LLOGIC BLOG 2019
EDITOR: ANDR´E NIES
The Logic Blog is a shared platform for • rapidly announcing results and questions related to logic • putting up results and their proofs for further research • parking results for later use • getting feedback before submission to a journal • foster collaboration.Each year’s blog is posted on arXiv a 2-3 months after the year has ended.Logic Blog 2018 (Link: http://arxiv.org/abs/1902.08725 )Logic Blog 2017 (Link: http://arxiv.org/abs/1804.05331 )Logic Blog 2016 (Link: http://arxiv.org/abs/1703.01573 )Logic Blog 2015 (Link: http://arxiv.org/abs/1602.04432 )Logic Blog 2014 (Link: http://arxiv.org/abs/1504.08163 )Logic Blog 2013 (Link: http://arxiv.org/abs/1403.5719 )Logic Blog 2012 (Link: http://arxiv.org/abs/1302.3686 )Logic Blog 2011 (Link: http://arxiv.org/abs/1403.5721 )Logic Blog 2010 (Link: http://dx.doi.org/2292/9821 ) How does the Logic Blog work?Writing and editing.
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Computability of Ergodic Convergence . In Andr´e Nies (editor),Logic Blog, 2012, Part 1, Section 1, available at http://arxiv.org/abs/1302.3686 .The logic blog, once it is on arXiv, produces citations on Google Scholar. a r X i v : . [ m a t h . L O ] F e b EDITOR: ANDR´E NIES
Contents
Part 1. Computability theory and randomness
31. Barmpalias and Nies: a randomness notion between ML andKL-randomness 32. Nies: For non-high states, strong Solovay tests are as powerful asgeneral quantum ML tests 43. Franklin and Turetsky: (non)convexity of difference randomdegrees 5
Part 2. Group theory and its connections to logic
64. Nies and Segal: Non-axiomatizability for classes of profinitegroups 64.1. Number of generators 64.2. Comparison with the discrete case 74.3. Finite axiomatisability implies that the set of primes that occuris finite 75. Nies, Perin and Segal: all f.g. profinite groups are ω -homogeneous 86. Nies and Stephan: Some properties ofinfinitely generated nilpotent-of-class-2 groups 96.1. Three properties of an elementary abelian group 96.2. Nilpotent of class 2 groups 106.3. FA-presentability 11 Part 3. Computability theory and set theory A and T and 1-generic sets, totally low sets, and not computing aMAD 168.5. C.e. MAD sets 178.6. The mass problem U corresponding to the ultrafilter number,and a strict reduction T < S U Part 4. Model theory and definability
OGIC BLOG 2019 3
Part Computability theory and randomness Barmpalias and Nies: a randomness notion between ML andKL-randomness
Jason Rute [27, Section 10] introduced possible strengthenings of Kolmo-gorov-Loveland (KL)-randomness that are still implied by ML-randomness.In fact he worked in the general setting of computable probability spaces,where he also defined a notion of computable randomness. The followingis one of his notions. It relies on the little explored notion of partial com-putable randomness; see [17, Ch. 7]. (A betting strategy can be undefinedoff the sequences it has to succeeds on, which allows for potentially indefinitewaiting before making a bet.)
Definition 1.1. Z ∈ N is called Rute random if Φ( Z ) is partial computablyrandom for each measure preserving Turing functional Φ.A pair R = (Φ , L ) of a m.p. Turing functional and a partial computablemartingale will be called a Rute betting strategy . We say that R is total ifboth Φ and L are total functions.To say that Φ is measure preserving means that λ Φ − ( C ) = λ C for eachmeasurable C ⊆ N . It suffices to require this for cylinders C = [ σ ] ≺ where σ is any string. Such Turing functionals are almost total: the domain is aconull Π set. So Φ( Z ) is defined for any Kurtz “random” Z . Its kernel {(cid:104) X, Y (cid:105) : Φ X = Φ Y ↓} is a Π equivalence relation with all classes null. Φis given by a binary tree of uniformly Σ sets U σ = Φ − ([ σ ] ≺ ) such that λU σ = 2 −| σ | .If Φ is total (i.e., a tt functional) then Φ is onto by a compactness ar-gument. In this case the kernel is Π . An easy example of a m.p. totalfunction is the shift functional T where T ( Z ) is Z with the first bit erased.For a more interesting example, view bit sequences as 2-adic numbers, andconsider a computable unit r ∈ Z ∗ (anything that ends in 1). Then Φ Z = rZ is a m.p. total Turing functional, with inverse Ψ Z = r − Z . This examplecan be generalised to sequences over a p -bit alphabet, for a prime p .Each scan rule in the sense of [13, 17] induces a measure preserving ttfunctional on Cantor space. So Rute randomness implies KL-randomness.The sets Φ − ( { X : L ( X ) > m } ) form a ML-test that succeeds on each Z on which the strategy does. So ML-randomness implies Rute randomness. Fact 1.2.
Given a Rute strategy R = (cid:104) Φ , L (cid:105) with Φ total but possibly L partial, there are total Rute strategies R , R such that if R succeeds on Z then one of the R i succeeds as well.Proof. We use the similar result from [13] that given a KL-strategy K (alsosee [17, Ex. 7.6.25]) we can build two total KL-strategies K , K that togethercover the success set of K .In our case K is simply L (with the identity scan rule). We obtain scanrules K i = (cid:104)S i , B i (cid:105) Let Φ i = S i ◦ Φ which are m.p. total Turing functionals.Now the R i given by Φ i , B i are as needed. (cid:3) Recall that each KL-betting strategy fails on some c.e. set (e.g [13]. Thisargument doesn’t quite go through. However the following is easy.
EDITOR: ANDR´E NIES
Proposition 1.3.
Each Rute strategy (Φ , L ) fails on a non-Kurtz random.Proof. We may assume that Φ is total, else it fails by not being defined.So we can also assume that L is total. Let R be a computable path alongwhich L is bounded (say in the n -th bet, along R the martingale L goes upby at most 2 − n ; this is a Σ event which has to happen on at least one 1-bitextension of a string). Then Φ − ( R ) is a Π null class on which the Rutestrategy fails. (cid:3) As a consequence, if we wanted to show that Rute = ML, we’d need atleast two strategies to cover all non-MLR randoms.
Question 1.4.
Determine lowness for Rute randomness. Note that it iscontained in Low ( MLR , CR ) which coincides with the K -trivials [15] . Nies: For non-high states, strong Solovay tests are aspowerful as general quantum ML tests
Recall that for a non-high bit sequence Z , Schnorr randomness impliesML-randomness (e.g. [17, Ch. 3]). Here we give a similar argument in thesetting of randomness for infinite sequences of qubits [18].We use the convention as in the final section of [18] that all states andprojections are elementary, i.e. the relevant matrices consist of algebraiccomplex numbers. So it is OK to directly use a state ρ as an oracle. (Ingeneral we would have to work with elementary approximations to the den-sity matrices ρ (cid:22) n , similarly as is done in computable analysis to deal with,say, sequences of reals). Definition 2.1 ([18], 3.14) . • A quantum Solovay test is an effective sequence (cid:104) G r (cid:105) r ∈ N of quantumΣ sets such that (cid:80) r τ ( G r ) < ∞ . • We say that the test is strong if the G r are given as projections; thatis, from r we can compute n r and a matrix of algebraic numbers in M n r describing G r = p r . • For δ ∈ (0 , ρ fails the quantum Solovay test at order δ if ρ ( G r ) > δ for infinitely many r ; otherwise ρ passes the qML test at order δ . • We say that ρ is quantum Solovay-random if it passes each quantumSolovay test (cid:104) G r (cid:105) r ∈ N at each positive order, that is, lim r ρ ( G r ) = 0.Tejas Bhojraj showed that q-Solovay tests are equivalent to qML tests(but the order at which the test succeeds changes). However, it is likely thatstrong q-Solovay tests are indeed more restricted, hence the significance ofthe following Resultatchen . Proposition 2.2.
Let ρ be a non-high state on the usual CAR algebra M ∞ .If ρ fails some qML test (cid:104) G r (cid:105) r ∈ N at order δ > , then it fails a strong Solovaytest (cid:104) q r (cid:105) at the same order. Moreover we can achieve that (cid:80) r Tr ( q r ) is acomputable real.Proof. Given a qML-test (cid:104) G r (cid:105) = (cid:104) p rn (cid:105) as defined in [18, 3.5] that ρ failsat order δ , the function f ( r ) = µn. Tr ( ρ (cid:22) n p rn ) > δ is total and satisfies f ≤ T ρ . Since ρ is not high, there is a computable function h such that OGIC BLOG 2019 5 ∃ ∞ r.h ( r ) ≥ f ( r ). Let now q r = p rh ( r ) . This is a strong Solovay test of therequired form, and ρ fails it at the same order δ . (cid:3) Franklin and Turetsky: (non)convexity of differencerandom degrees
The ML-random degrees can be divided into two sets: those which com-pute (cid:48) , and those which do not compute a PA-degree. The latter correspondto the difference random degrees. The first set is known to be convex: if d and d are MLR degrees above (cid:48) , then every degree d with d ≤ d ≤ d is an MLR degree. This is because every degree above (cid:48) contains a ran-dom. Still, it’s natural to ask whether convexity holds for the differencerandom degrees. Here we construct a counterexample. Our constructionsimply requires combining several existing results.First we need the following: Proposition 3.1.
Every ∆ MLR degree computes a K -trivial which doesnot have c.e. degree. One could prove this directly, or just invoke Kuˇcera’s result that every ∆ MLR degree bounds a noncomputable c.e. degree, Hirschfeldt et al.’s [10]results showing that said degree must be K -trivial, and then Yates’s resultthat every noncomputable c.e. degree bounds a minimal degree (and thenNies [15] that K -triviality is downwards closed under Turing reducibilty).And I guess Sacks’s density theorem to get that a minimal degree is not ac.e. degree.Next, we need the following result of Greenberg and Turetsky (see here: https://arxiv.org/abs/1707.00258 ): Proposition 3.2.
For every K -trivial A there is a c.e. K -trivial B ≥ T A such that every ML-random which computes A computes B . Of course, if A does not have c.e. degree, then B will be strictly above A .Finally we’ll need this: Proposition 3.3. If A does not compute B , then for almost every oracle Z , A ⊕ Z does not compute B .Proof. Suppose otherwise. Fix an e such that for positive measure of oracles Z , Φ e ( A ⊕ Z ) = B . Apply Lebesgue density and then majority vote to showthat A computes B , contrary to hypothesis. (cid:3) We’ll also need van Lambalgen’s theorem. We’re ready to build the coun-terexample.Fix X an incomplete ∆ random. Fix A ≤ T X a K -trivial which doesnot have c.e. degree. Fix B > T A such that every ML-random computing A computes B . For almost every oracle Y , X ⊕ Y is random, X ⊕ Y doesnot compute (cid:48) , and A ⊕ Y does not compute B , so fix such a Y .Our failure of convexity is provided by Y ≤ T A ⊕ Y ≤ T X ⊕ Y . Suppose A ⊕ Y were Turing equivalent to a random Z . Then A ⊕ Y ≥ T Z ≥ T A ⊕ Y ,and thus Z ≥ T A . Then Z ≥ T B , by choice of B . So A ⊕ Y ≥ T B , contraryto choice of Y . EDITOR: ANDR´E NIES
It’s known that convexity holds for hyperimmune-free MLR degrees. Infact, if B is ML-random and of hyperimmune free degree, and A ≤ T B isnoncomputable, then A is of ML-random Turing degree using that A ≤ tt B and an argument of Demuth. How common are failures outside those? Doesevery hyperimmune MLR degree bound a failure of convexity? Is every pairof hyperimmune MLR degrees with one strictly above the other a failure ofconvexity? Part Group theory and its connections to logic Nies and Segal: Non-axiomatizability for classes ofprofinite groups
Expressiveness of first-order logic for classes of groups remains an inter-esting topic because it straddles logic and group theory and connects theseareas in novel ways. For topological groups the topic is particularly inter-esting, because in first-order logic (say in the language of groups) one can’tdirectly talk about open sets: first-order means that one can only talk aboutelements of the structure.Here we show that some commonplace properties cannot be axiomatisedby a single sentence.4.1.
Number of generators.
The following results and proofs work withsmall changes also in the setting of discrete groups.
Proposition 4.1. (a)
Being d -generated cannot be expressed by a single first order sentencewithin the profinite groups. (b) Being finitely generated cannot be expressed by a single first ordersentence within the profinite groups.Proof.
Let C q denote the cyclic group of order q . By a result of Szmielevrelated to her proof that the theory of abelian groups in decidable (1950s),for each sentence φ in the language of groups the following holds: for almostall primes q , an abelian group G satisfies φ iff G × C q satisfies φ .Recall that (cid:98) Z is the profinite completion of Z , which is the Cartesianproduct of all the Z p for p a prime. It is 1-generated and has each finitecyclic group as a quotient (in fact it is the free profinite group of dimension1). If φ expresses being d -generated, then ( (cid:98) Z ) d | = φ . Let q be a prime asabove, then also ( (cid:98) Z ) d × C q | = φ . Yet, this group is not d -generated becauseit has C d +1 q as a quotient..A similar argument shows that no sentence φ can express being finitelygenerated among profinite groups. Modify the argument above. Replace( (cid:98) Z ) d × C q by the profinite group (cid:98) Z × ( C q ) ℵ which is not finitely generatedbut would also satisfy φ , for almost every prime q . This uses that φ canbe expressed as a Boolean combination of so-called Szmielev invariant sen-tences, which can only contain information about finitely many primes; seeHodges [11, Thm A.2.7]. (cid:3) A profinite group G has rank r if each closed subgroup of G is r -generated.For an abelian profinite group G , this is equivalent to the whole group being OGIC BLOG 2019 7 r -generated, by the structure theorem that G is a direct product of pro-cyclicgroups and finite cyclic groups. This implies: Corollary 4.2.
There is no first-order sentence expressing that a profinitegroup G has finite rank. On the other hand, the entire theory of a profinite group G containsthe information whether it is (topologically) finitely generated: Jarden andLubotzky [12] (also see [28, Thm. 4.2.3]) proved that if G, H are profinitegroups, G is f.g. and G ≡ H then G ∼ = H . Whether a profinite group is d -generated can also be recognized from the entire theory, because it meansthat each finite quotient is d -generated, and one can express in f.o. logicwhat the finite quotients are again by Jarden and Lubotzky [12].This leaves open the question whether a uniform set of axioms can expressthat a profinite group is f.g. Question 4.3.
Is there a set of sentences S in the language of group theorysuch that for profinite groups G , we have G | = S ⇔ G is f.g.? Comparison with the discrete case.
Let’s compare this for a mo-ment with the case of discrete groups where f.g. now means literally finitelygenerated.The elementary theory of a group G doesn’t determine whether G is f.g.:by an easy elementary chain argument, for each countable group G there is acountable group H such that G is an elementary submodel of H , in particularhas the same theory, and H is not f.g. [16, Section 4]. (This argument usingelementary chains of type ω doesn’t work for profinite groups. The reasonis that model theoretic constructions such as adding a constant, or union ofelementary chains, don’t work in the topological setting such as for profinitestructures.)We note that the theory T of f.g. groups is Π complete by Morozovand Nies [14]. Its models could be called “pseudo-f.g.”, in analogy with thepseudofinite groups. An example of such a group that is not f.g. is the freegroup of infinite dimension F ω .4.3. Finite axiomatisability implies that the set of primes that oc-cur is finite.
In the following “FA” is short for “finitely axiomatisable”.We begin with an observation of T. Scanlon. Rings will be commutativewith unit element. Z p denotes the ring (or sometimes group) of p -adicintegers . Proposition 4.4 (going back to Scanlon, 2016) . Let R be the profinite ring (cid:81) p ∈ S Z p where π is an infinite set of primes. Then R is not FA within theclass of profinite rings.Proof. This follows from the Feferman-Vaught theorem (see any large texton model theory, such as [11]). For every sentence φ in the language of ringsthere is a finite sequence of sentences ψ , . . . , ψ n in the language of rings anda formula θ ( x , . . . , x n ) in the language of Boolean algebras so that for anyindex set I and any family of rings R i indexed by I , letting An fun exercise to get used to the 3-adic integers is the following. Note that − . . . − = 1 in Z using the multipication algorithm fromelementary school. EDITOR: ANDR´E NIES X j := { i ∈ I : R i | = ψ j } ,we have (cid:81) i ∈ I R i | = φ if and only if P ( I ) | = θ ( X , . . . , X n ).Assume for a contradiction that a sentence φ shows that R is FA. By thepigeon hole principle, we can find two distinct primes r (cid:54) = q in π so that forall j ≤ n we have Z r | = ψ j ⇐⇒ Z q | = ψ j . Define R p := Z p if p (cid:54) = r and R r := Z q . Then R (cid:48) := (cid:81) R p | = φ but R (cid:48) (cid:54)∼ = R as, for example, r is a unit in R (cid:48) but not in R . (cid:3) UT ( R ) denotes the usual Heisenberg group over the ring R . It is wellknown that Z (UT ( R )) ∼ = ( R, +) because the centre consists of the matriceswith only a nonzero entry in the upper right corner (other than the diagonal). Proposition 4.5.
Let R be the profinite ring (cid:81) p ∈ π Z p where π is a set ofprimes. If UT ( R ) is FA within the profinite groups then π is finite.Proof. Note that UT ( R ) can be interpreted in R by a collection of first-order formulas that do not depend on R . Hence, for each sentence θ in thelanguage of groups, there is a sentence (cid:101) θ in the language of rings such thatfor each ring R , UT ( R ) | = θ ⇔ R | = (cid:101) θ .Assume for a contradiction that a sentence θ shows that UT ( R ) is FA.within the profinite groups. Applying the above Feferman-Vaught analysisto φ = (cid:101) θ , let r (cid:54) = q be primes as above, and let also R (cid:48) defined as above.Then R (cid:48) | = (cid:101) θ and hence UT ( R (cid:48) ) | = θ . However, UT ( R (cid:48) ) (cid:54)∼ = UT ( R ) becauseevery element in the centre of UT ( R (cid:48) ), but not in the centre of UT ( R ), isdivisible by r . (cid:3) A similar argument works for some other algebraic groups. For instance,for each n , GL n ( R ) can be interpreted in R by formulas not depending on R , and its centre consists of the scalar matrices αI n where α is a unit of R .As mentioned, the groups of units are not isomorphic for the rings R (cid:48) and R as above.5. Nies, Perin and Segal: all f.g. profinite groups are ω -homogeneous Let G be a profinite group (separable, as always) and g ∈ G k where k ∈ N .We want to show that the type of g (i.e., the first-order theory of ( G, g ))determines the orbit of g in G . In the case of countable groups such a resulthas been shown e.g. for the f.g. free groups [23]. The following was firstnoticed in discussions between Segal and Chloe Perin at an OW meeting in2015. Her Master’s student Noam Kolodner covered it in his thesis. Theorem 5.1.
Let
G, H be (topologically) f.g. profinite groups, k ∈ N , g ∈ G k and h ∈ H k . If ( G, g ) ≡ ( H, h ) then ( G, g ) ∼ = ( H, h ) . OGIC BLOG 2019 9
Here in fact it suffices to look at the Σ formulas satisfied by g in G todetermine the orbit. The case k = 0 is due to Jarden and Lubotzky [12].We need a lemma that is a straightforward extension of [8, Prop. 16.10.7].Let Im( G, g ) denote the set of pairs (
R, r ), R a finite group, r ∈ R k suchthat there is an epimorphism ( G, g ) (cid:32) ( R, r ), i.e. it sends g i to r i for i < k .(The symbol (cid:32) will denote such epimorphisms.) For a profinite group P , p ∈ P k and N (cid:47) P we will write (
P, p ) /N for the structure ( P/N, (cid:104)
N p i (cid:105) i Suppose that Im( H, h ) ⊆ Im( G, g ) . Then ( G, g ) (cid:32) ( H, h ) .Proof. Note that a f.g. profinite group P has only finitely many open sub-groups of each index. Let P n be the intersection of subgroups of index ≤ n .Then P n is a characteristic subgroup of finite index.Fix n . By hypothesis there is an open N (cid:47) G such that ( H, h ) /H n ∼ =( G, g ) /N . Then G n ≤ N , and hence ( G, g ) /G n (cid:32) ( H, h ) /H n . Let Φ n bethe set of witnessing epimorphisms.If φ ∈ Φ n +1 then φ ( G n /G n +1 ) ≤ H n /H n +1 , so φ induces a map (cid:101) φ ∈ Φ n .By Koenig’s Lemma applied to the tree where the n -th level is Φ n , there isan epimorphism ψ : ( G, g ) (cid:32) ( H, h ) in the inverse limit of the Φ n . (cid:3) Proof of the Theorem. We extend the proof in Segal [28, Thm. 4.2.3] of theJarden-Lubotzky theorem. We show that Im( H, h ) ⊆ Im( G, g ). By sym-metry, also Im( G, h ) ⊆ Im( H, g ). The lemma together with the Hopfianproperty of f.g. profinite groups now implies that ( G, g ) ∼ = ( H, h ).Suppose that G is topologically generated by d elements. Recall from[22] (also [28, Ch. 4]) that a group word w is called d -locally finite if the d -generated free group in the variety of groups satisfying w is finite. (As anexample, the word w = [ x, y ] z m is d -locally finite for each d because a f.g.abelian group of exponent m is finite.) A main technical result in [22] statesthat for such a word, there is f = f ( w, d ) such that w ( R ) = R ∗ fw for each d -generated finite group R , where R ∗ fw is the set of products of at most f many w values or their inverses. Note that R ∗ fw is definable by a ∃ formuladepending only on f and w .As in [28, Thm. 4.2.3] our hypothesis that ( G, g ) ≡ ( H, h ) implies that( G, g ) /G ∗ fw ∼ = ( H, h ) /H ∗ fw . Checking the formulas reveals that only ( G, g ) ≡ Σ ( H, h ) is needed.Suppose N (cid:47) H is open. We need to show that ( H, h ) /N is in Im( G, g ).Again as in [28, Thm. 4.2.3] pick a d -locally finite word w such that H ∗ fw ≤ N .Then ( G, g ) (cid:32) ( G, g ) /G ∗ fw ∼ = ( H, h ) /H ∗ fw (cid:32) ( H, h ) /N ,as required. (cid:3) Nies and Stephan: Some properties ofinfinitely generated nilpotent-of-class-2 groups Three properties of an elementary abelian group. Throughoutfix a prime p . Let G = F ( ω ) p denote the elementary abelian p group (i.e.,vector space over the field F p of infinite dimension). This group has thefollowing apparently unrelated properties. Definition 6.1. (1) G is word-automatic in the usual sense of Khoussainov and Nerode:finite automata can recognize the domain and the group operations.(2) G is ω -categorical: it is up to isomorphism the only countable modelof its theory.(3) G is pseudofinite: if a sentence φ holds in G then φ also holds in afinite group.Note that a word automatic group is called finite automata presentablein [16] in order to avoid confounding the notion with the better knownnotion of automatic groups due to Thurston. The group F ( ω ) p is ω -categoricalbecause for each n there are only finitely many n -types. To show that F ( ω ) p is pseudofinite, note that its theory is axiomatized by stating that the groupis infinite, has exponent p and is abelian. So we may assume that a sentence φ as above is a finite conjunction of these axioms, and so clearly has a finitemodel.6.2. Nilpotent of class 2 groups. Usually we will assume that p (cid:54) = 2. Wewant to study the three properties above for groups that are very close toabelian. Recall that a group G is nilpotent of class 2 if each commutator[ x, y ] is in the centre C ( G ). In other words, G is an extension of a group N by a group Q that is abelian, and N is contained in C ( G ) (i.e., G is a centralextension). It is not hard to show that such a group is entirely given by theabelian groups N , Q , and an alternating bilinear map L : Q × Q → N . Itdetermines a unique central extension G such that L ( N x, N y ) = [ x, y ] (notethat [ x, y ] only depends on the cosets of x, y ). In fact this is a special caseof the well known fact that an extension of abelian group N by Q is givenby an element of the second co-homology group H ( Q, N ).We will study our properties for three examples of infinitely generatedgroups of exponent p . We will always have Q = F ( ω ) p , and fix a basis (cid:104) x i (cid:105) i ∈ N of Q .Let L p be the free nilpotent-2 exponent p group of infinite rank. One has N ∼ = Q ; let y i,k ( i < k ) be free generators of N . It is given by the function φ ( x i , x k ) = y i,k if i < k . Thus L p = (cid:104) x i , y i,k | x pi , [ x i , x k ] y − i,k , [ y i,k , x r ]( i < k ) (cid:105) where the y i,k are actually redundant. G p is the group where Q = F ( ω ) p , N = C p (cyclic group of order p ) and φ ( x i , x k ) = 1 if i < k , − i > k (and of course 0 if i = k ).This group is extra-special in the sense of Higman and Hall. (This meansthat the centre is cyclic of order p , equals the derived subgroup, and thecentral quotient is an elementary abelian p -group.)Felgner [7, Section 4] has proved that G p is ω -categorical for p (cid:54) = 2. Inhis notation G p is G ( p, ≤ ) where ≤ is the ordering of ω . This is the onlycountably infinite extra-special group of exponent p .Felgner (p. 423) also provides a recursive axiom system for the theory of G p ( p odd as before). One expresses that the group is of exponent p , thatthe centre is cyclic of order p , contains the derived subgroup, and that thederived subgroup is non-trivial (and hence equals the centre). Further one OGIC BLOG 2019 11 expresses that the central quotient is infinite, using an infinite list of axioms.Note that for each odd k there isa (unique) extraspecial p -group of exponent p and order p k . (For k = 3 this is the upper triangular matrices over F p , forlarger k one takes central products of these.) So, finitely many axioms canbe satisfied in a finite model. Hence G p is pseudo-finite as already noted byFelgner.The group H p has N ∼ = Q , with free generators z k ( k > φ ( x i , x k ) = z k if i < k : H p = (cid:104) x i , z k | x pi , [ x i , x k ] z − k , [ z k , x r ] ( i < k, ≤ r ) (cid:105) .In a nilpotent group, each nontrivial normal subgroup intersects the centrenon-trivially. This implies that every proper quotient of G p is abelian; inparticular, G p is not residually finite. On the other hand L p and H p areresidually p -groups. To see this, take an element h (cid:54) = 1, where h depends ongenerators x , . . . , x n − . Then h (cid:54) = 1 in the finite quotient p -group where allthe generators x k , k ≥ n , become trivial. This means that both groups areembedded into their pro- p completions. Fact 6.2. None of the groups L p , G p , H p is abelian by finite.Proof. Let K be such a group and suppose M is a normal subgroup of finiteindex. There are k < r < s such that x r x − k ∈ M and x s x − k ∈ M . Then[ x r x − k , x s x − k ] = [ x r , x s ] x − k , and this commutator is not 1 in K . So M isnot abelian. (cid:3) FA-presentability. Any FA presentable structure has a decidable the-ory. Ershov [6] showed that a finitely generated nilpotent group has decid-able first-order theory if and only if it is virtually abelian. One the otherhand, Nies and Semukhin [20, Thm. 4.2] showed that an abelian group thatis an extension of finite index of an FA-presentable group is in itself FA-presentable. So a f.g. nilpotent group is FA-presentable iff it is abelian byfinite. Using Mal’cev coding, Nies and Thomas [21] extended this by showingthat each finitely generated subgroup of an FA-presentable group is abelian-by-finite. So the natural setting for finding interesting FA-presentable groupsis among the groups that are not finitely generated. Fact 6.3. G p is FA-presentable.Proof. Each element has the form c · q where c ∈ N , q ∈ Q . Note that q is given as a string α over the alphabet of digits 0 , . . . , p − 1, and c canbe stored in the state. In the following α, β denote such strings, which arethought of as extended by 0’s if necessary. Let [ α ] = (cid:81) i x α i i .It is easy to verify that, with arithmetic modulo p and component-wiseaddition of strings, [ α ][ β ] = [ α + β ] u (cid:80) k> α k ( (cid:80) i 1. Each such movecreates a factor u α k β i .A finite automaton processing α, β on two tracks can remember the cur-rent (cid:80) i Show that L p is not FA presentable. If only finitely many things do not commute with the others then it is notisomorphic to the group which we constructed. So it is a further exampleof such a group. The reason is quite easy: If x , .., x k do not commute witheach other and y , y , ... commute all with each other but not with x , .., x k then every basis has two things which commute. Fact 6.6. The pro- p completion (cid:98) H p of H p is B¨uchi presentable.Proof. Let (cid:98) Q be the pro- p completion of Q , which is C ωp (cartesian power)viewed as a compact group. (cid:98) H p is a central extension of (cid:98) Q by itself wherethe first (cid:98) Q has the topological generators z k , the second has the x i , and wehave [ x i , x k ] = z k for i < k , as before.An element h of the group (cid:98) H p is represented by a pair of infinite strings α, γ such that h = [ α ] (cid:81) i z γ i i . So we can use the same FA as above, butworking on infinite strings, to check the group operation. (cid:3) Part Computability theory and set theory Greenberg and Nies: cardinal characteristics andcomputability Nicholas Rupprecht, in his 2010 PhD thesis supervised by Blass [25],studied the connection of cardinal characteristics of the continuum in settheory and highness properties from computability theory. He developedthis connection systematically as coming from a single source. Somewhatunfortunately, the thesis is somewhat technically written and the notationis hard to access, which may be a reason for the fact that no immediatereaction by either of the two communities to this unexpected connectionresulted.The cardinal characteristics given by null sets correspond to highnessproperties defined in terms of algorithmic randomness. For instance, non( N ),the least size of a non-null set of reals, corresponds to the strength of anoracle that computes a Schnorr test containing all recursive bit sequences.Rupprecht published a single paper mainly on the computability theoreticimpact of that notion [26]. Rupprecht called this property “Schnorr cover-ing”, which is confusing because it does not correspond to the characteristic OGIC BLOG 2019 13 cov( N ) (the least size of a class of null sets with union R ); later on, Bren-dle et al. suggested the term “weakly Schnorr engulfing” [4]. We note thatRupprecht had attended the NSF FRG-funded school on computability andrandomness at Gainesville in 2008, where Nies gave a tutorial on random-ness; this may have prompted him to make the connection between measuretheoretic cardinal invariants and randomness. He started a career as an in-vestor shortly after his defence. According to LinkedIn his last job was asthe Global Director of Operations at Virtu Financial. Blass mentioned hedidn’t work there any longer. Apparently he has disappeared from view.A quote from Brendle et al. [4]: the analogy occurred implicitly in thework of Terwijn and Zambella [31], who showed that being low for Schnorrtests is equivalent to being computably traceable. (These are lowness prop-erties, saying the oracle is close to being computable; we obtain highnessproperties by taking complements.) This is the computability theoreticanalog of a result by Bartoszy´nski [2] that the cofinality of the null sets(how many null sets does one need to cover all null sets?) equals the dom-ination number for traceability. (Curiously, Terwijn and Zambella alludedto some connections with set theory in their work. However, it was notBartoszy´nski’s work, but rather work on rapid filters by Raisonnier [24].)Their proof bears striking similarity to Bartoszy´nski’s; for instance, bothproofs use measure-theoretic calculations involving independence. See alsothe book references [1, 2.3.9] and [17, 8.3.3]. (End quote.)8. Lempp, Miller, Nies and Soskova: Analogs in computabilityof combinatorial cardinal characteristics Brief summary. Our basic objects are infinite sets of natural numbers. Inset theory, the MAD number is the least cardinality of a maximal almostdisjoint class of sets of natural numbers. The ultrafilter number is the leastsize of an ultrafilter base.We study computability theoretic analogs of these cardinals. In our ap-proach, all the basic objects will be infinite computable sets. A class of suchbasic objects is encoded as the set of “columns” of a set R, which allows usto study the Turing complexity of the class.We show that each non-low oracle computes a MAD class R, give a fini-tary construction of a c.e. MAD set (compatible with permitting), and onthe other hand show that a 1-generic below the halting problem does notcompute a MAD class. We also provide some initial results on ultrafilterbases in this setting.The original impetus for this work resulted from some interactions of Nieswith Brendle, Greenberg and others at the August meeting on set theory ofthe reals at Casa Matem´atica Oaxaca in early August, which was organisedby Hrusak, Brendle, and Steprans. The computability results reported beloware due to Lempp, Miller, Nies and Soskova (in prep.), based on joint workthat happened during Nies’ visit in Madison end of August, 2 weeks after theend of the CMO meeting. Downey’s visit in Madison in September furtheradvanced it. Set theory. Let b be the unbounding number (the least size of a classof functions on ω that is not dominated by a single function).Several cardinal characteristics are based on cardinals of subclasses of[ ω ] ω / = ∗ , i.e. the infinite subsets of ω under almost equality. One of them iscalled the almost disjointness number, denoted a . This is the minimal sizeof a maximal almost disjoint (MAD) family of subsets of ω . The followingis well known. Fact 8.1. b ≤ a .Proof. Suppose that the infinite cardinal κ is less than b . Let V be an almostdisjoint family of size κ . We show that V is not MAD. Let (cid:104) r n (cid:105) , n ∈ ω , bea sequence of pairwise distinct elements of V . Given y ∈ V let f y ( n ) = max( y ∩ r n )with the convention that max ∅ = 0. Let g be an increasing function suchthat g ( n ) ∈ r n for each n and ∀ y ∈ V∀ ∞ n [ g ( n ) > f y ( n )]. It is clear that thefamily V ∪ { range( g ) } is almost disjoint. (cid:3) Other cardinal characteristics are based on properties of subclasses of theinfinite sets modulo almost equality: • the ultrafilter number u (the least size of a set with upward closurea free ultrafilter on ω ), • the tower number t (the minimum size of a linearly ordered subsetthat can’t be extended by putting a new element below all givenelements), • the independence number i and several others. See e.g. [3] or the talk by Diana Montoya at the CIRMmeeting on descriptive set theory in June 2018. This talk contained a Hassediagram of these cardinals with ZFC inequalities, reproduced here with per-mission:Note that r and s are the unreaping and splitting numbers, respectively.They are given by relations, and their analogs in computability have beenstudied [4]. Furthermore, e is the escaping number due to Brendle andShelah. Its analog has recently been considered in computability theoryby Ivan Ongay Valverde and Paul Tveite, two PhD students from Madison.Finally, h is the distributivity number. For another diagram see Soukup [30].8.2. Analogous mass problems in computability. For a set F , F [ n ] denotes the column { x : (cid:104) x, n (cid:105) ∈ F } . OGIC BLOG 2019 15 We will also denote this by F n if no confusion is possible.A systematic way to translate these characteristics into highness proper-ties of oracles is as follows. We consider subclasses of Rec / = ∗ \{ } , identifiedwith classes of infinite recursive sets in the context of almost inclusion. Sucha subclass C is encoded by a set F such that C = C F = { F [ n ] : n ∈ N } .We view these classes as mass problems. We compare them via Muchnikreducibility ≤ W and the stronger, uniform Medvedev reducibility ≤ S .8.3. The mass problems A and T . We say that (the class C F describedby) F ⊆ N is a almost disjoint, AD in brief, if each F n is infinite, and F n ∩ F k is finite for each n (cid:54) = k . Definition 8.2. The mass problem A corresponding to the almost disjoint-ness number a is the class of sets F such that C F is maximal almost disjoint(MAD) at the recursive sets. Namely, C F is AD, and for each infinite recur-sive set R there is n such that R ∩ F n is infinite.We need to get this out of the way: Fact 8.3. No MAD set F is computable.Proof. Suppose F is AD and computable. Let r − = 0, and r n be the leastnumber r > r n − such that r ∈ F n − (cid:83) i We say that G ⊆ N is a tower (or C G is a tower) if for each n we have G n +1 ⊆ ∗ G n and G n − G n +1 is infinite.By a function associated to G we mean an increasing function p satisfyingthe conditions p ( n ) ∈ (cid:84) i ≤ n G i . Definition 8.5. The mass problem T corresponding to the tower number t is the class of sets G such that C G is a tower that is maximal in the recursivesets. Namely, for each infinite recursive set R there is n such that R − G n is infinite. Fact 8.6. No tower is c.e. Otherwise there is a computable associated function p . The range of p would extend the tower. Fact 8.7. A ≤ S T ≤ S A .Proof. To check that A ≤ S T , given a set G let Diff( G ) be the set D suchthat D n = G n − G n +1 . Clearly the operator Diff can be seen as a Turingfunctional. If G is a maximal tower then D = Diff( G ) is MAD. For, if R is infinite recursive then R − G n is infinite for some n , and hence R ∩ D i isinfinite for some i < n .For T ≤ S A , given a set F let G = Cp( F ) be the set such that x ∈ G n ↔ ∀ i < n [ x (cid:54)∈ F n ] . Again Cp is a Turing functional. If F is AD then G is a tower, and if F isMAD then G is a maximal tower. (cid:3) Recall that a characteristic index for a set M is an e such that χ M = φ e . Proposition 8.8. Suppose F ∈ A . While each F n is computable, ∅ (cid:48) is notable to compute, from input n , a characteristic index for F n .Proof. Assume the contrary. Then there is a computable function f suchthat φ lim s f ( n,s ) is the characteristic function of F n .Let (cid:98) F be defined as follows. Given n, x compute the least s > x such that φ f ( n,s ) ,s ( x ) ↓ . If the value is not 0 put x into (cid:98) F n .Clearly (cid:98) F is computable and F n = ∗ (cid:98) F n for each n . So (cid:98) F is MAD, contraryto Fact 8.3. (cid:3) This research started at the CMO by looking at an analog of b ≤ a . Thisisn’t quite the right approach as we will see. At least, it is misleading.The analog of b is the property of an oracle C to be high , namely ∅ (cid:48)(cid:48) ≤ T C (cid:48) ;for detail on this see [26, 4]. (To see this as a mass problem, take thefunctions eventually dominating all computable functions, instead.) Wewant to show that an analog of the relationship b ≤ a holds: every highoracle computes a set in A . The inversion that occurs here is part of thegenerally accepted setup of analogy between the two areas of logic. Onecan show that each high oracle C computes a set in A . The proof is astraightforward application of a dominating function computed by C .Lempp, Miller, Nies and Soskova (in prep.) have strengthened the result:each non-low oracle computes a set in T , and hence in A . The result isuniform in the sense of mass-problems. Let NonLow denote the class oforacles X such that X (cid:48) (cid:54)≤ T ∅ (cid:48) . Theorem 8.9. T ≤ S NonLow .Proof. In the following x, y, z denote binary strings; we identify x with thenumber 1 x via the binary expansion. Define a Turing functional Φ for theMedvedev reduction, as follows. Φ Z = G , where for each nG n = { x : | x | = s ≥ n ∧ Z (cid:48) s (cid:22) n = x (cid:22) n } . It is clear that for each n we have G n +1 ⊆ ∗ G n and G n − G n +1 is infinite.Also, for each n , for large s the string Z (cid:48) s (cid:22) n settles, so G n is computable.Suppose now that R is an infinite set such that R ⊆ ∗ G n for each n . Then Z (cid:48) ( k ) = lim x ∈ G, | x | >k x ( k ), and hence Z (cid:48) ≤ T R (cid:48) . So if Z ∈ NonLow then R cannot be computable, and hence Φ Z ∈ T . (cid:3) and -generic sets, totally low sets, and not computinga MAD. We provide a type of sets which doesn’t compute a MAD set.Joe Miller and Mariya Soskova showed (during the weekend after Andrehad left) that if L is ∆ and 1-generic, then L does not compute a MADfamily. Downey’s visit in Madison before the Lempp ‘60 conference broughtanother interesting lowness property into the game: totally low oracles. Thisproperty, derived from Joe’s and Mariya’s proof, is in between 1-generic ∆ ,and not computing a MAD. No separation is known at present.Note that for a ∆ oracle L , ∅ (cid:48)(cid:48) can compute from e an index for Φ Le ,provided that it’s computable. To be totally low means that ∅ (cid:48) suffices. Definition 8.10. We call an oracle L totally low if ∅ (cid:48) can compute from e an index for Φ Le whenever Φ Le is computable. In other words, there is afunctional Ψ a such that OGIC BLOG 2019 17 Φ Le computable ⇒ Ψ( ∅ (cid:48) ; e ) is an index for it.No assumption is made on the convergence of Ψ( ∅ (cid:48) ; e ) in case Φ Le is not acomputable function. A total function is computable iff its graph is. Amoment’s thought using this fact shows that we may restrict ourselves to e such that Φ Le is { , } -valued. Clearly, being totally low is closed downwardunder ≤ T .By Prop 8.8, a totally low oracle D does not compute a MAD set. Inparticular by Theorem 8.9, D is low (as the name suggests) Theorem 8.11. Suppose L is ∆ and -generic. Then L is totally low.Proof. Suppose F = Φ Le and F is a computable set. Let S e = { σ : ( ∃ τ (cid:23) σ )( ∃ τ (cid:23) σ )( ∃ p )[Φ τ e ( p ) (cid:54) = Φ τ e ( p )] } . Suppose that S e is dense along L . We claim that the set C e = { τ : ( ∃ p )[Φ τe ( p ) (cid:54) = F ( p )] } is also dense along L : i.e. for every k there is some τ (cid:23) L (cid:22) k such that τ ∈ C e . Indeed, let σ (cid:23) L (cid:22) k be a member of S e and let τ and τ and p witness that. Let τ i where i = 1 or 2 be such that Φ τ i e ( p ) (cid:54) = F ( p ). Then τ i (cid:23) L (cid:22) k is in C e . The set C e is c.e. and hence L meets C e , contradictingour assumption that F = Φ Le .It follows that S e is not dense along L . In other words, there is some least k e such that there is no splitting of Φ e above L (cid:22) k . On input e the oracle ∅ (cid:48) can compute k e and L (cid:22) k e . This allows ∅ (cid:48) to find an index for F , given bythe following procedure. To compute F ( p ), find the least τ (cid:23) L (cid:22) k e suchthat Φ τe ( p ) ↓ (in | τ | many steps). Such a τ exists because Φ Le ( p ) ↓ . By ourchoice of k e it follows that Φ τe ( p ) = Φ Le ( p ) = F ( m ). (cid:3) By EX one denotes the class of computable functions f so that an index canbe learned in the limit: there is a machine M taking values f (0) , f (1) , . . . such that lim s M ( f (0) , . . . , f ( s ) exists and is an index for M . To relativizethis to oracle X , allow M access to X (but the functions are still com-putable). Recall that A is low for EX learning if EX A = EX . Slaman andSolovoy [29] showed that A is low for EX-learning ⇔ A is ∆ and 1-generic.We summarize the known implications of lowness notions.∆ and 1-generic ⇒ Totally low ⇒ computes no MAD ⇒ low.The last arrow doesn’t reverse by what follows; the others might.8.5. C.e. MAD sets. It may come as a surprise that a set in A can be c.e.,and in fact can be built by a priority construction with finitary requirements,akin to Post’s construction of a simple set. Such a construction is compatiblewith permitting. The result to follow is part of the joint work mentionedabove. Theorem 8.12. For each incomputable c.e. set A , there is a MAD c.e. set R ≤ T A . Proof. Let (cid:104) V e (cid:105) e ∈ N be a u.c.e. sequence of sets such that V e = W e and V e +1 = N for each e . We will build an auxiliary c.e. set S ≤ T A and let thec.e. set R ≤ T A be defined by R [ e ] = S [2 e ] ∪ S [2 e +1] . The role of the V e +1 isto make the sets S [2 e +1] , and hence the R [ e ] infinite. The construction alsoensures that S , and hence R , is AD, and that (cid:83) n S [ n ] is coinfinite.We will write S e for S [ e ] . We provide a stage-by-stage construction tomeet the requirements P n , n = (cid:104) e, k (cid:105) given by P n : V e − (cid:91) i 0. For each n < s such that P n is not satisfied, n = (cid:104) e, k (cid:105) , if thereis x ∈ V e,s − (cid:83) i Each S e is computable, being enumerated in an increasingfashion.Each P n is active at most once. This ensures that (cid:83) e S e is coinfinite:for each N , if x < N enters this union then this is due to the action of arequirement P n with n ≤ N , so there are at most N many such x .To see that a requirement P n , n = (cid:104) e, k (cid:105) , is met, suppose that its hypoth-esis holds. Then there are potentially infinitely many candidates x that cango into S e . Since A is incomputable, one of them will be permitted.Now, by the choice of V e +1 , each S e +1 , and hence each R e is infinite.By construction, for e < m we have | S e ∩ S m | ≤ m . So the family describedby S and therefore also the one described by R is almost disjoint.To show R is MAD, it suffices to verify that if V e is infinite then V e ∩ R p isinfinite for some p . If all the P e,k are satisfied during the construction thenwe let p = e . Otherwise let k be least such that P n is never satisfied where n = (cid:104) e, k (cid:105) . Then its hypothesis fails, so V e ⊆ ∗ (cid:83) i T ≤ S A in Fact 8.7, we obtain Corollary 8.13. For each incomputable c.e. set A , there is a co-c.e. set G ≤ T A such that G describes a maximal tower, i.e. G ∈ T . Namely, G = Cp( A ) i.e., x ∈ G n ↔ ∀ i < n [ x (cid:54)∈ A n ] . Fact 8.14. From a co-c.e. tower G one can uniformly compute an infiniteco-c.e. set B such that B ⊆ ∗ G n for each n . Moreover, the index for B isalso obtained uniformly.Proof. We may assume that G n ⊇ G n +1 for each n . Let γ n,s be the n -thelement of G n,s . Clearly γ n,s is monotonic in s and strictly monotonic in n .Let γ n = lim s γ n,s . The set B = { γ n : n ∈ ω } is as required. (cid:3) Since a totally low set computes no MAD set, we obtain: Corollary 8.15. No incomputable, c.e. set L is totally low. OGIC BLOG 2019 19 Downey and Nies in October (during Nies’ Wellington visit where Niesgave a seminar talk about this stuff) have given a direct proof of this. As-sume for contradiction that L is incomputable, c.e., and totally low. DefineTuring functionals Γ e , uniformly in e . By the recursion theorem, we knowin advance a computable function p such Γ Xe ( n ) = Φ Xp ( e ) ( n ) for each X, n .We meet for each e the requirement P e : Γ Le is computable but Φ ∅ (cid:48) e ( p ( e )) is not an index for it.Write α ( e, s ) for Φ ∅ (cid:48) e ( p ( e ))[ s ]. Construction. Assign an increasing sequence of followers x e ∈ ω [ e ] to P e .Initially we set Γ Le ( x e ) = 0 with use x e . Once φ α ( e,s ) ( x e ) = 0, we call x e certified for e . If L at a certain stage t permits a certified x e that is largerthan any follower that is fulfilled, we change the value Γ Le ( x e ) to 1, and call x e fulfilled . Verification. Since followers are enumerated in an increasing fashion, Γ Le is computable. There is a final value lim s α ( e, s ), which hence is an indexfor Γ Le . After α ( e, s ) has stabilized to a value v , infinitely many followersget certified. So L permits at a stage t a certified follower for P e . Thisfollower gets fulfilled at stage t , which causes the index v to become incorrect.Contradiction.8.6. The mass problem U corresponding to the ultrafilter number,and a strict reduction T < S U .Definition 8.16. Let U be the class of sets F such that F is tower as inDef. 8.4, and C F is an ultrafilter base within the recursive sets: for eachrecursive set R there is n such that F n ⊆ ∗ R or F n ⊆ ∗ R .Since U ⊆ T we trivially have T ≤ S U via the identity reduction. Sinceour properties of (codes for) classes are arithmetical, each of the mass prob-lems introduced contains an element computable from ∅ ( n ) for sufficientlylarge n . For instance, ∅ (cid:48)(cid:48) computes an oracle in U . To see this, take any r -cohesive set C . By definition of r -cohesiveness, the recursive sets R suchthat C ⊆ ∗ R is cofinite form an ultrafilter. If e.g. C is r -maximal, we canwrite this ultrafilter as C F for some F ≤ T ∅ (cid:48)(cid:48) .For the next fact we follow the upcoming work mentioned above due toLempp, Miller, Nies and Soskova. In the cardinal setting we have t ≤ u ,which suggests that each maximal tower should compute an ultrafilter base.But this is not true by the following. Proposition 8.17. No ultrafilter base R is computably dominated.Proof. Let p ( n ) be an associated function as in Def. 8.4, i.e. an increasingfunction p such that p ( n ) ∈ (cid:84) i ≤ n R i . Then p ≤ T R . Assume that there is acomputable function f ≥ p . The conditions n = 1 and n k +1 = f ( n k ) definea computable sequence. So the set E = (cid:91) i [ n i , n i +1 )is computable. Clearly R n (cid:54)⊆ ∗ E and R n (cid:54)⊆ ∗ E for each n . So R is not anultrafilter base. (cid:3) On the other hand, each non-low set computes a set in A ≡ S T as shownabove, and of course a computably dominated oracle can be non-low.8.7. Co-c.e. ultrafilter bases. Recall that no tower, and hence no ultra-filter base, is c.e. Fact 8.18. Every co-c.e. ultrafilter base G is high.Proof. Let B be a co-c.e. set as obtained in Fact 8.14. Clearly B ⊆ ∗ R or B ⊆ ∗ R for each computable R . Hence B is r -maximal, and therefore high.Since B ≤ T G , G is also high. (cid:3) Let us say for the moment that a tower F is a ultrafilter base for c.e. sets if for each computably enumerable set R we have F n ⊆ ∗ R or F n ⊆ ∗ R . ByFact 8.14 there is no co-c.e. ultrafilter base for c.e. sets. To build a co-c.e.ultrafilter base we need to make use of the fact that we are given c.e. indexfor a computable set and also one for its complement. Theorem 8.19. There is a co-c.e. ultrafilter base F .Proof. We build the co-c.e. tower F by providing uniformly co-c.e. sets F e , e ∈ N that form a descending sequence: F e ⊃ ∞ F e +1 . If we remove x from F e at a stage s we also remove it from all F i for i > e , without furthermention.Let (cid:104) ( V e, , V e, ) (cid:105) be an effective listing of the pairs of disjoint c.e. sets.The construction will ensure that the following requirements are met. M e : F e \ F e +1 is infinite. P e : V e, ∪ V e, = N ⇒ F e +1 ⊆ ∗ V e, ∨ F e +1 ⊆ ∗ V e, .This suffices to establish that F is an ultrafilter base.The tree of strategies is T = { , , } < ∞ . Each string α ∈ T of length e is associated with M e and also P e . We write α : M e and α : P e to indicatethat we view α as a strategy of the respective type. Streaming. For each string α ∈ T , | α | = e , at each stage of the constructionwe have a set S α , thought of as a stream of numbers used by α . Each time α is initialized, S α is made empty and its content removed from F e +1 . Also, S α is enlarged only at stages at which α appears to be on the true path. Wewill verify the following properties.(1) S ∅ = ω ;(2) if α is not the empty node then S α is a subset of S α − (where α − isthe immediate predecessor of α );(3) At every stage S γ ∩ S β = ∅ for incomparable strings γ and β ;(4) at the time a number x first enters S α , x is in F e +1 ; and(5) if α is along the true path of the construction then S α is an infinitecomputable set.Thus S α can be thought of as a set d. c. e. uniformly in α ; S α is finite if α is to the left of the true path of the construction; an infinite computable setif S α is along the true path; and empty if α is to the right of the true path. Construction. Stage 0. Let δ be the empty string. Let F e = N for each e . OGIC BLOG 2019 21 Stage s + 1. Let S ∅ ,s = [0 , s ). Substage e < s . We suppose that α = δ s +1 (cid:22) e has been defined.Strategy α : M e removes every other element of S α from F e +1 . Let S (cid:48) α denotethe set of remaining numbers. More precisely, if we have S α = { r < . . . 2. Let α (cid:98) α has a reserved, unprocessed number x .2. [0 , x ] ⊆ V e, ∪ V e, and x ∈ V e, .Let s be greatest stage < s at which α was initialized. Add x to S α andremove from F e +1 all numbers in the interval ( s , x ) which are not in S α .Declare that α has processed x . Let α (cid:98) , x ] ⊆ V e, ∪ V e, and x ∈ V e, .Let s be greatest stage < s at which α was initialized or α (cid:98) x to S α and remove from F e +1 all numbers in the interval ( s , x )which are not in S α . Declare that α has processed x . Let α (cid:98) Otherwise. Let s be greatest stage < s at which α was initialized or α (cid:98) S α = S (cid:48) α ∩ ( s , s ). Let α (cid:98) δ s +1 ( e ) = i where α (cid:98) i , 0 ≤ i ≤ 2, has been declared eligible toact next. Verification. By construction and our convention above, F e ⊇ F e +1 for each e , and F is co-c.e.Let g ∈ ω denote the true path, namely the leftmost path in { , , } ω such that ∀ e ∃ ∞ s [ g (cid:22) e (cid:22) δ s ]. In the following, given e let α = g (cid:22) e , and let s α be the largest stage s such that α is initialized at stage s . We verify anumber of claims. Claim 8.20. The “streaming” properties (1)-(5) hold. (1,2) hold by construction.(3) Assume this fails for incomparable γ, β , so x ∈ S γ ∩ S β at stage s . Wemay as well assume that γ = α (cid:98) i and β = α (cid:98) k where i < k . By construction k ≤ k = 2. Since x ∈ S α (cid:98) i adn i ≤ x was reservedby α at some stage t ≤ s . So x can never go into S α (cid:98) by the initializationof α (cid:98) x was reserved.(4) is true by construction.(5) holds by definition of the true path, and because S α is enumerated inincreasing fashion at stages ≥ s α . Claim 8.21. (i) F e ⊆ ∗ S α . (ii) S α ⊆ ∗ F e . The claim is verified by induction on e . It holds for e = 0 because F = S ∅ = N . Suppose the claim is true for e . To verify it for e + 1, let γ = g (cid:22) e +1 ,and let s γ be the largest stage s such that γ is initialized at stage s .(i) Suppose x ∈ F e +1 . Then x ∈ F e , so inductively x ∈ S α for almost allsuch x . By construction, any element x that isn’t promoted to S γ is alsoremoved from F e +1 , unless x is the last element α reserves. However, inthat case necessarily γ = α (cid:98) 2, so this leads to at most one new element in F e +1 \ S γ .(ii) Suppose x ∈ S γ . Then x ∈ S α , so inductively x ∈ F e for almost all such x . At stage s ≥ s γ an element x of S α can not be removed from F e +1 bya strategy β > L α because S β ∩ S α = ∅ because of (3) verified above, andsince β can only remove elements from S β . So x can only be removed by astrategy tied to α .If α : M e removes x from F e +1 , then x (cid:54)∈ S (cid:48) α , contradiction. So by con-struction the only way x can be removed from F e +1 is by α : P e , which fora sufficiently large x means that x does not get promoted to S γ either. Claim 8.22. M e is met, namely, F e \ F e +1 is infinite. To see this, recall α = g (cid:22) e . By the foregoing claim, the action of α : M e removes infinitely many elements of S α ⊆ F e from F e +1 . Claim 8.23. P e is met. Suppose the hypothesis of P e holds. Then every number α reserves even-tually gets processed. So either g ( e ) = 0, in which case F e +1 ⊆ ∗ V e, byClaim 8.21, or g ( e ) = 1, in which case F e +1 ⊆ ∗ V e, by Claim 8.21. (cid:3) Added Jan. 2020: Lempp, Miller, Nies and Soskova (in prep.) have shownthat both U and the analog of the independence number are Medvedevequivalent to the mass problem of dominating functions. The paper will beposted. 9. Yu Liang Concerning the Joint coding theorem: a handwritten proof byHarrington. The following theorem has appeared in [5]. Theorem 9.1. Given two uncountable Σ -sets A , A , for any real z , thereare reals x ∈ A and x ∈ A so that x ⊕ x ≡ h z ⊕ O . On 11/July/2019, Steve Simpson sent us a manuscript by Harringtondated on 9/1975 in which Theorem 9.1 was already proved. Harrington’sproof is model theoretic and certainly deserves to be studied in detail. Themanuscript follows. I Theo Yer, L e + /J O ; / l :i J e -f,.__ ,, c,.;, c,, v,--/2 ~ j a f ' rea/J { real=- J c-J./e+ . , p I C<__,, I I • F ; r '2 I I r e cJ /1 h ? .hyp cf2- -/-Ae1-e e V ci_,7{ .] a O / Cl:t j . t . ' Cl O €.. A ) C?:1.. ~ i5 ~ f; P <. Clo Cl.1_ >, P r Q O f' l L e+ ev.L -= / J./- n ()I/ - r~C vrJ / ve OJ...c///f c'J /, Le+ l'.13 be. fAe .J A2/I
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The following statement is independent from ZF C : forany real x , the set A x = { y | y ≥ h x } is Borel.Proof. Clearly if V = L , then the set A x = { y | y ≥ h x } is Borel for anyreal x .Now suppose that ω = ( ω ) L and there is an L -random real r . We provethat A r is not Borel. Otherwise, there is a countable ordinal α so that ∀ z ( z ∈ A r = ⇒ r ∈ L α [ z ]) . Let y ∈ L be a real so that ω y > α . By Martin’s theorem relative to y ,there is a ∆ ( y )-Sacks generic real z be a so that z ⊕ y ≥ h r . Then r is L α [ y ⊕ z ]-random and so r (cid:54)∈ L α [ y ⊕ z ] but r ≤ h y ⊕ z , a contradiction. (cid:3) Note : Actually it can be proved that A x is Borel if and only if x ∈ L . Theidea is as follows: Suppose that α is a countable ordinal and x (cid:54)∈ L . Let β > α be a countable admissible ordinal. Then there is a real y so that ω y = β but x (cid:54)≤ h y . Then one can prove that there is a real z ≥ T y so that z > h y but x (cid:54)∈ L β [ z ]. Part Model theory and definability First-order logic, computability, and enriched structures A structure in mathematical logic is a non-empty domain (a set) withsome relations and functions defined on it. Examples are graphs, and groups.First-order logic is made for dealing with them: the relations and functionsare described by atomics formulas, the quantifiers range over the set. Whenthe domain is countably infinite, the notions of computability also work wellwith such structures: via some coding, one can see the domain elementsas inputs to machines, and one requires that the relations and functions becomputable.Many kinds of structures that mathematicians study are not of this kind.Often they study “enriched structures”: a non-empty domain, some relationsand functions on it, and something else, some extra structure. Examples are:(1) metric structures: there is a distance function producing a topology,the relations are closed, the functions continuous.(2) Hilbert spaces, with a C -valued inner product.(3) C ∗ -algebras, which are Banach ∗ -algebras satisfying || xx ∗ || = || x || .(4) tracial von Neumann algebras N , such as the hyperfinite II -factor R .The extra structure is a trace , a positive functional sending 1 N to 1,corresponding to the (normalised) trace of matrices(5) Lie groups: groups on a manifold.(6) Profinite groups: a group on a compact, totally disconnected spacewith group operations continuous.And these are often the most relevant for applications elsewhere: e.g. Hilbertspaces and operator algebras are used in quantum physics. Lie groups alsoplay a major role in modern physics. OGIC BLOG 2019 37 Remark 10.1. The last two examples suggest that the proposed view maybe that of a logician. The extra structure can also be viewed as the primaryone, with the algebraic structure sitting on top of it. For instance, Liegroups are group objects in the category of manifolds (which has Cartesianproducts); these group objects form a new category where the morphismsare the Lie group homomorphisms. Ditto for profinite groups, which aregroup objects on compact totally disconnected topological spaces.How do we extend the methods of first-order logic, and of computability,to such structures? There are two pathways.A. The first pathway, not followed here, is to also extend the language, orto adapt the notion of computability to the new setting. For the language,this has been done in many of the cases above. E.g., continuous logic hasbeen introduced to deal with metric structures, and also with tracial vonNeumann algebras. Relations are now real valued, and the distance functionplays the role that formerly equality had played. For topological groups, onecan use a two-sorted language: one sort for the elements, and another forthe open sets, say.For computability, there are Blum-Shub-Smale (BSS) machines that di-rectly work on reals, as an example.B. The second pathway is to retain the bare tools of first-order logic and com-putability developed for the naked structures. The approach to the enrichedstructures will then necessarily be indirect. We still express things about,or compute with, the elements of the domain with their basic relationships;the expressivity is enhanced only because the semantics is different. For in-stance, one can study the expressivity of first-order logic in profinite groups.If the theory determines the group within a class of groups, this is calledquasi-axiomatisable (QA) relative to the class. Jarden and Lubotzky [12](also see [28, Thm. 4.2.3]) showed that each topologically f.g. profinite groupis QA within the profinite groups. Nies, Segal and Tent [19] show that formany profinite groups there is in fact a single sentence determining it up totopological isomorphism within the class of profinite groups. This propertyis called finitely axiomatizable (FA) relative to the class. Examples of FAgroups include SL n ( Z p ) for any odd prime p not dividing n .In the computability setting, the problem is that many of the structuresare now uncountable, e.g. Polish metric spaces. We have to reduce theseenriched structures to a countable core, which can be done by choosing adense countable subset D . In this way we can define computable metricspaces: each element is given a the limit of a certain fast converging Cauchysequence of elements of D . For profinite groups the most natural way toextend computability to the whole structure is by asking that the group bean effective inverse limit of finite groups with onto projections. This is e.g.the case for the additive group of p -adics, Z p = proj lim n C p n . One alsogets an ultrametric out of this, relative to which the group operations arecomputable.In the case of the von Neumann hyperfinite II factor R it is easy todetermine the canonical computable structure as a metric space. R canbe introduced as the completion of the ∗ -algebra (cid:83) n M n ( C ) with respectto norm given by the inner product (cid:104) A, B (cid:105) = τ n ( B ∗ A ), where τ n is the normalized trace given by τ n ( C ) = 2 − n (cid:80) i C ii ; see e.g. the Goldbring 2013notes [9].Consider a class C of enriched structures. To structures in C can be al-gebraically isomorphic, and fully isomorphic, i.e. also the extra structure ispreserved. How much stronger is the second of these notions? For Polishgroups (topological groups where the topology is Polish), any Baire mea-surable isomorphism is continuous. So, if the Axiom of Determinacy (AD)holds (which contradicts the axiom of choice), then there is no difference.So, to build profinite groups that are only algebraically isomorphic one needsto use the axiom of choice. For instance let A be the product of all cyclicgroups of order p n , n ∈ N − { } . Then A is algebraically isomorphic to A × Z p by Cor 3.2 in Kiehlmann (J. Group Theory 16 (2013), 141-157),which uses AC. They are not topologically isomorphic because the torsionsubgroup is dense in A but not in A × Z p . As alternative to Kiehlmann’sproof one can use Ulm invariants for abelian p groups. One can show thedual fact in abelian torsion groups, that (cid:76) n C p n ∼ = (cid:76) n C p n + C p ∞ . To makeUlm invariants work one again needs AC (in the guise of the well orderingtheorem).If an enriched structure M is quasi-axiomatizable within C then eachalgebraic isomorphim with another structure in C is automatically a fullisomorphism. 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