De Rijke introduced a unary interpretability logic $$\textbf{il}$$, and proved that $$\textbf{il}$$ is the unary counterpart of the binary interpretability logic $$\textbf{IL}$$. In this paper, we find the unary counterparts of the sublogics of $$\textbf{IL}$$.

We define an axiom schema for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic,, and use some of its basic model theory to establish some independence results for various axioms of set theory over. Then we ask: given a model M of, is there a model of whose ordinal arithmetic is isomorphic to M? We show that the answer is yes if.

By a result of L.D. Beklemishev, the hierarchy of nested applications of the $$\Sigma _1$$ -collection rule over any $$\Pi _2$$ -axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true $$\Pi _2$$ -sentences, S, we construct a sound $$(\Sigma _2 \! \vee \! \Pi _2)$$ -axiomatized theory T extending S such that the (...) hierarchy of nested applications of the $$\Sigma _1$$ -collection rule over T is proper. Our construction uses some results on subrecursive degree theory obtained by L. Kristiansen. (shrink)

The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge on (...) intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article offers some reflections on the relation between knowledge, proof, and provability, as inspired by the knower paradox and its solutions. (shrink)

After some generalities about connections between functions and relations in Sections 1 and 2 recalls the possibility of taking the semantic values of ‐ary Boolean connectives as ‐ary relations among truth‐values rather than as ‐ary truth functions. Section 3, the bulk of the paper, looks at correlates of these truth‐value relations as applied to formulas, and explores in a preliminary way how their properties are related to the properties of “logical relations” among formulas such as equivalence, implication (entailment) and contrariety (...) (logical incompatibility), concentrating for illustrative purposes on binary logical relations such as those just listed. To avoid an excess of footnotes, some points have been deferred to an Appendix as “Longer Notes”. (shrink)

We prove a strengthened version of Shavrukov’s result on the non-isomorphism of diagonalizable algebras of two $\Sigma _1$ -sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms from the algebra $(\mathfrak {L}_T, \Box _T\Box _T)$ onto the algebra $(\mathfrak {L}_T, \Box _T)$. The case of bimodal diagonalizable algebras (...) is also considered. We give several examples of pairs of theories with isomorphic diagonalizable algebras but non-isomorphic bimodal diagonalizable algebras. (shrink)

ABSTRACT We use Gödel’s incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel’s incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel’s incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel’s incompleteness theorems.

By Solovay’s celebrated completeness result [31] on formal provability we know that the provability logic ${\textbf {GL}}$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable axiomatisation. Japaridze generalised this result in [22] by considering a polymodal version ${\mathsf {GLP}}$ of ${\textbf {GL}}$ with modalities $[n]$ for each natural number n referring to ever increasing notions of provability. Modern treatments of ${\mathsf {GLP}}$ tend to interpret the $[n]$ provability notion as “provable in (...) a base theory T together with all true $\Pi ^0_n$ formulas as oracles.” In this paper we generalise this interpretation into the transfinite. In order to do so, a main difficulty to overcome is to generalise the syntactical characterisations of the oracle formulas of complexity $\Pi ^0_n$ to the hyper-arithmetical hierarchy. The paper exploits the fact that provability is $\Sigma ^0_1$ complete and that similar results hold for stronger provability notions. As such, the oracle sentences to define provability at level $\alpha $ will recursively be taken to be consistency statements at lower levels: provability through provability whence the name of the paper. The paper proves soundness and completeness for the proposed interpretation for a wide class of theories, namely for any theory that can formalise the recursion described above and that has some further very natural properties. Some remarks are provided on how the recursion can be formalised into second order arithmetic and on lowering the proof-theoretical strength of these systems of second order arithmetic. (shrink)

Truth is often considered to be a logico-linguistic tool for expressing indirect endorsements and infinite conjunctions. In this article, I will point out another logico-linguistic function of truth: to enable and validate what I call a blind argument, namely, an argument that involves indirectly endorsed statements. Admitting this function among the logico-linguistic functions of truth has some interesting consequences. In particular, it yields a new type of so-called conservativeness argument, which poses a new type of threat to deflationism about truth.

This paper engages the question ‘Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?’ within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof and reception of Gödel’s Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be, but also in which (...) Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be. (shrink)

Recent work on formal theories of truth has revived an approach, due originally to Tarski, on which syntax and truth theories are sharply distinguished—‘disentangled’—from mathematical base theories. In this paper, we defend a novel philosophical constraint on disentangled theories. We argue that these theories must be epistemically stable: they must possess an intrinsic motivation justifying no strictly stronger theory. In a disentangled setting, even if the base and the syntax theory are individually stable, they may be jointly unstable. We contend (...) that this flaw afflicts many proposals discussed in the literature; we defend a new, stable disentangled theory, double second-order arithmetic. (shrink)

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. We 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. We 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. (shrink)

Tanaka proved a powerful generalization of Friedman’s self-embedding theorem that states that given a countable nonstandard model \\) of the subsystem \ of second order arithmetic, and any element m of \, there is a self-embedding j of \\) onto a proper initial segment of itself such that j fixes every predecessor of m. Here we extend Tanaka’s work by establishing the following results for a countable nonstandard model \\ \)of \ and a proper cut \ of \:Theorem A. The (...) following conditions are equivalent: \ is closed under exponentiation. There is a self-embedding j of \\) onto a proper initial segment of itself such that I is the longest initial segment of fixed points of j.Theorem B. The following conditions are equivalent: \ is a strong cut of \ and \ There is a self-embedding j of \\) onto a proper initial segment of itself such that \ is the set of all fixed points of j. (shrink)

We link two concepts from the literature, namely hard sequences for the satisfiability problem sat and so‐called pseudo proof systems proposed for study by Krajíček. Pseudo proof systems are elements of a particular nonstandard model constructed by forcing with random variables. We show that the existence of mad pseudo proof systems is equivalent to the existence of a randomized polynomial time procedure with a highly restrictive use of randomness which produces satisfiable formulas whose satisfying assignments are probably hard to find.

The conservativeness argument poses a dilemma to deflationism about truth, according to which a deflationist theory of truth must be conservative but no adequate theory of truth is conservative. The debate on the conservativeness argument has so far been framed in a specific formal setting, where theories of truth are formulated over arithmetical base theories. I will argue that the appropriate formal setting for evaluating the conservativeness argument is provided not by theories of truth over arithmetic but by those over (...) subject matters ‘richer’ than arithmetic, such as set theory. The move to this new formal setting provides deflationists with better defence and brings a broader perspective to the debate. (shrink)

We show that a miniaturised version of Maclagan's theorem on monomial ideals is equivalent to and classify a phase transition threshold for this theorem. This work highlights the combinatorial nature of Maclagan's theorem.

We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

We introduce a notion ofsyntactical truth predicate(s.t.p.) for the second order arithmeticPA2. An s.t.p. is a setTof closed formulas such that:(i)T(t=u) if and only if the closed first order termstanduare convertible, i.e., have the same value in the standard interpretation(ii)T(A→B) if and only if (T(A) ⇒T(B))(iii)T(∀xA) if and only if (T(A[x←t]) for any closed first order termt)(iv)T(∀X A) if and only if (T(A[X← ∆]) for any closed set definition ∆ = {x∣D(x)}).S.t.p.'s can be seen as a counterpart to Tarski's notion (...) of (model-theoretical)validityand have main model properties. In particular, their existence is equivalent to the existence of anω-model ofPA2, this fact being provable inPA2with arithmetical comprehension only. (shrink)

We show that many principles of first-order arithmetic, previously only known to lie strictly between [Formula: see text]-induction and [Formula: see text]-induction, are equivalent to the well-foundedness of [Formula: see text]. Among these principles are the iteration of partial functions of Hájek and Paris, the bounded monotone enumerations principle by Chong, Slaman, and Yang, the relativized Paris–Harrington principle for pairs, and the totality of the relativized Ackermann–Péter function. With this we show that the well-foundedness of [Formula: see text] is a (...) far more widespread than usually suspected. Further, we investigate the [Formula: see text]-iterated version of the bounded monotone iterations principle, and show that it is equivalent to the well-foundedness of the -height [Formula: see text]-tower [Formula: see text]. (shrink)

A semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church's Thesis and a variant of a modal set theory (...) due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra's generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott's graph model (for the untyped lambda calculus) which witnesses the failure of the stability of non-identity. (shrink)

By a classical result of Kotlarski, Krajewski and Lachlan, pathological satisfaction classes can be constructed for countable, recursively saturated models of Peano arithmetic. In this paper we consider the question of whether the pathology can be eliminated; we ask in effect what generalities involving the notion of truth can be obtained in a deflationary truth theory (a theory of truth which is conservative over its base). It is shown that the answer depends on the notion of pathology we adopt. It (...) turns out in particular that a certain natural closure condition imposed on a satisfaction class—namely, closure of truth under sentential proofs—generates a nonconservative extension of a syntactic base theory (Peano arithmetic). (shrink)

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence (...) relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic. (shrink)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? Once the question has been properly formulated, the answer turns out to be about as elegant as one could want: Adding a theory of truth to a finitely axiomatized theory T is more or less equivalent to a (...) kind of abstract consistency statement. A large part of the interest of the paper lies in the way syntactic theories are 'disentangled' from object theories. (shrink)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? It turns out that, in a wide range of cases, we can get some nice answers to this question, but only if we work in a framework that is somewhat different from those usually employed in discussions of (...) axiomatic theories of truth. These results are then used to address a range of philosophical questions connected with truth, such as what Tarski meant by "essential richness" and the so-called conservativeness argument against deflationism. -/- This draft dates from about 2009, with some significant updates having been made around 2011. Around then, however, I decided that the paper was becoming unmanageable and that I was trying to do too many things in it. I have therefore exploded the paper into several pieces, which will be published separately. These include "Disquotationalism and the Compositional Principles", "The Logical Strength of Compositional Principles", "Consistency and the Theory of Truth", and "What Is Essential Richness?" You should probably read those instead, since this draft remains a bit of a mess. Terminology and notation are inconsistent, and some of the proofs aren't quite right. So, caveat lector. I make it public only because it has been cited in a few places now. (shrink)

The ‘Turing barrier’ is an evocative image for 0′, the degree of the unsolvability of the halting problem for Turing machines—equivalently, of the undecidability of Peano Arithmetic. The ‘barrier’ metaphor conveys the idea that effective computability is impaired by restrictions that could be removed by infinite methods. Assuming that the undecidability of PA is essentially depending on the finite nature of its computational means, decidability would be restored by the ω-rule. Hypercomputation, the hypothetical realization of infinitary machines through relativistic and (...) quantium models, would thus be capable of breaking the Turing barrier. The speculation is unfounded in principle, apart from issues of physical realizability: The point is that the ω-rule does not cope with all objects entailing the undecidability of PA. As long as the system is consistent, the computational boundary can be established by paradox-like constructions, such as Gödel-Rosser’s and Yablo’s, which are refractory to infinite induction, and do stand as the barrier’s buttresses. (shrink)

Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...) and principles of mathematical and transfinite induction as well as the status of the latter with respect to various foundational characterizations of number theory. (shrink)

The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.

In this paper, we investigate the phenomenon ofspeed-upin the context of theories of truth. We focus on axiomatic theories of truth extending Peano arithmetic. We are particularly interested on whether conservative extensions of PA have speed-up and on how this relates to a deflationist account. We show that disquotational theories have no significant speed-up, in contrast to some compositional theories, and we briefly assess the philosophical implications of these results.

It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition. The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's ‘predicative arithmetic’ program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly (...) as a restricted form of logicism. (shrink)

We transform the proof of the second incompleteness theorem given in [3] to a proof-theoretic version, avoiding the use of the arithmetized completeness theorem. We give also new proofs of old results: The Arithmetical Hierarchy Theorem and Tarski's Theorem on undefinability of truth; the proofs in which the construction of a sentence by means of diagonalization lemma is not needed.

Continuing the earlier research from [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001] we show that for the price of multiplying the number of parts by 3 we may construct partitions all of whose homogeneous sets are much smaller than in [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 4981–5001]. We also show that the Paris–Harrington independent statement remains unprovable if the number of colors is (...) restricted to 2, in fact, the statement is unprovable in IΣb. Other results concern some lower bounds for partitions of pairs. (shrink)

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy (...) of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity, while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic. Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the~$\omega$-rule or to the non-computability and non-continuity of probability assignments. Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments, such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated. In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar. (shrink)

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

We study the theories I∇n, L∇n and overspill principles for ∇n formulas. We show that IEn ⇒ L∇n ⇒ I∇n, but we do not know if I∇n L∇n. We introduce a new scheme, the growth scheme Crγ, and we prove that L∇n ⇒ Cr∇n⇒ I∇n. Also, we analyse the utility of bounded collection axioms for the study of the above theories.

In this paper we will state and prove some comparative theorems concerning PRA and IΣ1. We shall provide a characterization of IΣ1 in terms of PRA and iterations of a class of functions. In particular, we prove that for this class of functions the difference between IΣ1 and PRA is exactly that, where PRA is closed under iterations of these functions, IΣ1 is moreover provably closed under iteration. We will formulate a sufficient condition for a model of PRA to be (...) a model of IΣ1. This condition is used to give a model-theoretic proof of Parsons’ theorem, that is, IΣ1 is Π2-conservative over PRA. We shall also give a purely syntactical proof of Parsons’ theorem. Finally, we show that IΣ1 proves the consistency of PRA on a definable IΣ1-cut. This implies that proofs in IΣ1 can have non-elementary speed up over proofs in PRA. (shrink)

In this paper we naturally define when a theory has bounded quantifier elimination, or is bounded model complete. We give several equivalent conditions for a theory to have each of these properties. These results provide simple proofs for some known results in the model theory of the bounded arithmetic theories like CPV and PV1. We use the mentioned results to obtain some independence results in the context of intuitionistic bounded arithmetic. We show that, if the intuitionistic theory of polynomial induction (...) on strict $\Pi_2^{b}$ formulas proves decidability of $\Sigma_1^b$ formulas, then P=NP. We also prove that, if the mentioned intuitionistic theory proves ${\rm LMIN}(\Sigma_{1}^{b})$ , then P=NP. (shrink)

We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

In this paper we study with proof-theoretic methods the function(al) s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functional provably recursive from $RCA_0 + COH + \Pi _1^0 - CP$ are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that $WKL_0 + \Pi _1^0 - (...) CP + COH$ is $\Pi _1^0 $ -conservative over PRA. Recent work of the first author showed that $\Pi _1^0 + CP + COH$ is equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey's theorem for pairs and two colors $\left( {RT_2^2 } \right)$ we obtain the upper bounded that the type 2 functionals provable recursive relative to $RCA_0 + \sum\nolimits_2^0 {IA + RT_2^2 } $ are in T₁. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T₁-bounds from proofs that use RT2..... Moreover, this yields a new proof of the fact that $WKL_0 + \sum\nolimits_2^0 {IA} + RT_2^2 $ is $\Pi _1^0 $ -conservative over $RCA_0 + \sum\nolimits_2^0 { - IA} $ . The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using $\Pi _1^0 $ -comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for $RT_2^2 $. (shrink)

Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ? -rule in his 1931 paper ?Die Grundlegung der elementaren Zahlenlehre?. The main question we discuss here is whether the finitist (meta-)mathematician would (...) be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ?Die Widerspruchsfreiheit der reinen Zahlentheorie? (1936). As to Hilbert and Bernays 1939, we end on a ?critical? note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original finitism have fallen victim to sheer proof-theoretic pragmatism. (shrink)

The paper presents a particular example of a formula which is a standard tautology of Łukasiewicz but not its general tautology; an example of a model in which the formula is not true is explicitly constructed. Analogous example of a formula and its model is given for product logic.

Definitional and axiomatic theories of truth -- Objects of truth -- Tarski -- Truth and set theory -- Technical preliminaries -- Comparing axiomatic theories of truth -- Disquotation -- Classical compositional truth -- Hierarchies -- Typed and type-free theories of truth -- Reasons against typing -- Axioms and rules -- Axioms for type-free truth -- Classical symmetric truth -- Kripke-Feferman -- Axiomatizing Kripke's theory in partial logic -- Grounded truth -- Alternative evaluation schemata -- Disquotation -- Classical logic -- Deflationism (...) -- Reflection -- Ontological reduction -- Applying theories of truth. (shrink)

We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

Storrs McCall continues the tradition of Lucas and Penrose in an attempt to refute mechanism by appealing to Gödel’s incompleteness theorem. That is, McCall argues that Gödel’s theorem “reveals a sharp dividing line between human and machine thinking”. According to McCall, “[h]uman beings are familiar with the distinction between truth and theoremhood, but Turing machines cannot look beyond their own output”. However, although McCall’s argumentation is slightly more sophisticated than the earlier Gödelian anti-mechanist arguments, in the end it fails badly, (...) as it is at odds with the logical facts. (shrink)

We show that we can interpret concatenation theories in arithmetical theories without coding sequences by identifying binary strings with \(2\times 2\) matrices with determinant 1.