On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural extension of BLT is definitionally equivalent with ZF.

We show that numerous distinctive concepts of constructive mathematics arise automatically from an “antithesis” translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically “constructivize” classical definitions, handling the resulting bookkeeping automatically.

Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko’s Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style entailment relation over a single-conclusion one.

Two famous negative results about da Costa’s paraconsistent logic ${\mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed ${\mathscr {C}}_1$ seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic ${\mathscr {C}}_1$ . On the one hand, we strengthen the negative results about ${\mathscr {C}}_1$ by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, ${\mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of ${\mathscr {C}}_1$ covered in the literature. We prove that for extensions ${\mathcal {S}}$ such as ${\mathcal {C}ilo}$ [26], every algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ contains a Boolean subalgebra, and for extensions ${\mathcal {S}}$ such as , , or [16, 53], every subdirectly irreducible algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety ${\mathsf {Alg}}^*({\mathcal {S}})$ and the intrinsic variety $\mathbb {V}({\mathcal {S}})$ , with , , and .

O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable $C^{\infty }$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable $C^{\infty }$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.

Abstract prepared by Rodrigo Figueiredo.

E-mail: rodrigo@ime.usp.br

URL: https://doi.org/10.11606/T.45.2019.tde-28042019-181150

In this survey we discuss work of Levin and V’yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V’yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal {A}$ and $\mathcal {B}$ , $\mathcal {A}$ is below $\mathcal {B}$ in this ordering if $\mathcal {A}\setminus \mathcal {B}$ is negligible. The degree structure associated with this ordering, the Levin–V’yugin degrees (or $\mathrm {LV}$ -degrees), can be shown to be a Boolean algebra, and in fact a measure algebra. We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin–V’yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V’yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V’yugin’s about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree.

The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations.

Suppose that we have a method which estimates the conditional probabilities of some unknown stochastic source and we use it to guess which of the outcomes will happen. We want to make a correct guess as often as it is possible. What estimators are good for this? In this work, we consider estimators given by a familiar notion of universal coding for stationary ergodic measures, while working in the framework of algorithmic randomness, i.e., we are particularly interested in prediction of Martin-Löf random points. We outline the general theory and exhibit some counterexamples. Completing a result of Ryabko from 2009 we also show that universal probability measure in the sense of universal coding induces a universal predictor in the prequential sense. Surprisingly, this implication holds true provided the universal measure does not ascribe too low conditional probabilities to individual symbols. As an example, we show that the Prediction by Partial Matching (PPM) measure satisfies this requirement with a large reserve.

We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ (that is, the ones that are closed under Borel preimages) and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep (namely, that every non-selfdual Wadge class can be expressed as the result of a Hausdorff operation applied to the open sets). The exposition is self-contained, except for facts from classical descriptive set theory.

Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.

Locality is a property of logics, whose origins lie in the works of Hanf and Gaifman, having their utility in the context of finite model theory. Such a property is quite useful in proofs of inexpressibility, but it is also useful in establishing normal forms for logical formulas. There are generally two forms of locality: (i’) if two structures $\mathfrak {A}$ and $\mathfrak {B}$ realize the same multiset of types of neighborhoods of radius d, then they agree on a given sentence $\Phi $ . Here d depends only on $\Phi $ ; (ii’) if the d-neighborhoods of two tuples $\vec {a}_1$ and $\vec {a}_2$ in a structure $\mathfrak {A}$ are isomorphic, then $\mathfrak {A} \models \Phi (\vec {a}_1) \Leftrightarrow \Phi (\vec {a}_2)$ . Again, d depends on $\Phi $ , and not on $\mathfrak {A}$ . Form (i’) originated from Hanf’s works. Form (ii’) came from Gaifman’s theorem. There is no doubt about the usefulness of the notion of locality, which as seen applies to a huge number of situations. However, there is a deficiency in such a notion: all versions of the notion of locality refer to isomorphism of neighborhoods, which is a fairly strong property. For example, where structures simply do not have sufficient isomorphic neighborhoods, versions of the notion of locality obviously cannot be applied. So the question that immediately arises is: would it be possible to weaken such a condition and maintain Hanf/Gaifman-localities? Arenas, Barceló, and Libkin establish a new condition for the notions of locality, weakening the requirement that neighborhoods should be isomorphic, establishing only the condition that they must be indistinguishable in a given logic. That is, instead of requiring $N_d(\vec {a}) \cong N_d(\vec {b})$ , you should only require $N_d(\vec {a}) \equiv _k N_d(\vec {b})$ , for some $k \geq 0$ . Using the fact that logical equivalence is often captured by Ehrenfeucht–Fraïssé games, the authors formulate a game-based framework in which logical equivalence-based locality can be defined. Thus, the notion defined by the authors is that of game-based locality. Although quite promising as well as easy to apply, the game-based framework (used to define locality under logical equivalence) has the following problem: if a logic $\mathcal {L}$ is local (Hanf-, or Gaifman-, or weakly) under isomorphisms, and $\mathcal {L}'$ is a sub-logic of $\mathcal {L}$ , then $\mathcal {L}'$ is local as well. The same, however, is not true for game-based locality: properties of games guaranteeing locality need not be preserved if one passes to weaker games. The question that immediately arises is: is it possible to define the notion of locality under logical equivalence without resorting to game-based frameworks? In this thesis, I present a homotopic variation for locality under logical equivalence, namely a Quillen model category-based framework for locality under k-logical equivalence, for every primitive-positive sentence of quantifier-rank k.

Abstract prepared by Hendrick Maia.

E-mail: hendrickmaia@gmail.com

URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/334956

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to $\exists xA(x)$ , or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent ( ${\Sigma}_1^0$ -correct). Here we show that if the result is supposed to be provable within S, a statement about all ${\Pi}_2^0$ statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel’s but arises naturally out of the Hilbert program itself.

The interplay of philosophical ambitions and technical reality have given birth to rich and interesting approaches to explain the oft-claimed special character of mathematical and logical knowledge. Two projects stand out both for their audacity and their innovativeness. These are logicism and proof-theoretic semantics. This dissertation contains three chapters exploring the limits of these two projects. In both cases I find the formal results offer a mixed blessing to the philosophical projects.

Chapter 1. Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. I re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. I conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.

Chapter 2. There have been several recent results bringing into focus the super-intuitionistic nature of most notions of proof-theoretic validity. But there has been very little work evaluating the consequences of these results. In this chapter, I explore the question of whether these results undermine the claim that proof-theoretic validity shows us which inferences follow from the meaning of the connectives when defined by their introduction rules. It is argued that the super-intuitionistic inferences are valid due to the correspondence between the treatment of the atomic formulas and more complex formulas. And so the goals of proof-theoretic validity are not undermined.

Chapter 3. Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This chapter resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The chapter further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.

Abstract prepared by Will Stafford extracted partially from the dissertation.

E-mail: stafford@flu.cas.cz

URL: https://escholarship.org/uc/item/33c6h00c

When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as $\mathbb Q_{p}\langle X\rangle $ , $\mathbb Z_{p}\langle X\rangle $ , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg.

Abstract prepared by Madeline G. Barnicle.

E-mail: barnicle@math.ucla.edu

URL: https://escholarship.org/uc/item/6t02q9s4

The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which one still has control from the ground model, and has shown that these axiomatics necessarily lead to the usual concepts of genericity and of forcing extensions, and also that one can infer from them the usual recursive definition of forcing predicates. In this paper, we present a more general such approach, covering both class forcing and set forcing, over various base theories, and we provide additional details regarding the formal setting that was outlined in [3].

Invariance criteria are widely accepted as a means to demarcate the logical vocabulary of a language. In previous work, I proposed a framework of “semantic constraints” for model-theoretic consequence which does not rely on a strict distinction between logical and nonlogical terms, but rather on a range of constraints on models restricting the interpretations of terms in the language in different ways. In this paper I show how invariance criteria can be generalized so as to apply to semantic constraints on models. Some obviously unpalatable semantic constraints turn out to be invariant under isomorphisms. I shall connect the discussion to known counter-examples to invariance criteria for logical terms, and so the generalization will also shed light on the current existing debate on logicality. I analyse the failure of invariance to fulfil its role as a criterion for logicality, and argue that invariance conditions should best be thought of as merely methodological meta-constraints restricting the ways the model-theoretic apparatus should be used.

In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example, linear independence in vector spaces yields such a relation.

Some important classes in the stability hierarchy are stable, simple, and NSOP $_1$ , each being contained in the next. For each of these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example, simplicity is equivalent to admitting a certain independence relation, which must then be unique.

All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic, and even a very general category-theoretic framework. In this thesis we continue this work.

We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple, or NSOP $_1$ -like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing, and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.

Switching frameworks, we generalise Kim-dividing in NSOP $_1$ theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP $_1$ if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.

Abstract prepared by Mark Kamsma.

E-mail: mark@markkamsma.nl.

URL: https://markkamsma.nl/phd-thesis.

We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, and for any $n\geq k+1$ ,

$$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \end{align*}$$

$$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}}(T) \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, \end{align*}$$

$T_0$

$\mathrm {TI}_{\varPi ^1_n}(\varepsilon _0)^{-}$

$\varepsilon _0$

$\varPi ^1_n$

The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal almost disjoint family. An ultrafilter $\mathcal {U}$ on $\omega $ is called a P-point if every countable $\mathcal {B\subseteq U}$ there is $X\in $ $\mathcal {U}$ such that $X\setminus B$ is finite for every $B\in \mathcal {B}.$ This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them.

In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under $\mathfrak {s\leq a}.$ None of the results in this chapter are due to the author.

The second chapter is dedicated to a principle of Sierpiński. The principle $\left ( \ast \right ) $ of Sierpiński is the following statement: There is a family of functions $\left \{ \varphi _{n}:\omega _{1}\longrightarrow \omega _{1}\mid n\in \omega \right \} $ such that for every $I\in \left [ \omega _{1}\right ] ^{\omega _{1}}$ there is $n\in \omega $ for which $\varphi _{n}\left [ I\right ] =\omega _{1}.$ This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set $X=\left \{ f_{\alpha }\mid \alpha <\omega _{1}\right \} \subseteq \omega ^{\omega }$ such that for every $g:\omega \longrightarrow \omega $ there is $\alpha $ such that if $\beta>\alpha $ then $f_{\beta }\cap g$ is infinite (sets with that property are referred to as $\mathcal {IE}$ -Luzin sets ). Miller showed that the principle of Sierpiński implies that non $\left ( \mathcal {M}\right ) =\omega _{1}.$ He asked if the converse was true, i.e., does non $\left ( \mathcal {M}\right ) =\omega _{1}$ imply the principle $\left ( \ast \right ) $ of Sierpiński? We answer his question affirmatively. In other words, we show that non $\left ( \mathcal {M}\right ) =\omega _{1}$ is enough to construct an $\mathcal {IE}$ -Luzin set. It is not hard to see that the $\mathcal {IE}$ -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non $\left ( \mathcal {M}\right ) =\omega _{1}$ and every $\mathcal {IE}$ -Luzin set is meager.

The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non $\left ( \mathcal {M}\right ) $ or at least cov $\left ( \mathcal {M}\right ) $ (it is known that the definability is an important requirement, otherwise $\mathfrak {a}$ would be a counterexample). Although the veracity of this conjecture is still an open problem, we were able to obtain some partial results: The conjecture is false for “Borel invariants of $\omega _{1}^{\omega }$ ” nevertheless, it is true for a large class of definable invariants. This is part of a joint work with Michael Hrušák and Jindřich Zapletal.

In the fourth chapter we present a survey on destructibility of ideals and MAD families. We prove several classic theorems, but we also prove some new results. For example, we show that every almost disjoint family of size less than $\mathfrak {c}$ can be extended to a Cohen indestructible MAD family is equivalent to $\mathfrak {b=c}$ (this is part of a joint work with Michael Hrušák, Ariet Ramos, and Carlos Martínez). A MAD family $\mathcal {A}$ is Shelah–Steprāns if for every $X\subseteq \left [ \omega \right ] ^{<\omega }\setminus \left \{ \emptyset \right \} $ either there is $A\in \mathcal {I}\left ( \mathcal {A}\right ) $ such that $s\cap A\neq \emptyset $ for every $s\in X$ or there is $B\in \mathcal {I}\left ( \mathcal {A}\right ) $ that contains infinitely many elements of X (where $\mathcal {I}\left ( \mathcal {A}\right ) $ denotes the ideal generated by $\mathcal {A}$ ). This concept was introduced by Raghavan which is connected to the notion of “strongly separable” introduced by Shelah and Steprāns. We prove that Shelah–Steprāns MAD families have very strong indestructibility properties: Shelah–Steprāns MAD families are indestructible for “many” definable forcings that does not add dominating reals (this statement will be formalized in the fourth chapter). According to the author’s best knowledge, this is the strongest notion (in terms of indestructibility) that has been considered in the literature so far. In spite of their strong indestructibility, Shelah–Steprāns MAD families can be destroyed by a ccc forcing that does not add unsplit or dominating reals. We also consider some strong combinatorial properties of MAD families and show the relationships between them (This is part of a joint work with Michael Hrušák, Dilip Raghavan, and Joerg Brendle).

The fifth chapter is one of the most important chapters in the thesis. A MAD family $\mathcal {A}$ is called $+$ -Ramsey if every tree that branches into $\mathcal {I}\left ( \mathcal {A}\right ) $ -positive sets has an $\mathcal {I}\left ( \mathcal {A}\right ) $ -positive branch. Michael Hrušák’s first published question is the following: Is there a $+$ -Ramsey MAD family? It was previously known that such families can consistently exist. However, there was no construction of such families using only the axioms of ZFC. We solve this problem by constructing such a family without any extra assumptions.

In the fourth and fifth chapters, we introduce several notions of MAD families, in the sixth chapter we prove several implications and non implications between them. We construct (under $\mathsf {CH}$ ) several MAD families with different properties.

In the seventh chapter we build models without P -points. We show that there are no P -points after adding Silver reals either iteratively or by the side by side product. These results have some important consequences: The first one is that is its possible to get rid of P -points using only definable forcings. This answers a question of Michael Hrušák. We can also use our results to build models with no P -points and with arbitrarily large continuum, which was also an open question. These results were obtained with David Chodounský.

Abstract prepared by Osvaldo Guzmán González

E-mail : oguzman@matmor.unam.mx

URL : https://arxiv.org/abs/1810.09680

Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR $_{0}$ (and so $\Pi _{1}^{1}$ -CA $_{0}$ or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They seem to be typical applications of ACA $_{0}$ but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity.

This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA $_{0}$ they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes ( $\Pi _{1}^{1}$ , r- $\Pi _{1}^{1}$ , and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA $_{0}$ in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA $_{0}$ but over ACA $_{0}$ is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic.