aa r X i v : . [ m a t h . L O ] J u l MONADIC SECOND ORDER LIMIT LAWS FORNATURAL WELL ORDERINGS
ANDREAS WEIERMANN
Abstract.
By combining classical results of B¨uchi, some elemen-tary Tauberian theorems and some basic tools from logic and com-binatorics we show that every ordinal α with ε ≥ α ≥ ω ω satisfiesa natural monadic second order limit law and that every ordinal α with ω ω > α ≥ ω satisfies a natural monadic second order Cesarolimit law. In both cases we identify as usual α with the class ofsubstructures { β : β < α } .We work in an additive setting where the norm function N as-signs to every ordinal α the number of occurrrences of the symbol ω in its Cantor normal form. This number is the same as the numberof edges in the tree which is canonically associated with α .For a given α with ω ≤ α ≤ ε the asymptotic probabilityof a monadic second order formula ϕ from the language of linearorders is lim n →∞ { β<α : Nβ = n ∧ β | =Φ } { β<α : Nβ = n } if this limit exists. If this limitexists only in the Cesaro sense we speak of the Cesaro asympoticprobability of ϕ .Moreover we prove monadic second order limit laws for the or-dinal segments below below Γ (where the norm function is ex-tended appropriately) and we indicate how this paper’s results canbe extended to larger ordinal segments and even to certain impred-icative ordinal notation systems having notations for uncountableordinals. We also briefly indicate how to prove the correspondingmultiplicative results for which the setting is defined relative to theMatula coding.The results of this paper concerning ordinals not exceeding ε have been obtained partly in joint work with Alan R. Woods. Introduction
This paper concerns logical limit laws for infinite ordinals. It is basedon methods and techniques from the theory of logical limit laws forclasses of finite structures and fundamental results about the monadictheory of ordinals by B¨uchi. For the analytic part we make use oftechniques developed by Bell, Burris and Compton [1, 2, 8].In 2001 the author discussed the possibility of logical limit laws forordinals with Kevin Compton at an AOFA-workshop in Tatihoo and
Mathematics Subject Classification.
Key words and phrases.
Schur’s Tauberian theorem, Hua’s Tauberian theorem,B¨uchi’s definability results, asymptotic density, ordinals, monadic second orderlogic, limit law, natural well orderings.
Compton very kindly suggested among other things to contact AlanWoods because Woods proved very general results about limit lawsfor finite trees [20]. This initiated a very fruitful interaction betweenWoods and the author over the years. Tragically Woods passed awayuntimely in december 2011.The cooperation with Woods led to a first publication about firstorder zero one laws and limit laws for ordinals [19] based on a mixtureof results from [18], [20] and [8].In this article we move our focus from first order logic to monadicsecond order logic. B¨uchi already provided a very explicit descriptionof the ordinal spectrum of monadic second order sentences. In thisarticle it is shown how this description can be combined with resultsfrom [1, 2, 8] and [18] to prove monadic second order limit laws andmonadic second order Cesaro limit laws using elementary Tauberianmethods.Alan Woods had originally in mind to prove the monadic secondorder results regarding the ordinals not exceeding ε by using Shelah’stheory of additive colourings [16]. We believe that this will prove usefulin future research which among other things could lead to an automatafree proof of this paper’s results.2. Some basic definitions
Let ε be the least ordinal ξ such that ξ = ω ξ where ω refers to thefirst infinite ordinal. The ordinal ε plays an important role in the prooftheory of first order arithmetic and related contexts (see, for example,[15]) but in this article we consider this ordinal (which is identifiedwith its segment of smaller elements) as an object which stands juston itself.By seminal work of Cantor we know that every ordinal α < ε canbe written uniquely as ω α + · · · + ω α n where α ≥ . . . ≥ α n . Thisnormal form allows to associate finite trees canonically to the ordinalsbelow ε .Indeed, by writing out the α i hereditarily in a similar fashion everyordinal is associated with a unique term representation. To each α < ε we can therefore canonically associate a finite tree T ( α ) in a recursivemanner as follows. T (0) is the singleton tree consisting only of itsroot and if α has Cantor normal form ω α + · · · + ω α n (meaning that α ≥ . . . ≥ α n ) and if the T ( α i ) are already constructed then let T ( α )be the tree with immediate subtrees T ( α i ) connected to a new root. T ( α ) is then a rooted non planar finite tree.Let N ( α ) be the number of edges in T ( α ) which is one plus thenumber of nodes in T ( α ). Then N N ( α ) = n + N ( α ) + . . . + N ( α n ) if α has Cantor normal form ω α + · · · + ω α n . In other words N α is the number of occurrences of ω in the ordinal α . SO LIMIT LAWS FOR NATURAL WELL ORDERINGS 3
The norm function N is additive in the sense that N ( α ) = N ( ω α ) + . . . + N ( ω α n ). For ω ≤ β ≤ ε let c β ( n ) := { α < β : N ( α ) = n } . Then c β ( n ) is a well defined natural number. Morever using techniques from[2, 8] it has been shown in [18] that c β ( n ) ∈ RT if β < ε . The latterrefers to terminology borrowed from [8] and means lim n → ∞ c β ( n ) c β ( n +1) =1 so that the radius of convergence of the associated generating functionis 1.To introduce limit laws for ordinals we work with relational (monadicsecond order) languages L < which come equipped with exactly onerelation symbol for the less than relation. We neither do allow constantsnor function symbols. For any L < sentence ϕ the semantics of α | = ϕ is defined in the natural way.For an L < sentence ϕ we can define δ βϕ ( n ) = { α<β : α | = ϕ ∧ N ( α )= n } c β ( n ) . If δ βϕ = lim n →∞ δ βϕ ( n ) exists for all ϕ under consideration then β ful-fills a monadic second order limit law and when lim n →∞ δ βϕ ( n ) existsin the Cesaro sense for all ϕ under consideration (which means thatlim n →∞ n P ni =0 δ βϕ ( i ) exists) then β fulfills a monadic second order Ce-saro limit law. Clearly limit laws yield Cesaro limit laws but not viceversa.For describing a corresponding multiplicative setting one might usethe Matula coding M : ε → ω which is defined as follows. Let M (0) =1 and M ( α ) = p M ( α ) · · · p M ( α n ) if α has Cantor normal form ω α + · · · + ω α n . Here p i refers to the i -th prime number starting with p = 2. Thiscoding is multiplicative in the sense that M ( α ) = M ( ω α ) · . . . · M ( ω α n ).For a sentence ϕ we can define ∆ βϕ ( n ) = { α<β : α | = ϕ ∧ M ( α ) ≤ n } C β ( n ) . Monadicsecond order Cesaro limit laws and monadic second order limit laws canbe defined accordingly in the multiplicative setting.3. Cesaro limit densities for semi linear subsets ofordinals below ω ω For this section let us fix an infinite β < ω ω . We first assume β = ω r +1 . We call a subset L of β linear if there exists a double sequence a r , b r , . . . , a , b of non negative integers such that L = { α = ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) : ( ∀ i ≤ r )[ l i < ω ] } . We call a subset L of β semi linear if it is a finite union of linear subsetsof β .For a semi linear subset L ⊆ β let D L ( n ) = { α ∈ L : N ( α ) = n } { α < β : N ( α ) = n } . We will show that C − lim →∞ D L ( n ) exists in the Cesaro sense. Thiswill be useful to show monadic second order limit laws for β in section6. ANDREAS WEIERMANN
In bypassing let us remark that in general the standard limit lim →∞ D L ( n )does not alway exist. We can take β = ω and L = { α = 2 · l : l < ω } .Then D L (2 n ) = 1 and D L (2 n + 1) = 0.Let us first show that Cesaro limits exist for linear subsets of β . Lemma 1.
Let β = ω r +1 . Suppose that L = { α = ω r · ( b r · l r ) + · · · + ω · ( b · l ) : ( ∀ i ≤ r )[ l i < ω ] } . If there exists an i ≤ r such that b i = 0 then lim n →∞ D L ( n ) = 0 .Proof. We can regard β , hence the set of ordinals less than β , as anadditive number system in the sense of [8] with set of primes givenby { ω j : 0 ≤ j ≤ r } . The norm function for this number system isgiven by N and the addition function is provided by the natural sumof ordinals.Let L ′ = { α = ω r · l r + · · · + ω i +1 · l i +1 + ω i · ω i − · l i − + · · · + ω · l :( ∀ i ≤ r )[ l i < ω ] } . Then L ′ can be considered as a partition set withsmall exponent 0 for the partition element { ω i } .Since c β ( n ) ∈ RT we conclude that lim n →∞ { α ∈ L ′ : Nα = n } c β ( n ) = 0 byCompton’s theorem 4.2 in [8].Since L ⊆ L ′ we see that lim n →∞ D L ( n ) = 0. (cid:3) So we are left with the case that all the b i are different from zero.Let us first recall Hua’s theorem (see, for example, theorem 2.48 in[8]). Theorem 1.
Suppose the additive number system A has finite rank r with d the gcd of supp ( p ( n )) . Let supp ( p ( n )) = { d , . . . , d k } and let p ( d i ) = m i . Then a ( nd ) ∼ d r ( r − Q d mii · n r − as n → ∞ . Lemma 2.
Let β = ω r +1 and L = { α = ω r · ( b r · l r ) + · · · + ω · ( b · l ) :( ∀ i ≤ r )[ l i < ω ] } where no b i is zero. Let d := gcd (( r + 1) · b r , . . . , · b ) . Then lim n →∞ D L ( d · n ) = d Q ri =1 b i . Proof.
Since all b i are non zero we can regard L as an additive numbersystem with primes ω i · b i (0 ≤ i ≤ r ). Then L has rank r + 1. More-over, d = gcd( supp ( p ( n )) where supp ( p ( n )) = { N ( ω r · b r ) , N ( ω r − ) · b r − , . . . , N ( ω · b ) } . Then d = gcd(( r + 1) · b r , . . . , { α ∈ L : N α = n · d } ∼ d r +1 r ! · Q ri =0 ( i +1) · b i · n r .Also β itself can be seen as an additive number system with primes ω i (0 ≤ i ≤ r ). Then β has rank r + 1. Since gcd( N ω r , . . . , N ω ) = 1a second application of Hua’s result yields c β ( n ) ∼ r ! · Q ri =0 ( i +1) · n r .Putting things together we find lim n →∞ D L ( d · n ) = lim n →∞ { α ∈ L : Nα = nd } c β ( nd ) = d Q ri =1 b i . SO LIMIT LAWS FOR NATURAL WELL ORDERINGS 5 (cid:3)
Lemma 3.
Let β = ω r +1 and L = { α = ω r · ( b r · l r ) + · · · + ω · ( b · l ) :( ∀ i ≤ r )[ l i < ω ] } where no b i is zero. Let d := gcd (( r + 1) · b r , . . . , · b ) . Then C − lim n →∞ D L ( d · n ) = Q ri =1 b i .Proof. Let 0 < e < d . Then there will be no α ∈ L such that N α = nd + e . Otherwise such an α has the form α = ω r · ( b r · l r )+ · · · + ω · ( b · l ).From N α = nd + e we would conclude N ( ω r · b r ) · l r + · · · + N ( ω · b ) · l − nd = e. Then by theorem 4.1 in [17] the number gcd( N ( ω r · b r ) , . . . , N ( ω · b ) , d ) would divide e which is absurd. Therefore D L ( n · d + e ) = 0. Combining this with lim n →∞ D L ( d · n ) = d Q ri =1 b i we findthat C − lim n →∞ D L ( n ) = lim n →∞ n P ni =1 D L ( i ) = Q ri =1 b i . (cid:3) We now consider semi linear sets where the a i might be non zero. Lemma 4.
Let β = ω r +1 and let L = { α = ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) : ( ∀ i ≤ r )[ l i < ω ] } be a semi linear subset of β . (1) If some b i = 0 then lim n →∞ D L ( n ) = 0 . (2) If all b i are non zero then C − lim n →∞ D L ( d · n ) = Q ri =1 b i .Proof. Let L ′ = { α = ω r · ( b r · l r ) + · · · + ω · ( b · l ) : ( ∀ i ≤ r )[ l i < ω ] } .Then α = ω r · ( a r + b r · l r )+ · · · + ω · ( a + b · l ) ∈ L has norm n if α ′ = ω r · ( b r · l r )+ · · · + ω · ( b · l ) ∈ L ′ has norm n − N ( ω r · a r ) − . . . − N ( ω · a ).Hence D L ( n + ( r + 1) · a r + · · · + a ) = D L ′ ( n ). Therefore the previousresults obviously carry over from L ′ to L since c β ( n ) ∈ RT . (cid:3) Let us now consider semi linear subsets.
Lemma 5.
Let β = ω r +1 . If L and L ′ are linear subsets of β then L ∩ L ′ is either empty or again a linear set. The same conclusion holdsfor any finite intersection of linear sets.Proof. The second assertion follows from the first by an obvious induc-tion.Now assume that L = { α = ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) :( ∀ i ≤ r )[ l i < ω ] } and L ′ = { α = ω r · ( a ′ r + b ′ r · l r ) + · · · + ω · ( a ′ + b ′ · l ) :( ∀ i ≤ r )[ l i < ω ] } .Then α ∈ L ∩ L ′ iff α = ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) = ω r · ( a ′ r + b ′ r · l ′ r ) + · · · + ω · ( a ′ + b ′ · l ′ ) iff for all j ≤ r we have a j + b j l j = a ′ j + b ′ j l ′ j iff for all j ≤ r we have a j − a ′ j = b ′ j l ′ j − b j l j .So if there would exist an j ≤ r such that gcd( b j , b ′ j ) does not divide a j − a ′ j then by theorem 4.1 of [17] we would obtain L ∩ L ′ = ∅ . So letus assume that gcd( b j , b ′ j ) does divide a j − a ′ j for all j ≤ r .For the moment let us fix a j ≤ r . Assume that ( ⋆ ) a j + b j l j = a ′ j + b ′ j l ′ j . Let h j := gcd( b j , b ′ j ). Assume l ∗ j is the minimal non negative ANDREAS WEIERMANN left hand side of a non negative solution l j , l ′ j of ( ⋆ ). Let l ′∗ j be the righthand side of such a solution. By theorem 4.1 in [17] the set of all integersolution of ( ⋆ ) has the form ( l ∗ j , l ′∗ j ) + Z ( b ′ j h j , b j h j ). But since we are onlyinterested in non negative solutions and since ( l ∗ j , l ′∗ j ) is left minimalall non negative solutions of ( ⋆ ) have the form ( l ∗ j , l ′∗ j ) + N ( b ′ j h j , b j h j ). Atypical right hand side of ( ⋆ ) thus has the form a j + b j ( l ∗ j + l j b ′ j h j ) = a j + b j l ∗ j + l j b j b ′ j h j . This analysis can be done for the other j ’s as welland therefore L ∩ L ′ = { α = ω r · ( a r + b r l ∗ r + b r b ′ r h r l r ) + · · · + ω · ( a + b l ∗ + b b ′ h l ) : ( ∀ i ≤ r )[ l i < ω ] } which is a linear set. (cid:3) Lemma 6.
Let β = ω r +1 . Then Cesaro limit densities exist for allsemi linear subsets of β. Proof.
Cesaro limits distribute over finite sums. The counting functionfor a given semi linear set can be calculated as a finite sum of countingfunctions of linear sets (a sum with possibly negative integer coeffi-cients). This follows from the inclusion exclusion principle stating thatfor finite sets A i with i ranging over a finite set I we have that | [ i ∈ I A i | = n X k =1 ( − k − X ( I ∈{ ,...,n } k ) | \ i ∈ I A i | . (cid:3) Finally let us consider a general β of the form β = ω r · c r + · · · + ω · c .Then the set of ordinals α less than β can be written as a disjoint unionover sets L j,k j := { α = ω r · c r + · · · + ω j · ( c j − k j )+ ω j − l j − + · · · + ω · l :( ∀ i ≤ j )[ l i < ω ] } where j ≤ r and k j < c j Then α = ω r · c r + · · · + ω j · ( c j − k j ) + ω j − l j − + ω · l ∈ L j,k j ⇐⇒ ω j − l j − + · · · + ω · l < ω r .If L = { α = ω r · ( a r + b r · l r )+ · · · + ω · ( a + b · l ) : ( ∀ i ≤ r )[ l i < ω ] } isa linear subset of β then b r = 0 since c r < ω . Moreover a r = c r − k r forsome k r . Let L ′ = { α = ω r − · ( a r − + b r − · l r − ) + · · · + ω · ( a + b · l ) :( ∀ i ≤ r − l i < ω ] } . Then L ′ is a linear subset of ω r .Hence { α ∈ L : N α = N ( ω r · a r ) + n } = { α ∈ L ′ : N α = n } .Since c β ( n ) ∈ RT we conclude that the Cesaro density for L existssince the Cesaro limit for L ′ exists. Since by the same proof as beforealso in this situation the intersection of linear sets is either empty oragain a linear set we have proved. Theorem 2.
Let ω ≤ β < ω ω . Then Cesaro limit densities exist forall semi linear subsets of β. Theorem 3.
Let ω ≤ β < ω ω . Then the Cesaro limit densities arerational for all semi linear subsets of β. SO LIMIT LAWS FOR NATURAL WELL ORDERINGS 7
Proof.
The limit densities resulting in our setting from Hua’s theoremfor linear sets are rational numbers. The resulting limiting densitiesfor semilinear sets are formed by taking finite sums with integer coef-ficients. This yields the assertion. (cid:3) Limit laws for semi linear sets of ordinals stretchingabove ω ω For this section let us fix an ordinal β with ε > β ≥ ω ω . Letus first concentrate on the case where β of the form β = ω γ where γ ≥ ω . Then β is closed under ordinal addition and it forms with thestandard norm function an additive number system with respect to thenatural sum operation. This additive number system has primes givenby { ω α : α < γ } and it has an infinite rank.As before, we call a subset L of β linear if there exists non negativeinteger r , the length of L , and a double sequence a r , b r , . . . , a , b ofnon negative integers such that L = { α = ω r +1 · α + ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) : ( ∀ i ≤ r )[ l i < ω ] ∧ α < β } . We call a subset L of β semi linear if it is a finite union of linear subsets of β .Obviously, if a linear set is defined with respect to length r and adouble sequence a r , b r , . . . , a , b then for any s ≥ r we can put a l = 0and b l = 1 for r < l ≤ s to obtain a representation using the sequence a s , b s , . . . , a , b . So in forming finite unions and intersections of linearsets we can always assume that the linear sets have the same lengths.For a semi linear subset L ⊆ β let D L ( n ) = { α ∈ L : N ( α ) = n } { α < β : N ( α ) = n } . We will show that lim →∞ D L ( n ) always exists in the usual sense.This will be useful to show monadic second order limit laws for thesegment of ordinals determined by β in section 6. Lemma 7.
Let β = ω γ where ε > γ ≥ ω . Suppose that L = { α = ω r +1 · α + ω r · ( b r · l r ) + · · · + ω · ( b · l ) : ( ∀ i ≤ r )[ l i < ω ] ∧ α < β } .If there exists an i ≤ r such that b i = 0 then lim n →∞ D L ( n ) = 0 .Proof. As before we can regard β , hence the set of ordinals less than β , as an additive number system in the sense of [8] with set of primesgiven by { ω j : 0 ≤ j < γ } . The norm function for this number systemis given by N and the addition function is provided by the natural sum.Let L ′ = { α = ω r +1 · α + ω r · l r + · · · + ω i +1 · l i +1 + ω i · ω i − · l i − + · · · + ω · l : α < β ∧ ( ∀ i ≤ r )[ l i < ω ] } . Then L ′ can be considered asa partition set with small exponent 0 for the partition element { ω i } .Since c β ( n ) ∈ RT we conclude that lim n →∞ { α ∈ L ′ : Nα = n } c β ( n ) = 0 byCompton’s theorem 4.2 in [8].Since L ⊆ L ′ we see that lim n →∞ D L ( n ) = 0. (cid:3) ANDREAS WEIERMANN
So we are left with the case that all the b i are different from zero.Let us now recall Schur’s theorem (theorem 3.42 in [8]). Theorem 4.
Let S ( x ) , T ( x ) be two power series such that for some ρ ≥ T ( x ) ∈ RT ρ , and (2) S ( x ) has radius of convergence ρ s greater than ρ .Then lim [ x n ]( S ( x ) · T ( x )[ x n ] T ( x ) = S ( ρ ) . Here [ x n ] T ( x ) refers to the n -th coefficient of the power series T ( x )and [ x n ] S ( x ) is defined correspondingly. T ( x ) ∈ RT ρ means thatlim n →∞ [ x n ] T ( x )[ x n +1 ] T ( x ) = ρ . Lemma 8.
Let ω ω ≤ β = ω γ < ε and L β := { α : α = ω r +1 · α + ω r · ( b r · l r ) + · · · + ω · ( b · l ) : α < β ∧ ( ∀ i ≤ r )[ l i < ω ] } where all b i are different from zero. Let l β ( n ) = { α ∈ L β : N α = n } . Then lim n →∞ D L ( n ) = lim n →∞ l β ( n ) c β ( n ) = S (1) = b r · ... · b .Proof. Let l β ( n ) := { α ∈ L β : N α = n } . The set L β can be seen asan additive number system with primes in the set ˜ P := { ω ξ : γ > ξ >r } ∪ { ω i · b i : i ≤ r } .Let S ( x ) = (1 + · · · + x ( r +1)( b r − ) · . . . · (1 + · · · + x · ( b − ) and T ( x ) = P l β ( n ) x n . By theorem 2.20 in [8] we obtain for real numbers x < S ( x ) · T ( x )= S ( x ) · X l β ( n ) x n = S ( x ) Y p ∈ ˜ P (1 − x N ( p ) ) − = ( Y ω ξ : γ>ξ>r (1 − x N ( ω ξ ) ) − ) · S ( x ) · (1 − x N ( ω r · b r ) ) − · . . . · (1 − x N ( ω · b ) ) − = Y ω ξ : γ>ξ>r (1 − x N ( ξ )+1 ) − · (1 − x r +1 ) − · . . . · (1 − x ) − = X c β ( n ) x n . The radius of convergence of S ( x ) is infinite, hence bigger than 1 andso Schur’s Tauberian Theorem is applicable and yields lim n →∞ c β ( n ) l β ( n ) = S (1) = b r · . . . · b . By taking inverses this yields lim n →∞ D L ( n ) =lim n →∞ l β ( n ) c β ( n ) = b r · ... · b . (cid:3) We now consider semi linear sets where the a i might be non zero. Lemma 9.
Let ω ω ≤ β = ω γ < ε and let L = { α = ω r +1 · α + ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) : α < β ∧ ( ∀ i ≤ r )[ l i < ω ] } be asemi linear subset of β . (1) If some b i = 0 then lim n →∞ D L ( n ) = 0 . SO LIMIT LAWS FOR NATURAL WELL ORDERINGS 9 (2)
If all b i are non zero then C − lim n →∞ D L ( d · n ) = Q ri =1 b i .Proof. Let L ′ = { α = ω r +1 · α + ω r · ( b r · l r ) + · · · + ω · ( b · l ) : α <β ∧ ( ∀ i ≤ r )[ l i < ω ] } .Then α = ω r +1 · α + ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) ∈ L hasnorm n if α ′ = ω r +1 · α + ω r · ( b r · l r ) + · · · + ω · ( b · l ) ∈ L ′ has norm n − N ( ω r · a r ) − . . . − N ( ω · a ). Hence D L ( n + ( r + 1) · a r + · · · + a ) = D L ′ ( n ).Since c β ( n ) ∈ RT we see c β ( n + ( r + 1) · a r + · · · + a ) ∼ c β ( n ) as n → ∞ .Therefore the previous results easily carry over from L ′ to L . (cid:3) Let us now consider semi linear subsets of β = ω γ . Lemma 10.
Let ω ω ≤ β = ω γ < ε . If L and L ′ are linear subsets of β then L ∩ L ′ is either empty or again a linear set. The same conclusionholds for any finite intersection of linear sets.Proof. The proof from the last section carries over immediately. (cid:3)
Lemma 11.
Let ω ω ≤ β = ω γ < ε . Then limit densities exist for allsemi linear subsets of β. Proof.
Standard limits distribute over finite sums. The counting func-tion for a given semi linear set can be calculated as a finite sum countingfunctions of linear sets (a sum with possibly negative integer coeffi-cients). This follows from the inclusion exclusion principle. (cid:3)
Finally let us consider a general β of the form β = ω γ · d + · · · + ω γ s d s .Then the set of ordinals α less than β can be written as a disjoint unionover sets L j,k j := { α = ω γ · d + · · · + ω γ j ( d j − k j ) + δ where δ < ω γ j .Then α = ω γ · d + · · · + ω γ j ( d j − k j ) + δ ∈ L j,k j ⇐⇒ δ < ω γ j .Since c β ( n ) ∈ RT we conclude that the density for L exists for alllinear sets.Since by the same proof as before also in this situation the intersec-tion of linear sets is either empty or again a linear set we have proved. Theorem 5.
Let ω ω ≤ β < ε . Then limit densities exist for all semilinear subsets of β. Theorem 6.
Let ω ω ≤ β < ε . Then the limit densities are rationalnumbers for all semi linear subsets of β. Proof.
The limit densities resulting in our setting from Schur’s theoremfor linear sets are rational numbers. The resulting limiting densities forsemi linear sets are formed by taking finite sums with integer coeffi-cients. This yields the assertion. (cid:3) Limit laws for ordinals at least a big as ε For this section let us fix ε ≤ β = ω γ . A typical choice wouldbe β = ε or β = Γ (see, for example, [15] for a definition) or theBachmann Howard ordinal (see, for example, [5] for a definition). Weassume that c β ( n ) ∈ RT ρ for ρ <
1. This is our standing assumptionand will be true for all notations systems known from the literature.For ε this follows because enumerating ordinals below ε comes downto counting finite rooted non planar trees. The generating function forcounting these trees has radius of convergence smaller than one by [13].Counting ordinals below Γ comes down to counting 2-trees. Herethe norm function satisfies N ( ϕαβ + γ ) = 1 + N α + N β + N γ where ϕαβ + γ is in Cantor normal form and ϕ denotes the binary fixed pointfree Veblen function. The resulting generating function for countingthese trees has radius of convergence smaller than one by [10, 4].In general the generating function for counting such ordinals has ra-dius of convergence smaller than one by the general theory for countingtrees as for example explained in [3, 9].The ordinals below β form again an additive number system withprimes given by { ω α : α < γ } . This system has an infinite rank.We can define linear and semilinear sets as in the last section andthe analysis proceeds exactly as before. The only difference is that weare now no longer in the RT case.So we call a subset L of β linear if there exists a non negative inte-ger r , the length of L , and a double sequence a r , b r , . . . , a , b of nonnegative integers such that L = { α = ω r +1 · α + ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) : ( ∀ i ≤ r )[ l i < ω ] ∧ α < β } . We call a subset L of β semi linear if it is a finite union of linear subsets of β .For a semi linear subset L ⊆ β let D L ( n ) = { α ∈ L : N ( α ) = n } { α < β : N ( α ) = n } . We will show that lim →∞ D L ( n ) exists in the usual sense. This willagain be useful to show monadic second order limit laws for the segmentof ordinals determined by β . In contrast to the previous section let usremark that L = { α = ω r +1 · α + ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) :( ∀ i ≤ r )[ l i < ω ] ∧ α < β } and there exists an i ≤ r such that b i = 0does not imply lim n →∞ D L ( n ) = 0. The reason is that we cannot applyCompton’s theorem 4.2 in [8].In the new setting we still can apply Schur’s theorem and it will alsocover the case where some b i = 0. Lemma 12.
Let ε ≤ β = ω γ and assume that β forms an additivenumber system with respect to a natural norm function which extendsthe norm function for the ordinals less than ε and where the primesare given by the set P := { ω α : α < γ } . Assume that the associated SO LIMIT LAWS FOR NATURAL WELL ORDERINGS 11 generating function has radius of convergence ρ strictly smaller thanone. Let L β := { α : α = ω r +1 · α + ω r · ( b r · l r )+ · · · + ω · ( b · l ) : ( ∀ i ≤ r )[ l i < ω ] ∧ α < β } and let l β ( n ) = { α ∈ L β : N α = n } . Moreover let S ( x ) := Y i : b i > (1 + · · · + x ( i +1)( b i − ) · Y i : b i =0 − x i . Then lim n →∞ D L ( n ) = lim n → ∞ l β ( n ) c β ( n ) = 1 S ( ρ ) . Proof.
The set L β := { α : α = ω r +1 · α + ω r · ( b r · l r ) + · · · + ω · ( b · l ) :( ∀ i ≤ r )[ l i < ω ] ∧ α < β } can be seen as an additive number systemwith primes in the set ˜ P := { ω ξ : γ > ξ > r } ∪ { ω i · b i : i ≤ r } .Let T ( x ) := P ∞ n =0 { α ∈ L β : N α = n } x n .By theorem 2.20 in [8] we obtain S ( x ) · T ( x )= S ( x ) Y p ∈ ˜ P (1 − x N ( p ) ) − = ( Y ω ξ : γ>ξ>r (1 − x N ( ω ξ ) ) − ) · S ( x ) · (1 − x N ( ω r · b r ) ) − · . . . · (1 − x N ( ω · b ) ) − = Y ω ξ : γ>ξ>r (1 − x N ( ξ )+1 ) − · (1 − x r +1 ) − · . . . · (1 − x ) − · = X c β ( n ) x n . The radius of convergence of S ( x ) is at least as big as 1 and so Schur’sTauberian Theorem is applicable and yields lim n →∞ c β ( n ) l β ( n ) = S ( ρ ). Bytaking inverses this yields lim n →∞ D L ( n ) = lim n →∞ l β ( n ) c β ( n ) = S ( ρ ) . (Notethat S ( ρ ) is defined since the radius of convergence of S is strictlybigger than one.) (cid:3) We now consider semi linear sets where the a i might be non zero. Lemma 13.
Let ε ≤ β = ω γ and assume that β forms an additivenumber system with respect to a natural norm function which extendsthe norm function for the ordinals less than ε and where the primesare given by the set P := { ω α : α < γ } . Assume that the associatedgenerating function has radius of convergence ρ strictly smaller thanone. Let L = { α = ω r +1 · α + ω r · ( a r + b r · l r ) + · · · + ω · ( a + b · l ) :( ∀ i ≤ r )[ l i < ω ] ∧ α < β } be a semi linear subset of β . Then thelimiting density for L exists.Proof. Let L ′ = { α = ω r +1 · α + ω r · ( b r · l r )+ · · · + ω · ( b · l ) : ( ∀ i ≤ r )[ l i <ω ] ∧ α < β } . Then α = ω r +1 · α + ω r · ( a r + b r · l r )+ · · · + ω · ( a + b · l ) ∈ L has norm n if α ′ = ω r +1 · α + ω r · ( b r · l r )+ · · · + ω · ( b · l ) ∈ L ′ has norm n − N ( ω r · a r ) − . . . − N ( ω · a ). Hence D L ( n + ( r + 1) · a r + · · · + a ) = D L ′ ( n ).Since c β ( n ) ∈ RT ρ we see c β ( n +( r +1) · a r + · · · + a ) ∼ ρ ( r +1) · a r + ··· + a · c β ( n ) as n → ∞ .Therefore the previous results easily carry over from L ′ to L .Let us now consider semi linear subsets of β = ω γ . Lemma 14.
Let ω ω ≤ β = ω γ < ε . If L and L ′ are linear subsets of β then L ∩ L ′ is either empty or again a linear set. The same conclusionholds for any finite intersection of linear sets. Proof. The proof from the last section carries over immediately. (cid:3)
Lemma 15.
Let ω ω ≤ β = ω γ < ε . Then limit densities exist for allsemi linear subsets of β. Proof. Standard limits distribute over finite sums. The countingfunction for a given semi linear set can be calculated as a finite sumcounting functions of linear sets (a sum with possibly negative integercoefficients). This follows again from the inclusion exclusion principle. (cid:3)
Finally let us consider a general β of the form β = ω γ · d + · · · ω γ s d s .Then the set of ordinals α less than β can be written as a disjoint unionover sets L j,k j := { α = ω γ · d + · · · ω γ j ( d j − k j ) + δ : δ < ω γ j } .Then α = ω γ · d + · · · ω γ j ( d j − k j ) + δ ∈ L j,k j ⇐⇒ δ < ω γ j .Since c β ( n ) ∈ RT ρ we find that the density for L exists for all linearsets.Since by the same proof as before also in this situation the intersec-tion of linear sets is either empty or again a linear set we have proved. Theorem 7.
Let ε ≤ β = ω γ . Then limit densities exist for all semilinear subsets of β. Proof. (cid:3)
Let Q ( ρ ) be the least field containing ρ . Theorem 8.
Let ε ≤ β = ω γ . Then the limit densities are elementsof Q ( ρ ) for all semi linear subsets of β. Proof.
The limit densities resulting in our setting from Schur’s theoremfor linear sets are in Q ( ρ ) . The resulting limiting densities for semilinear sets are formed by taking finite sums with integer coefficients.This yields the assertion. (cid:3) Finally let us briefly discuss the case where β is the Howard Bach-mann ordinal. Every ordinal from the standard notation system forthe Howard ordinal (when it is built on the ϑ function) can be wewritten as a finite multiset over terms of the form D α and D α SO LIMIT LAWS FOR NATURAL WELL ORDERINGS 13 where α is again such an ordinal. (See, for example, [5] for a proof.)This means that in the Flajolet Sedgewick notation [9] we find OT =Mult( { D , D } × OT ). Every countable ordinal in OT can be written amultiset over terms of the form D α with α ∈ OT . This means that forcounting these numbers we need to count the class CT = Mult( { D } × OT ). For the induced generating functions this means that OT ( x ) =exp( P ∞ i =1 x i · · OT ( x i ) i ) and that CT ( x ) = exp( P ∞ i =1 x i · OT ( x i ) i ). These willbe tree generating functions with radius of convergence < Monadic second order (Cesaro) limit laws for naturalwell orderings
Let sgn( α ) be 1 if α > α ) be 0 otherwise. Let usrecall the well known theorem by B¨uchi (theorem 4.8 in [6]) on thecountable spectrum of monadic second order sentences. Theorem 9.
Let ϕ be a monadic second order sentence in the languageof linear orders. Then there exists a finite number r and there exist afinite set K , an element a ∈ K , a subset W ⊆ K and operations F , . . . , F r +1 on K such that for all countable ordinals α of the form α = ω r +1 · α + ω r · k r + . . . + ω · k we have the equivalence: α | = ϕ iff F k · · · F k r r F sgn( α ) r +1 ( a ) ∈ W .Moreover F r +1 = F r +1 and i < j ≤ r yields F j F i = F j . So let us assume a monadic second order sentence ϕ in the languageof linear orders is given and choose p , K, a, W, F , . . . , F r +1 accordingto B¨uchi’s theorem. Then clearly for every i we find numbers a i and b i such that F a i i = F a i + b i i since F i : K → K and K is finite. For α = ω r +1 · α + ω r · k r + . . . + ω · k we then find that α | = ϕ iff α | = ϕ where α = ω r +1 · α + ω r · k r + . . . + ω · k . Here k i = k i if k i < a i and k i = a i + µc : k i − a i = c ( mod b i ). So the spectrum of ϕ consistsof a semi linear set for which we have proved that limit densities (for β ≥ ω ω ) or Cesaro limit densities (for ω ≤ β < ω ω ) exist.This yields for infinite ordinals β < ω ω a monadic second order Ce-saro limit law and for β ≥ ω ω a monadic second order limit law. More-over the proofs yield that these limits will always be rational numberswhen β < ε .7. Weak Monadic second order limit laws for ordinals inthe presence of addition and multiplication
We can define h ω, + , ·i in the weak monadic second order languageover any infinite structure h α, + i . To define · we use the well knownfact that x divides y is definable on the smallest limit element by thefollowing description: There is a finite set X such that x ∈ X and suchthat for every not maximal element v ∈ X we have v + x ∈ X and and such that y is the maximal element of X and x is the minimal elementin X and for every element v in X which is not x there exists a w ∈ X such that w + x = v . Then we can define squaring over ω by y, y + 1both divide x + y and for all z < x + y if y divides z then then it is notthe case that y + 1 divides u . Then multiplication is defined by x · y = z if there are u, v, w ( u = x ∧ v = y ∧ w = ( x + y ) ∧ w = u + v + z + z ).This yields that no algorithm can separate ϕ ( from the weak monadicsecond order language) with δ βn ( ϕ ) → δ βn ( ϕ ) → β ≥ ω .By additional results of B¨uchi’s about the ordinal spectrum of weakmonadic second order logc we obtain from the previous results of thispaper weak monadic second order limit laws for ω ≤ β ≤ ε .Weak monadic second order limit laws lead to first order limit lawswith respect to L ( <, +) for classes of structures { ω α : α < β } where ω ≤ β ≤ ε . These can be inherited because by a theorem of Ehren-feucht for any choice of δ, δ ′ the structure h ω ω ω · δ + α i is W M SO ( <, +)elementarily equivalent with h ω ω ω · δ ′ + α i since h ω ω · δ + α i is W M SO ( < )elementarily equivalent with h ω ω · δ ′ + α i (for a proof confer, e.g. [14]).Moreover this leads to first order limit laws for L ( <, + , · ) for classes ofstructures { ω ω α : α < β } where ω ≤ β ≤ ε . This is again a conse-quence of another result of Ehrenfeucht since the structure h ω ω ω · δ + α i is W M SO ( <, +) elementarily equivalent with h ω ω ω · δ ′ + α i . Final remarks: (1) The corresponding results for the multiplicative setting whichare defined relative to the Matula coding (when ordinals notexceeding ε are concerned) follow from our previous analy-sis together with Theorem 9.53 (the multiplicative version ofSchur’s theorem) and Theorem 8.30 (the multiplicative versionof Hua’s theorem) in [8].(2) We have shown monadic second order limit laws for β ≥ ε inthe additive setting. We intend to investigate whether similarresults holds in the multiplicative setting.(3) A very exciting extension of our work concerns logical limitlaws for uncountable ordinals. Corresponding density notionscan be induced by working with ordinal notation systems for theBachmann Howard ordinal and farer reaching notation systems.We believe that monadic second order limit laws will hold for allordinals (including the uncountable ones) from such a notationsystem. For this B¨uchi’s theorem for ordinals less than ω seemsto be applicable [7]. Recent results by Itay Neeman [11, 12] seemto pave the way to study limit laws for ordinals above ω butwe quit at this point. References [1] J.P. Bell. Sufficient conditions for zero-one laws. Trans. Amer. Math. Soc. 354(2002), no. 2, 613–630
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