for a term t is a finite sequence of terms ( t 1 , . . . , t ℓ ) such that t ℓ : ≡ t and, for each k ∈ { 1 , . . . , ℓ } , the term t k is either • a variable symbol, • the constant symbol 0, • St j for some j < k , or • + t i t j or · t i t j or Et i t j for some i, j < k . Example. (0 , v 1 , Sv 1 , +0 Sv 1 ) is term construction sequence for the +0 Sv 1 . Lemma. Every term t has a term construction sequence of length at most the number of symbols in t . (Easy proof by induction.)
∆ -Definition of Terms = { � t � : t is a term } Definition. A term construction sequence for a term t is a finite sequence of terms ( t 1 , . . . , t ℓ ) such that t ℓ : ≡ t and, for each k ∈ { 1 , . . . , ℓ } , the term t k is either • a variable symbol, • the constant symbol 0, • St j for some j < k , or • + t i t j or · t i t j or Et i t j for some i, j < k . Key to defining Terms : We will write a ∆-formula defining the set TermConSeq = { ( c, a ) : c = � � t 1 � , . . . , � t ℓ � � and a = � t ℓ � where ( t 1 , . . . , t ℓ ) is a term construction sequence } .
∆ -Definition of Terms = { � t � : t is a term } TermConSeq ( c, a ) : ≡ � Codenumber ( c ) ∧ ( ∃ ℓ < c ) Length ( c, ℓ ) ∧ IthElement ( a, ℓ, c ) ∧ � ( ∀ k ≤ ℓ )( ∃ e k < c ) IthElement ( e k , k, c ) ∧ Variable ( e k ) �� ∨ e k = 2 10 } “ e k is � Se j � ” } } “ e k is � 0 � ” � �� � Se j ] e k = 2 12 · 3 ∨ ( ∃ j < k )( ∃ e j < c )[ IthElement ( e j , j, c ) ∧ ∨ · · · Key to defining Terms : We will write a ∆-formula defining the set TermConSeq = { ( c, a ) : c = � � t 1 � , . . . , � t ℓ � � and a = � t ℓ � where ( t 1 , . . . , t ℓ ) is a term construction sequence } .
∆ -Definition of Terms = { � t � : t is a term } Now there is an obvious way to define Term ( a ): Term ( a ) : ≡ ( ∃ c ) TermConSeq ( c, a ) . To make this a ∆-formula, we need an upper bound on c as a function of a .
∆ -Definition of Terms = { � t � : t is a term } Now there is an obvious way to define Term ( a ): Term ( a ) : ≡ ( ∃ c ) TermConSeq ( c, a ) . To make this a ∆-formula, we need an upper bound on c as a function of a . Suppose a = � t � . Another easy lemma by induction: The number of symbols in t is at most a . Therefore, there exists a term construction sequence ( t 1 , . . . , t ℓ ) for t with length ≤ a . We may assume that each t k is a subterm of t , so that � t k � ≤ � t � = a for all k ∈ { 1 , . . . , ℓ } . Let c := � � t 1 � , . . . , � t ℓ � � . We have c = 2 � t 1 � +1 3 � t 2 � +1 · · · ( p ℓ ) � t ℓ � +1 ≤ ( p ℓ ) � t 1 � + ··· + � t ℓ � + ℓ ≤ ( p ℓ ) ℓa + ℓ ≤ ( p a ) a 2 + a . Easy fact: The a th prime number p a is at most a a . (In fact, p a ≤ 2 a 2 using the Prime Number Theorem: a (log a + log log a − 1) < p a < a (log a + log log a ) for all a ≥ 6.) We conclude that c ≤ a a ( a 2 + a ) ≤ a 2 a 3 .
∆ -Definition of Terms = { � t � : t is a term } Now there is an obvious way to define Term ( a ): Term ( a ) : ≡ ( ∃ c ) TermConSeq ( c, a ) . To make this a ∆-formula, we need an upper bound on c as a function of a . We may therefore take Term ( a ) : ≡ ( ∃ c ≤ Ea · SS 0 EaSSS 0 ) TermConSeq ( c, a ) . � �� � a 2 a 3
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