Construction of the proofs, by means of which the formalization of the higher inferences is made possible and the consistency problem is becoming accessible in a general way. Finitist arithmetic involves induction and primitive recursion [ 3 ] from the outset, and the central metamathematical arguments all proceed straightforwardly by induction on proof figures. Here we find the first significant consistency proof—from a finitist perspective.
Here we just note that it involves two logical rules, namely, modus ponens and substitution for individual, function and statement variables in axioms. The non-logical axioms concern identity, zero and successor, and recursion equations that define primitive recursive functions. The resulting syntactic configurations, a Beweisfigur , contains now only numerical formulae that are built up from equations or inequations between numerals and Boolean connectives; these formulae can be effectively determined to be true or false.
The induction principle can be directly incorporated into these considerations when it is formulated as a rule for quantifier-free statements. These proof theoretic considerations are striking and important as they involve for the first time genuine transformations of formal derivations. Nevertheless, they are preliminary as they concern a quantifier-free theory that is a part of finitist mathematics and need not be secured by a consistency proof. The strategy was direct and started to emerge already in First, introduce functional terms by the transfinite axiom [ 5 ].
Using the epsilon terms, quantifiers can now be eliminated from proofs in quantificational logic, thus transforming them into quantifier-free ones. Finally, the given derivation allows one, so it was conjectured, to determine numerical values for the epsilon terms. In his Leipzig talk of September , published in , Hilbert discussed this Ansatz for eliminating quantifiers and reducing the transfinite case to that of the quantifier-free theory. Ackermann continued in section III of his at the very spot where Hilbert and Bernays had left off. His paper, submitted to Mathematische Annalen in March of , and the corrective work he did in led to the conviction that the consistency of elementary arithmetic had been established.
The corrective work had been done to address difficulties von Neumann had pointed out, but was not published by Ackermann; it was only presented in the second volume of Hilbert and Bernays pp. Let F be a theory containing exclusively such principles, like primitive recursive arithmetic PRA ; the principles of PRA consist of the Peano axioms for zero and successor, the defining equations for all primitive recursive functions defined in note 3 , and quantifier-free induction.
Now the significance of a consistency proof in F can be articulated as follows:. Theorem 1. On the basis of your results one must now conclude, however, that that proof cannot be formalized within the system Z [of elementary number theory]; this must in fact hold true even if the system is restricted so that, of the recursive definitions, only those for addition and multiplication are retained. If we take a theory, which is constructive in the sense that each existence assertion made in the axioms is covered by a construction, and if we add to this theory the non-constructive notion of existence and all the logical rules concerning it, e.
His methods not only applied to PM but to any formal system that contains a modicum of arithmetic.
Every consistent and effectively axiomatized theory that allows for the development of basic parts of arithmetic cannot prove its own consistency. This came to be known as the second incompleteness theorem. For details on these theorems and their history see appendix A. As a matter of fact, contemporary characterizations of finitist mathematics have elementary arithmetic as an upper bound.
The semi-formal calculi, which articulate the broader framework, are based on rules that reflect mathematical practice, but also define the meaning of logical connectives. Adding the tertium non datur in the form. The axioms comprise the usual equations for zero, successor, addition, multiplication, exponentiation, and the less-than relation.
These principles together with classical logic constitute the theory of first order arithmetic or first order number theory , also known as Dedekind-Peano arithmetic , PA ; together with intuitionist logic they constitute intuitionistic first order arithmetic commonly known as Heyting-arithmetic , HA.
Thus, PA is consistent relative to HA. This result is technically of great interest and had a profound effect on the perspective concerning the relationship between finitism and intuitionism: finitist and intuitionist mathematics were considered as co-extensional; this theorem showed that intuitionist mathematics is actually stronger than finitist mathematics. Thus, if the intuitionist standpoint is taken to guarantee the soundness of HA , then it guarantees the consistency of PA. The corresponding connection between classical and intuitionist logic had been established already by Kolmogorov who not only formalized intuitionist logic but also observed the translatability of classical into intuitionist logic.
The connection between PA and HA is paradigmatic and leads to the notion of proof-theoretic reduction. Any finite object such as a string of symbols or an array of symbols can be coded via a single natural number in such a way that the string or array can be retrieved from the number when we know how the coding is done.
Typical finite objects include formulae in a given language and also proofs in a theory. Talk about formulae or proofs can then be replaced by talking about predicates of numbers that single out the codes of formulae and proofs, respectively. We then say that the concepts of formula and proof have been arithmetized and thereby rendered expressible in the language of PRA. Definition 1. The complexity of formulae of PRA is stratified as follows.
Thus the complexity of a formula is measured in terms of quantifier alternations. For his further finitist investigations Gentzen introduced new calculi that were to become of utmost importance for proof theory: natural deduction and sequent calculi. As we noted above, Gentzen had already begun in to be concerned with the consistency of full elementary number theory.
As the logical framework he used, what we now call, natural deduction calculi. They evolved from an axiomatic calculus that had been used by Hilbert and Bernays since and introduced an important modification of the calculus for sentential logic. Hilbert and Bernays introduced this new logical formalism for two reasons, i to be able to better and more easily formalize mathematics, and ii to bring out the understanding of logical connectives in methodological parallel to the treatment of geometric concepts in Foundations of geometry.
The methodological advantages of this calculus are discussed in Bernays The starting formulae can be chosen in quite different ways. A great deal of effort has been spent, in particular, to get by with a minimal number of axioms, and the limit of what is possible has indeed been reached. However, for the purposes of logical investigations it is better to separate out, in line with the axiomatic procedure for geometry, different axiom groups in such a way that each of them expresses the role of a single logical operation.
Bernays had investigated in his Habilitationsschrift rule based calculi. However, in the given context, the simplicity of the metamathematical description of calculi seemed paramount, and in Bernays p. Gentzen was led to a rule-based calculus with introduction and elimination rules for every logical connective. The truly distinctive feature of this new type of calculus was for Gentzen, however, making and discharging assumptions.
This feature, he remarked, most directly reflects a crucial aspect of mathematical argumentation. Gentzen discovered a remarkable fact for the intuitionist calculus, having observed that proofs can have peculiar detours of the following form: a formula is obtained by an I-rule and is then the major premise of the corresponding E-rule.
For conjunction such a detour is depicted as follows:. Clearly, a proof of B is already contained in the given derivation. Theorem 2. Focusing on normal proofs, Gentzen proved then that the complexity of formulae in such proofs can be bounded by that of assumptions and conclusion. Corollary 2. To be able to formulate it [the Hauptsatz ] in a direct way, I had to base it on a particularly suitable logical calculus. The calculus of natural deduction turned out not to be appropriate for that purpose. In his thesis Gentzen introduced a form of the sequent calculus and his technique of cut elimination.
As this is a tool of utmost importance in proof theory, an outline of the underlying ideas will be discussed next. The sequent calculus can be generalized to so-called infinitary logics and is central for ordinal analysis. The Hauptsatz is also called the cut elimination theorem. In point of fact, one could limit this axiom to the case of atomic formulae A. We have structural rules of the form. A special case of the structural rule, known as contraction , occurs when the lower sequent has fewer occurrences of a formula than the upper sequent.
In the rules for logical operations, the formulae highlighted in the premises are called the minor formulae of that inference, while the formula highlighted in the conclusion is the principal formula of that inference. The other formulae of an inference are called side formulae. The Cut rule differs from the other rules in an important respect. With the rules for introducing connectives, one sees that every formula that occurs above the line occurs below the line either directly, or as a subformula of a formula below the line.
That is also true for the structural rules. But in the case of the Cut rule, the cut formula A vanishes. The proof of the cut elimination theorem is rather intricate as the process of removing cuts interferes with the structural rules. It is contraction that accounts for the high cost of eliminating cuts. The sequent calculus we have been discussing allows the proof of classically, but not intuitionistically correct formulae, for example, the law of excluded middle. The cut elimination theorem is also provable for this intuitionist variant. In either case, the Hauptsatz has an important corollary that parallels that of the Normalization theorem for intuitionist logic and expresses the subformula property.
This Corollary has another direct consequence that explains the crucial role of the Hauptsatz for obtaining consistency proofs. The foregoing results are solely concerned with pure logic. Formal theories that axiomatize mathematical structures or serve as formal frameworks for developing substantial chunks of mathematics are based on logic but have additional axioms germane to their purpose. If they are of the latter kind, such as first-order arithmetic or Zermelo-Fraenkel set theory , they will assert the existence of mathematical objects and their properties.
What happens when we try to apply the procedure of cut elimination to theories? Axioms are usually detrimental to this procedure. It breaks down because the symmetry of the sequent calculus is lost. In general, one cannot remove cuts from deductions in a theory T when the cut formula is an axiom of T.
However, sometimes the axioms of a theory are of bounded syntactic complexity. Then the procedure applies partially in that one can remove all cuts that exceed the complexity of the axioms of T. This gives rise to partial cut elimination. It is a very important tool in proof theory. For example, it can be used to analyze theories with restricted induction such as fragments of PA ; cf.
Sieg They had been obtained at least in principle for fragments of elementary number theory; in practice, Gentzen did not include the quantifier-free induction principle. Having completed his dissertation, Gentzen went back to investigate natural deduction calculi and obtained in his first consistency proof for full first-order arithmetic. This proof was published only in ; it was subsequently analyzed most carefully in Tait and Buchholz Here we just mention that Bernays extensively discussed transfinite induction in Grundlagen der Mathematik II.
The main issue for Bernays was the question, is it still a finitist principle? We will see, how far current techniques lead us and what foundational significance one can attribute to them. Cut elimination fails for first-order arithmetic i. Gentzen, however, found an ingenious way of dealing with purported contradictions in arithmetic. In Gentzen b he showed how to effectively transform an alleged PA -proof of an inconsistency the empty sequent in his sequent calculus into another proof of the empty sequent such that the latter gets assigned a smaller ordinal than the former.
Ordinals are a central concept in set theory as well as in proof theory. This is the first time we talk about the transfinite and ordinals in proof theory. Ordinals have become very important in advanced proof theory. The concept of an ordinal is a generalization of that of a natural number. Such an ordering is called a well-ordering of X. However, 0 plays a unique role. Let us first state some precise definitions and a Cantorian theorem. Definition 3. These are the successor elements of A , with a being the successor of b.
In set theory a set is called transitive just in case all its elements are also subsets. It follows that each ordinal is the set of predecessors. According to the trichotomy above, there is a least ordinal which is just the empty set and all other ordinals are either successor or limit ordinals.
Fact 3. The following states the definitions just to convey the flavor:. In essence Cantor defined the first ordinal representation system in Natural ordinal representation system s are frequently derived from structures of the form. Theorem 3. For instance a coding function. Further define:. Unsurprisingly, the above notion has certain intensional aspects and hinges on the naturality of the representation system for a discussion see Rathjen a: section 2. Hence PA is consistent.
As it turned out, the obstacles to cut elimination, inherent to PA , could be overcome by moving to a richer proof system, albeit in a drastic way by going infinite. This richer system allows for proof rules with infinitely many premises.
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The price to pay will be that deductions become infinite objects, i. Thus free variables are discarded and all terms will be closed. All formulae of this system are therefore closed, too. We will notate this by. The surreal sum and product of two ordinals coincide with the Hessenberg sum and product, and Cantor's normal form of ordinals has a natural extension to the surreals.
Techniques of Admissible Recursion Theory
In  we proved that there is a meaningful way to take both the derivative and the integral anti-derivative of a surreal number, hence in particular on an ordinal number. The derivative of the ordinal number omega is 1, the derivative of a real number is zero, and the derivative of the sum and product of two surreal numbers obeys the expected rules. More difficult is to understand what is the derivative of an ordinal power of omega, for instance the first epsilon-number, but this can be done in a way that reflects the formal properties of the derivation on a Hardy field germs of non-oscillating real functions.
In  we showed that many surreal numbers can indeed be interpreted as germs of differentiable functions on the surreals themselves, so that the derivative acquires the usual analytic meaning as a limit.
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It is still open whether we can interpret all the surreals as differentiable functions, possibly changing the definition of the derivative. To appear in the Journal of the European Mathematical Society. This will be a three part tutorial: 1. General introduction to Stone duality 2. Applications in semantics 3. Applications in formal languages: automata and beyond. The notion of generic-case complexity was introduced by Kapovich, Myasnikov, Schupp, and Shpilrain to study problems with high worst-case complexity that are nevertheless easy to solve in most instances. They also introduced the notion of generic computability, which captures the idea of having a partial algorithm that halts for almost all inputs, and correctly computes a decision problem whenever it halts.
Jockusch and Schupp began the general computability-theoretic investigation of generic computability and also defined the notion of coarse computability, which captures the idea of having a total algorithm that might make mistakes, but correctly decides the given problem for almost all inputs although this notion had been studied earlier in Terwijn's dissertation. Two related notions, which allow for both failures to answer and mistakes, have been studied by Astor, Hirschfeldt, and Jockusch although one of them had been considered in the 's by Meyer and by Lynch. All of these notions lead to notions of reducibility and associated degree structures.
I will discuss recent and ongoing work in the study of these reducibilities. Gandhi, Khoussainov, and Liu introduced and studied a generalized model of finite automata able to work over arbitrary structures. The model mimics finite automata over finite structures, but has an additional ability to perform in a restricted way operations attached to the structure under consideration.
As one relevant area of investigations for this model Gandhi et al. In the talk we pick up this suggestion and consider their automata model as a finite automata variant in the BSS model of real number computation. We study structural properties as well as un- decidability results for several questions inspired by the classical finite automata model.
Symbolic Dynamics is the study of subshifts, sets of infinite words given by local constraints.
Subshifts constitute a shift-invariant version of Pi classes of sets and are intimately linked to automata theory in dimension one and tiling theory in higher dimensions. One of its distinguished feature is that subshifts of finite type, which are equivalent to tilings of the discrete space by Wang tiles, already exhibit a large range of uncomputable behaviours, as evidence by Berger in the 60s and popularized by Robinson and his so-called Robinson tiling in the 70s. While these results could be deemed negative, a recent approach due to Hochman show that various quantities and invariants defined for subshifts can be completely understood and characterized using various concepts from computability theory.
The goal of this talk is to show a striking resemblance between these recent results and the embedding theorems pioneered by Higman in the 60s for combinatorial group theory. To do this, I will present a framework in which these theorems can be written using the exact same vocabulary, and show how the easy part of the theorems follow from the exact same proof. I will discuss how computability-theoretic methods can be used to prove theorems in topology.
A naive approach to developing the methods of homological algebra for difference and differential fields, rings and modules quickly encounters numerous obstacles, such as the failure of the hom-tensor duality. We will conclude by applying these techniques to study the cohomology of difference algebraic groups and discuss potential model-theoretic consequences. Well-known examples of amenable groups are finite groups, solvable groups and locally compact abelian groups. In this talk we will consider automorphism groups of certain Hrushovski's generic structures.
I will discuss novel applications of continuous logic to ergodic theory, particularly to the study of rigidity phenomena associated with strongly ergodic actions of countable groups. Kreisel has suggested that squeezing arguments, originally formulated for the informal concept of first order validity, should be extendable to second order logic, although he points out obvious obstacles. We develop this idea in the light of more recent advances and delineate the difficulties across the spectrum of extensions of first order logics by generalised quantifiers and infinitary logics.
In particular we argue that if the relevant informal concept is read as informal in the precise sense of being untethered to a particular semantics, then the squeezing argument goes through in the second order case. Consideration of weak forms of Kreisel's squeezing argument leads naturally to reflection principles of set theory. See attachment below. I have corresponded with Mirna Dzamonja, who invited me to submit a paper for LC However, the discussion on the nature of this method is still open.
There are 1 those who have seen it as a synthetic method, i. Each of these views has highlighted aspects of the way Hilbert conceived and practiced the axiomatic method, so they can be harmonized into an image better suited to the function the method was called to fulfill: i.
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We present some results about Frege proof complexities. Then we show, that all balanced tautologies in disjunctive normal form also have Frege proofs with polynomially bounded sizes. Both Turing reducibility and hyperarithmetical reducibility are important in the field of effective descriptive set theory. The even more general notion of degrees of constructibility is studied in set theory. Computability theory for digital computation is well developed.
Computability theory is less well developed for analog computation that occurs in analog computers , analog signal processing , analog electronics , neural networks and continuous-time control theory , modelled by differential equations and continuous dynamical systems Orponen ; Moore There are close relationships between the Turing degree of a set of natural numbers and the difficulty in terms of the arithmetical hierarchy of defining that set using a first-order formula.
One such relationship is made precise by Post's theorem.
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Similarly, Tarski's indefinability theorem can be interpreted both in terms of definability and in terms of computability. Recursion theory is also linked to second order arithmetic , a formal theory of natural numbers and sets of natural numbers. The fact that certain sets are computable or relatively computable often implies that these sets can be defined in weak subsystems of second order arithmetic. The program of reverse mathematics uses these subsystems to measure the noncomputability inherent in well known mathematical theorems.
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The field of proof theory includes the study of second-order arithmetic and Peano arithmetic , as well as formal theories of the natural numbers weaker than Peano arithmetic. One method of classifying the strength of these weak systems is by characterizing which computable functions the system can prove to be total see Fairtlough and Wainer For example, in primitive recursive arithmetic any computable function that is provably total is actually primitive recursive , while Peano arithmetic proves that functions like the Ackermann function , which are not primitive recursive, are total.
Not every total computable function is provably total in Peano arithmetic, however; an example of such a function is provided by Goodstein's theorem. The field of mathematical logic dealing with computability and its generalizations has been called "recursion theory" since its early days.
Robert I. Soare , a prominent researcher in the field, has proposed Soare that the field should be called "computability theory" instead. He argues that Turing's terminology using the word "computable" is more natural and more widely understood than the terminology using the word "recursive" introduced by Kleene. Many contemporary researchers have begun to use this alternate terminology. Not all researchers have been convinced, however, as explained by Fortnow  and Simpson.
Rogers has suggested that a key property of recursion theory is that its results and structures should be invariant under computable bijections on the natural numbers this suggestion draws on the ideas of the Erlangen program in geometry. The idea is that a computable bijection merely renames numbers in a set, rather than indicating any structure in the set, much as a rotation of the Euclidean plane does not change any geometric aspect of lines drawn on it. Since any two infinite computable sets are linked by a computable bijection, this proposal identifies all the infinite computable sets the finite computable sets are viewed as trivial.
According to Rogers, the sets of interest in recursion theory are the noncomputable sets, partitioned into equivalence classes by computable bijections of the natural numbers. The main professional organization for recursion theory is the Association for Symbolic Logic , which holds several research conferences each year. The interdisciplinary research Association Computability in Europe CiE also organizes a series of annual conferences.
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