More infinity for a better finitism

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed in around 1995 by Patrick Suppes and Richard Sommer. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes. Here, we discuss the inherent problems and limitations of the classical nonstandard framework and propose a much-needed refinement of ERNA, called ERNA^A, in the spirit of Karel Hrbacek’s stratified set theory. We study the metamathematics of ERNA^A and its extensions. In particular, we consider several transfer principles, both classical and ‘stratified’, which turn out to be related. Finally, we show that the resulting theory allows for a truly general, elegant and elementary treatment of basic analysis.     

Interpretations of the alternative set theory

Summary We show an axiom A such that there is no nontrivial interpretation of the alternative set theory (AST) inAST+A keeping , sets and the class of all standard natural numbers. Furthermore, there is no interpretation ofAST inAST without the prolongation axiom, but there is an interpretation ofAST in the theory having the prolongation axiom and the basic set-theoretical axioms only.     

Wellfoundedness proofs by means of non-monotonic inductive definitions II: First order operators

In this paper, we give two proofs of the wellfoundedness of a recursive notation system for Π_N-reflecting ordinals. One is based on distinguished classes, and the other is based on -inductive definitions.     

Nonstandard arithmetic and recursive comprehension

First order reasoning about hyperintegers can prove things about sets of integers. In the author’s paper Nonstandard Arithmetic and Reverse Mathematics, Bulletin of Symbolic Logic 12 (2006) 100-125, it was shown that each of the “big five” theories in reverse mathematics, including the base theory , has a natural nonstandard counterpart. But the counterpart of has a defect: it does not imply the Standard Part Principle that a set exists if and only if it is coded by a hyperinteger. In this paper we find another nonstandard counterpart, , that does imply the Standard Part Principle.     

The axiomatic power of Kolmogorov complexity

The famous Gödel incompleteness theorem states that for every consistent, recursive, and sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates for being added as axioms, but how can we obtain them? One classical (and well studied) approach is to add to some theory T an axiom that claims the consistency of T  . In this paper we discuss another approach motivated by Chaitin's version of Gödel's theorem where axioms claiming the randomness (or incompressibility) of some strings are probabilistically added, and show that it is not really useful, in the sense that this does not help us prove new interesting theorems. This result answers a question recently asked by Lipton. The situation changes if we take into account the size of the proofs: randomly chosen axioms may help making proofs much shorter (unless NP=PSPACE$\mathrm{NP}=\mathrm{PSPACE}$).     

Consistency,optimality, and incompleteness

Assume that the problem _P0$P_0$ is not solvable in polynomial time. Let T   be a first-order theory containing a sufficiently rich part of true arithmetic. We characterize T∪{_ConT}$T\cup \{{\mathit{Con}}_T\}$ as the minimal extension of T   proving for some algorithm that it decides _P0$P_0$ as fast as any algorithm B$\mathbb{B}$ with the property that T   proves that B$\mathbb{B}$ decides _P0$P_0$. Here, _ConT${\mathit{Con}}_T$ claims the consistency of T. As a byproduct, we obtain a version of Gödel?s Second Incompleteness Theorem. Moreover, we characterize problems with an optimal algorithm in terms of arithmetical theories.     

A homeomorphism between strong measure zero sets whose graph is not of strong measure zero

Assuming Martin's axiom MA, we define a homeomorphism between two strong measure zero sets in the real line whose graph in not of strong measure zero in the plane. Using Michael's concentrated sets, we give also some refinements of this result, and we describe some singular subgroups of the group ZN.     

Epsilon substitution method for [Π0 1, Π0 1]-FIX

We formulate epsilon substitution method for a theory [Π⁰₁, Π⁰₁]-FIX for two steps non-monotonic Π⁰₁ inductive definitions. Then we give a termination proof of the H-processes based on Ackermann [1].     

A measure-theoretic proof of Turing incomparability

We prove that if S is an ω-model of weak weak König’s lemma and , is incomputable, then there exists , such that A and B are Turing incomparable. This extends a recent result of Ku?era and Slaman who proved that if S₀ is a Scott set (i.e. an ω-model of weak König’s lemma) and A∈S₀, A⊆ω, is incomputable, then there exists B∈S₀, B⊆ω, such that A and B are Turing incomparable.     

Second order theories with ordinals and elementary comprehension

We study elementary second order extensions of the theoryID ₁ of non-iterated inductive definitions and the theoryPA _Ω of Peano arithmetic with ordinals. We determine the exact proof-theoretic strength of those extensions and their natural subsystems, and we relate them to subsystems of analysis with arithmetic comprehension plusΠ ₁ ¹ comprehension and bar induction without set parameters. Research supported by the Swiss National Science Foundation     

Unprovability threshold for the planar graph minor theorem

This note is part of the implementation of a programme in foundations of mathematics to find exact threshold versions of all mathematical unprovability results known so far, a programme initiated by Weiermann. Here we find the exact versions of unprovability of the finite graph minor theorem with growth rate condition restricted to planar graphs, connected planar graphs and graphs embeddable into a given surface, assuming an unproved conjecture (*): ‘there is a number a>0 such that for all k≥3, and all n≥1, the proportion of connected graphs among unlabelled planar graphs of size n omitting the k-element circle as minor is greater than a’. Let γ be the unlabelled planar growth constant (27.2269≤γ<30.061). Let P(c) be the following first-order arithmetical statement with real parameter c: “for every K there is N such that whenever G₁,G₂,…,G_N are unlabelled planar graphs with |G_i|<K+c⋅log₂i then for some i<j≤N, G_i is isomorphic to a minor of G_j”. Then

1.

for every , P(c) is provable in IΔ₀+exp;

2.

for every , P(c) is unprovable in .

We also give proofs of some upper and lower bounds for unprovability thresholds in the general case of the finite graph minor theorem.     

Injective power objects and the axiom of choice

“The axiom of choice states that any set X of non-empty sets has a choice function—i.e. a function satisfying f(x)∈x for all x∈X. When we want to generalise this to a topos, we have to choose what we mean by non-empty, since in , the three concepts non-empty, inhabited, and injective are equivalent, so the axiom of choice can be thought of as any of the three statements made by replacing “non-empty” by one of these notions.It seems unnatural to use non-empty in an intuitionistic context, so the first interpretation to be used in topos theory was the notion based on inhabited objects. However, Diaconescu (1975) [1] showed that this interpretation implied the law of the excluded middle, and that without the law of the excluded middle, even the finite version of the axiom of choice does not hold! Nevertheless some people still view this as the most appropriate formulation of the axiom of choice in a topos.In this paper, we study the formulation based upon injective objects. We argue that it can be considered a more natural formulation of the axiom of choice in a topos, and that it does not have the undesirable consequences of the inhabited formulation. We show that if it holds for , then it holds in a wide variety of topoi, including all localic topoi. It also has some of the classical consequences of the axiom of choice, although a lot of classical results rely on both the axiom of choice and the law of the excluded middle. An additional advantage of this formulation is that it can be defined for a slightly more general class of categories than just topoi.We also examine the corresponding injective formulations of Zorn’s lemma and the well-order principle. The injective form of Zorn’s lemma is equivalent to the axiom of injective choice, and the injective well-order principle implies the axiom of injective choice.     

The strength of compactness in Computability Theory and Nonstandard Analysis

Compactness is one of the core notions of analysis: it connects local properties to global ones and makes limits well-behaved. We study the computational properties of the compactness of Cantor space $2^{\mathbb{N}}$ for uncountable covers. The most basic question is: how hard is it to compute a finite sub-cover from such a cover of $2^{\mathbb{N}}$? Another natural question is: how hard is it to compute a sequence that covers $2^{\mathbb{N}}$ minus a measure zero set from such a cover? The special and weak fan functionals respectively compute such finite sub-covers and sequences. In this paper, we establish the connection between these new fan functionals on one hand, and various well-known comprehension axioms on the other hand, including arithmetical comprehension, transfinite recursion, and the Suslin functional. In the spirit of Reverse Mathematics, we also analyse the logical strength of compactness in Nonstandard Analysis. Perhaps surprisingly, the results in the latter mirror (often perfectly) the computational properties of the special and weak fan functionals. In particular, we show that compactness (nonstandard or otherwise) readily brings us to the outer edges of Reverse Mathematics (namely ${\mathrm{\Pi }}_2^1{\text{-}\mathsf{\text{CA}}}_0$), and even into Schweber's higher-order framework (namely ${\mathrm{\Sigma }}_1^2$-separation).     

k-Hypermonogenic automorphic forms

In this paper we deal with monogenic and k-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between k-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form. In the second part of the paper we compute the Fourier expansion of the k-hypermonogenic Eisenstein series which provide us with the simplest non-vanishing examples of k-hypermonogenic automorphic forms.     

The double negation of the intermediate value theorem

In the context of intuitionistic analysis, we consider the set F consisting of all continuous functions ? from [0,1] to R such that ?(0)=0 and ?(1)=1, and the set I₀ consisting of ?’s in F where there exists x∈[0,1] such that . It is well-known that there are weak counterexamples to the intermediate value theorem, and with Brouwer’s continuity principle we have I₀≠F. However, there exists no satisfying answer to . We try to answer to this question by reducing it to a schema (which we call ) about intuitionistic decidability that asserts “there exists an intuitionistically enumerable set that is not intuitionistically decidable”. We also introduce the notion of strong Specker double sequence, and prove that the existence of such a double sequence is equivalent to the existence of a function ?∈F_mon where .