Concepts of L1 space, integrable functions and integrals are formalized in weak subsystems of second order arithmetic. They are discussed especially in relation with the combinatorial principle WWKL (weak-weak König's lemma and arithmetical comprehension. Lebesgue dominated convergence theorem is proved to be equivalent to arithmetical comprehension. A weak version of Lebesgue monotone convergence theorem is proved to be equivalent to weak-weak König's lemma.

The dominated convergence theorem implies that if is a sequence of functions on a probability space taking values in the interval [0, 1], and converges pointwise a.e., then converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of (...) the sequence and the underlying space. We prove a slight strengthening of Tao’s theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorov’s theorem, and introduce a new mode of convergence related to these notions. (shrink)

We analyze the pointwise convergence of a sequence of computable elements of L1 in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA0, each is equivalent to the assertion that every Gδ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs lemma relativized to the Turing jump of any set. It is also (...) equivalent to the conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ2 collection. (shrink)

The theorem of convergence of opinions is an important theorem in the subjective theory of probability.It demonstrates that the subjectivity of a prior probability will be substituted with the objectivity of a posterior probability as evidences increase.The theorem of convergence of opinions is regarded as the dynamic principle of rationality concerning the subjective probability,and therefore is used to resolve Hume's problem,i.e.,the problem of inductive rationality.However,Hacking convincingly argues that the theorem of convergence of opinions (...) is not about the convergence of a posterior probability Pre(h),but about the convergence of a conditional probability Pr(h/e).The subjective theory of probability implies a tacit acceptance of an equation that is Pre(h)=Pr(h/e),usually called "the rule of conditionalisation".Thus,the problem of rationality of induction is converted into one concerning the rule of conditionalisation.In this paper a new principle of rationality(the principle of minimum number of initial probabilities)is put forward,which,in combination with the concept of "local rationality",offers a justification for the rule of conditionalisation. (shrink)

The Bayesian theorem on convergence to the truth states that a rational inquirer believes with certainty that her degrees of belief capture the truth about a large swath of hypotheses with increasing evidence. This result has been criticized as showcasing a problematic kind of epistemic immodesty when applied to infinite hypotheses that can never be approximated by finite evidence. The central point at issue—that certain hypotheses may forever be beyond the reach of a finite investigation no matter how (...) large one’s reservoir of evidence—cannot be captured adequately within standard probability theory. As an alternative, I propose a nonstandard probabilistic framework that, by using arbitrarily small and large numbers, makes room for the type of fine-grained conceptual distinctions appropriate for a deeper analysis of convergence to the truth. This framework allows for the right kind of modesty about attaining truth in the limit. (shrink)

It is shown how martingale convergence theorems apply to coherent belief change in radical probabilist epistemology.

We generalize the Bolzano-Weierstrass theorem on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family.

Bayesians since Savage (1972) have appealed to asymptotic results to counter charges of excessive subjectivity. Their claim is that objectionable differences in prior probability judgments will vanish as agents learn from evidence, and individual agents will converge to the truth. Glymour (1980), Earman (1992) and others have voiced the complaint that the theorems used to support these claims tell us, not how probabilities updated on evidence will actually}behave in the limit, but merely how Bayesian agents believe they will behave, suggesting (...) that the theorems are too weak to underwrite notions of scientific objectivity and intersubjective agreement. I investigate, in a very general framework, the conditions under which updated probabilities actually converge to a settled opinion and the conditions under which the updated probabilities of two agents actually converge to the same settled opinion. I call this mode of convergence deterministic, and derive results that extend those found in Huttegger (2015b). The results here lead to a simple characterization of deterministic convergence for Bayesian learners and give rise to an interesting argument for what I call strong regularity, the view that probabilities of non-empty events should be bounded away from zero. (shrink)

We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integrals such as the Bounded Convergence Theorem, the Dominated Convergence Theorem, and the Second Mean Value Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integrable Fine-computable function form a Euclidian computable sequence of reals which converges effectively to zero. This property of convergence is the effectivization of the Walsh-Riemann-Lebesgue Theorem. The (...) article is closed with the effective version of Dirichlet's test. (shrink)

Must probabilities be countably additive? On the one hand, arguably, requiring countable additivity is too restrictive. As de Finetti pointed out, there are situations in which it is reasonable to use merely finitely additive probabilities. On the other hand, countable additivity is fruitful. It can be used to prove deep mathematical theorems that do not follow from finite additivity alone. One of the most philosophically important examples of such a result is the Bayesian convergence to the truth theorem, (...) which says that conditional probabilities converge to 1 for true hypotheses and to 0 for false hypotheses. In view of the long-standing debate about countable additivity, it is natural to ask in what circumstances finitely additive theories deliver the same results as the countably additive theory. This paper addresses that question and initiates a systematic study of convergence to the truth in a finitely additive setting. There is also some discussion of how the formal results can be applied to ongoing debates in epistemology and the philosophy of science. (shrink)

New results in ergodic theory show that averages of repeated measurements will typically diverge with probability one if there are random errors in the measurement of time. Since mean-square convergence of the averages is not so susceptible to these anomalies, we are led again to compare the mean and pointwise ergodic theorems and to reconsider efforts to determine properties of a stochastic process from the study of a generic sample path. There are also implications for models of time and (...) the interaction between observer and observable. (shrink)

The paper deals with proximal convergence and Leader's theorem, in the constructive theory of uniform apartness spaces.

We refine results of Gannon [6, Theorem 4.7] and Simon [22, Lemma 2.8] on convergence of Morley sequences. We then introduce the notion of eventual $NIP$, as a property of a model, and prove a variant of [15, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [17] on the model-theoretic meaning of Grothendieck’s double limit theorem.

Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the (...) fan theorem, weak König's lemma, in the original classical setting of reverse mathematics started by Friedman and Simpson. (shrink)

For a free filter F on $$\omega $$ ω, endow the space $$N_F=\omega \cup \{p_F\}$$ N F = ω ∪ { p F }, where $$p_F\not \in \omega $$ p F ∉ ω, with the topology in which every element of $$\omega $$ ω is isolated whereas all open neighborhoods of $$p_F$$ p F are of the form $$A\cup \{p_F\}$$ A ∪ { p F } for $$A\in F$$ A ∈ F. Spaces of the form $$N_F$$ N F constitute the (...) class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space $$N_F$$ N F carries a sequence $$\langle \mu _n:n\in \omega \rangle $$ ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that $$\mu _n(f)\rightarrow 0$$ μ n ( f ) → 0 for every bounded continuous real-valued function f on $$N_F$$ N F if and only if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, that is, the dual ideal $$F^*$$ F ∗ is Katětov below the asymptotic density ideal $${\mathcal {Z}}$$ Z. Consequently, we get that if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z, then: (1) if X is a Tychonoff space and $$N_F$$ N F is homeomorphic to a subspace of X, then the space $$C_p^*(X)$$ C p ∗ ( X ) of bounded continuous real-valued functions on X contains a complemented copy of the space $$c_0$$ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and $$N_F$$ N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck. (shrink)

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ℳ of subsets of E, of nets of vectors in K. We then apply this result to various class ℳ and setsK in the context of E-valued Bochner integrable functions defined (...) on a finite measure space. (shrink)

Inspired by a construction of the Tsirelson space , we prove a general theorem for omitting countably many positive formulas in normed spaces. This theorem can be used in functional analysis as a tool to guarantee the existence of complicated normed spaces without having to construct them. The proof of this result is based on the notion of approximate truth and on a study of the relationship between approximate truth and convergence in normed spaces. We illustrate the (...) power of this result with an application to functional analysis. (shrink)

Some conditions are given to ensure that for a jump homogeneous Markov process $\{X(t),t\ge 0\}$ the law of the integral functional of the process $T^{-1/2} \int^T_0\varphi(X(t))dt$ converges to the normal law $N(0,\sigma^2)$ as $T\to \infty$, where $\varphi$ is a mapping from the state space $E$ into $\bbfR$.

This paper utilizes a logical correspondence theorem (which has been proved elsewhere) for the justification of weak conceptions of scientific realism and convergence to truth which do not presuppose Putnam's no-miracles-argument (NMA). After presenting arguments against the reliability of the unrestricted NMA in Sect. 1, the correspondence theorem is explained in Sect. 2. In Sect. 3, historical illustrations of the correspondence theorem are given, and its ontological consequences are worked out. Based on the transitivity of the (...) concept of correspondence, a correspondence-based notion of convergence to truth is developed in Sect. 4. In the final Sect. 5 it is argued that the correspondence theorem together with the assumption of ' minimal realism' yields a justification of a weak version of scientific realism, which is then compared to metaphysical realism and to instrumentalism. (shrink)

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ℳ︁ of subsets of E, of nets of vectors in K. We then apply this result to various class ℳ︁ and setsK in the context of E‐valued Bochner integrable functions defined (...) on a finite measure space. (shrink)

One might think that if the majority of virtue signallers judge that a proposition is true, then there is significant evidence for the truth of that proposition. Given the Condorcet Jury Theorem, individual virtue signallers need not be very reliable for the majority judgment to be very likely to be correct. Thus, even people who are skeptical of the judgments of individual virtue signallers should think that if a majority of them judge that a proposition is true, then that (...) provides significant evidence that the proposition is true. We argue that this is mistaken. Various empirical studies converge on the following point: humans are very conformist in the contexts in which virtue signalling occurs. And stereotypical virtue signallers are even more conformist in such contexts. So we should be skeptical of the claim that virtue signallers are sufficiently independent for the Condorcet Jury Theorem to apply. We do not seek to decisively rule out the relevant application of the Condorcet Jury Theorem. But we do show that careful consideration of the available evidence should make us very skeptical of that application. Consequently, a defense of virtue signalling would need to engage with these findings and show that despite our strong tendencies for conformism, our judgements are sufficiently independent for the Condorcet Jury Theorem to apply. This suggests new directions for the debate about the epistemology of virtue signalling. (shrink)

Tacking by conjunction is a well-known problem for Bayesian confirmation theory. In the first section, disadvantages of existing Bayesian solution proposals to this problem are pointed out and an alternative solution proposal is presented: that of genuine confirmation. In the second section, the notion of GC is briefly recapitulated and three versions of GC are distinguished: full GC, partial GC and quantitative GC. In the third section, the application of partial GC to pure post-facto speculations is explained. In the fourth (...) section it is demonstrated that full GC is a necessary condition for Bayesian convergence to certainty based on the accumulation of conditionally independent pieces of evidence. It is found that whenever a hypothesis is equivalent to a disjunction of more fine-grained hypotheses conveying different probabilities to the evidence, then conditional independence of the evidence fails. This failure occurs typically for unspecific negations of hypotheses. A refined version of the convergence to certainty theorem that overcomes this difficulty is developed in the final section. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.

The history of uniform convergence is typically focused on the contributions of Cauchy, Seidel, Stokes, and Björling. While the mathematical contributions of these individuals to the concept of uniform convergence have been much discussed, Weierstrass is considered to be the actual inventor of today’s concept. This view is often based on his well-known article from 1841. However, Weierstrass’s works on a rigorous foundation of analytic and elliptic functions date primarily from his lecture courses at the University of Berlin (...) up to the mid-1880s. For the history of uniform convergence, these lectures open up an independent branch of development that is disconnected from the approaches of the previously mentioned authors; to my knowledge, Weierstraß never explicitly referred to Cauchy’s continuity theorem (1821 or 1853) or to Seidel’s or Stokes’s contributions (1847). In the present article, Weierstrass’s contributions to the development of uniform convergence will be discussed, mainly based on lecture notes made by Weierstrass’s students between 1861 and the mid-1880s. The emphasis is on the notation and the mathematical rigor of the introductions to the concept, leading to the proposal to re-date the famous 1841 article and thus Weierstrass’s first introduction of uniform convergence. (shrink)

We investigate the classes of ideals for which the Egoroff’s theorem or the generalized Egoroff’s theorem holds between ideal versions of pointwise and uniform convergences. The paper is motivated by considerations of Korch :269–282, 2017).

de Finetti's representation theorem of exchangeable probabilities as unique mixtures of Bernoullian probabilities is a special case of a result known as the ergodic decomposition theorem. It says that stationary probability measures are unique mixtures of ergodic measures. Stationarity implies convergence of relative frequencies, and ergodicity the uniqueness of limits. Ergodicity therefore captures exactly the idea of objective probability as a limit of relative frequency (up to a set of measure zero), without the unnecessary restriction to probabilistically (...) independent events as in de Finetti's theorem. The ergodic decomposition has in some applications to dynamical systems a physical content, and de Finetti's reductionist interpretation of his result is not adequate in these cases. (shrink)

We present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem, which presents a simple condition under which the converse holds. Our approach is inspired by proof theory, and in particular Gödel's functional interpretation, which we use to establish quantitative versions of both of these results.

This item was published as 'Appendix 3: An Implication of the k-option Condorcet jury mechanism for the probability of cycles' in List and Goodin (2001) http://eprints.lse.ac.uk/705/. Standard results suggest that the probability of cycles should increase as the number of options increases and also as the number of individuals increases. These results are, however, premised on a so-called "impartial culture" assumption: any logically possible preference ordering is assumed to be as likely to be held by an individual as any other. (...) The present chapter shows, in the three-option case, that given suitably systematic, however slight, deviations from an impartial culture situation, the probability of a cycle converges either to zero (more typically) or to one (less typically) as the number of individuals increases. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

LS Penrose was the first to propose a measure of voting power (which later came to be known as ‘the [absolute] Banzhaf index’). His limit theorem – which is implicit in Penrose (1952) and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely while the quota is pegged at half the total weight, then – under certain conditions – the ratio between the voting powers (as measured (...) by him) of any two voters converges to the ratio between their weights. We conjecture that the theorem holds, under rather general conditions, for large classes of variously defined weighted voting games, other values of the quota, and other measures of voting power. We provide proofs for some special cases. (shrink)

This article examines the benefits of incorporating religious reflection into the psychology of religion and vice versa. By applying Bayes’ theorem, we discover that scientists and theologians can collaborate without sharing prior beliefs. Instead, rationality requires updating our beliefs before data collection in response to the degree of surprise generated by the data. Moreover, although people who start with different beliefs may become more aligned after data collection, rationality does not entail a convergence to identical beliefs. To illustrate (...) the potential for growth in understanding from greater collaboration between theologians and scientists, I examine a longitudinal investigation of religion after a natural disaster. This case study illustrates how conversations between theological and psychological perspectives on religion can lead to a more comprehensive understanding of virtue cultivation, benefiting both science and theology. (shrink)

L S Penrose’s Limit Theorem – which is implicit in Penrose [7, p. 72] and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely and the relative quota is pegged, then – under certain conditions – the ratio between the voting powers of any two voters converges to the ratio between their weights. Lindner and Machover [4] prove some special cases of Penrose’s Limit Theorem. They (...) give a simple counter-example showing that the theorem does not hold in general even under the conditions assumed by Penrose; but they conjecture, in effect, that under rather general conditions it holds ‘almost always’ – that is with probability 1 – for large classes of weighted voting games, for various values of the quota, and with respect to several measures of voting power. We use simulation to test this conjecture. It is corroborated with respect to the Penrose–Banzhaf index for a quota of 50% but not for other values; with respect to the Shapley–Shubik index the conjecture is corroborated for all values of the quota. (shrink)

In this paper, we introduce the notion of S ∗ p -partial metric spaces which is a generalization of S-metric spaces and partial-metric spaces. Also, we give some of the topological properties that are important in knowing the convergence of the sequences and Cauchy sequence. Finally, we study a new common fixed point theorems in this spaces.

The notions of permutable and weak-permutable convergence of a series|$\sum _{n=1}^{\infty }a_{n}$|of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD-|$\mathbb {N}$|implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation (...) theorem for series holds but BD-|$\mathbb{N}$|does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD-|$\mathbb {N}$|. We show that this is the case when the property is weak-permutable convergence. (shrink)

Let T be a positive L₁-L∞ contraction. We prove that the following statements are equivalent in constructive mathematics. (1) The projection in L₂ on the space of invariant functions exists: (2) The sequence (Tⁿ)n∈N Cesáro-converges in the L₂ norm: (3) The sequence (Tⁿ)n∈N Cesáro-converges almost everywhere. Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem. As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This (...) answers a question posed by Bishop. (shrink)

Say that an agent is "epistemically humble" if she is less than certain that her opinions will converge to the truth, given an appropriate stream of evidence. Is such humility rationally permissible? According to the orgulity argument : the answer is "yes" but long-run convergence-to-the-truth theorems force Bayesians to answer "no." That argument has no force against Bayesians who reject countable additivity as a requirement of rationality. Such Bayesians are free to count even extreme humility as rationally permissible.

Given a convergence theorem in analysis, under very general conditions a model-theoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory.

The justificatory force of empirical reasoning always depends upon the existence of some synthetic, a priori justification. The reasoner must begin with justified, substantive constraints on both the prior probability of the conclusion and certain conditional probabilities; otherwise, all possible degrees of belief in the conclusion are left open given the premises. Such constraints cannot in general be empirically justified, on pain of infinite regress. Nor does subjective Bayesianism offer a way out for the empiricist. Despite often-cited convergence theorems, (...) subjective Bayesians cannot hold that any empirical hypothesis is ever objectively justified in the relevant sense. Rationalism is thus the only alternative to an implausible skepticism. (shrink)

A famous mathematical theorem says that the sum of an infinite series of numbers can depend on the order in which those numbers occur. Suppose we interpret the numbers in such a series as representing instances of some physical quantity, such as the weights of a collection of items. The mathematics seems to lead to the result that the weight of a collection of items can depend on the order in which those items are weighed. But that is very (...) hard to believe. A puzzle then arises: How do we interpret the metaphysical significance of this mathematical theorem? I first argue that prior solutions to the puzzle lead to implausible consequences. Then I develop my own solution, where the basic idea is that the weight of a collection of items is equal to the limit of the weights of its finite subcollections contained within ever-expanding regions of space. I show how my solution is intuitively plausible and philosophically motivated, how it reveals an underexplored line of metaphysical inquiry about quantities and locations, and how it elucidates some classic puzzles concerning supertasks. (shrink)

Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$. We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from $[0,1]$ to ${\mathbb N}$. Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, (...) while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$. Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$ -function, find reals in the unit interval that map to the same natural number. (shrink)

This paper studies a simple dynamical system of stock price fluctuation time series based on the rule of stock market. When the stock price fluctuation system is disturbed by external excitations, the system exhibits obviously chaotic phenomena, and its basic dynamic properties are analyzed. At the same time, a new fixed-time convergence theorem is proposed for achieving fixed-time control of stock price fluctuation system. Finally, the effectiveness of the method is verified by numerical simulation.

Chalmers has argued that Bayesianism supports the existence of a priori truths, since it entails that scrutability conditionals are not rationally revisable. However, as we argue, Chalmers's arguments leave open that every proposition is rationally deniable, which would be devastating for large parts of his philosophical program. We suggest that Chalmers should appeal to well-known convergence theorems to argue that ideally rational subjects converge on the truth of scrutability conditionals. However, our discussion reveals that showing that these theorems apply (...) in effect requires assuming scrutability. Consequently, Bayesianism doesn't conflict with Chalmers's scrutability framework, but it doesn't support it, either. (shrink)

Integration within constructive, especially intuitionistic mathematics in the sense of L. E. J. Brouwer, slightly differs from formal integration theories: Some classical results, especially Lebesgue's dominated convergence theorem, have tobe substituted by appropriate alternatives. Although there exist sophisticated, but rather laborious proposals, e.g. by E. Bishop and D. S. Bridges , the reference to partitions and the Riemann-integral, also with regard to the results obtained by R. Henstock and J. Kurzweil , seems to give a better direction. Especially, (...) convergence theorems can be proved by introducing the concept of “equi-integrability”. (shrink)

A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter a, which is obtained using the collocation and least-squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in this (...) study. In five examples, the numerical results of the present method have been compared with other methods. For the first time, a -polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed. (shrink)

In recent years, proof theoretic transformations that are based on extensions of monotone forms of Gödel’s famous functional interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for bounded closed and (...) convex subsets of an abstract Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω∗ definable from bar recursion of lowest type. While a case study on the analysis of strong convergence results that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an explicit bound on a metastable version of this theorem that is primitive recursive relative to Ω∗.In the current paper we for the first time give the construction of Ω∗ . As a corollary to the fine analysis of the use of bar recursion in this construction we obtain that Ω∗ elevates arguments in Tn at most to resulting functionals in Tn+2 . In particular, one can conclude from this that our bound on Baillon’s theorem is at least definable in T4. (shrink)