K denotes both the knowledge predicate satisfied by every currently known theorem and the finite set of all currently known theorems. The set K is time-dependent, publicly available, and contains theorems both from formal and constructive mathematics. Any theorem of any mathematician from past or present forever belongs to K. Mathematical statements with known constructive proofs exist in K separately and form the set K_c⊆K. We assume that mathematical sets are atemporal entities. They exist formally in ZFC theory although their (...) properties can be time-dependent (when they depend on K) or informal. The true statement "K is non-empty" is outside K forever because K is not a formal set. Paul Cohen proved in 1963 that the equality 2^{\aleph_0}=\aleph_1 is independent of ZFC. Before 1963, the statement "There is a known constructively defined integer n⩾1 such that 2^{\aleph_0}=\aleph_n" was outside K. Since 1963, this statement is outside K forever. The true statement "There exists a set X⊆{1,...,49} such that card(X)=6 and X never occurred as the winning six numbers in the Polish Lotto lottery" refers to the current non-mathematical knowledge and is outside K forever. In Martin-Löf's terminology, every currently known theorem is actually true whereas every theorem (known or unknown) is potentially true. Actual truth is knowledge dependent and tensed. Potential truth is knowledge independent and tenseless. Algorithms always terminate. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivially true statements on decidable sets X⊆N that belong to constructive and informal mathematics and refer to the current mathematical knowledge on X. For every such statement, its truth concerning the current mathematical knowledge is implied by a true statement of the form: (the conjunction of i conditions of the form α∈K)∧(the conjunction of j conditions of the form β∉K)∧(the conjunction of k conditions of the form γ∈K_c)∧(the conjunction of l conditions of the form δ∉K_c), where i,j,k,l∈N and α,β,γ,δ are mathematical statements. In constructive mathematics and the traditional Brouwerian intuitionism, the truth of a mathematical statement means that we know a constructive proof. Therefore, the truth of a mathematical statement depends on time, where the statement is formally stated in the classical mathematics without the predicate K. In this article, mathematical statements on decidable sets X⊆N refer to time because they refer to the current mathematical knowledge on X. They cannot be formally stated in the classical mathematics without the predicate K and their logical values may change in time. For a set X⊆N whose infiniteness is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(X)=ω" is outside K. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove the assumption of the above statement. We prove that the set X={n∈N: the interval [-1,n] contains more than 29.5+(11!/(3n+1))∙sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. We present a table that shows satisfiable conjunctions of the form #(Condition1)∧(Condition 2)∧#(Condition 3)∧(Condition 4)∧#(Condition 5), where # denotes the negation ¬ or the absence of any symbol. We prove that there exists a naturally defined set C⊆N which satisfies the following conditions (6)-(11). (6) A known and simple algorithm for every k∈N decides whether or not k∈C. (7) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(C)=ω" is outside K. (8) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(N\C)=ω" is outside K. (9) It is conjectured, though so far unproven, that C is infinite. (10) For every known algorithm A with no input, the statement "A returns an integer n satisfying card(C)<ω ⇒ C⊆(-∞,n]" is outside K. (11) For every known algorithm A with no input, the statement "A returns an integer m satisfying card(N\C)<ω ⇒ N\C⊆(-∞,m]" is outside K. The set C={k∈N: 2^{2^k}+1 is composite} is naturally defined and satisfies conditions (6)-(11). (shrink)

Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆[2,f(7)]. Let B denote the system of equations: {x_j!=x_k: i,k∈{1,...,9}}∪{x_i⋅x_j=x_k: i,j,k∈{1,...,9}}. The system of equations {x_1!=x_1, x_1 \cdot x_1=x_2, x_2!=x_3, x_3!=x_4, x_4!=x_5, x_5!=x_6, x_6!=x_7, x_7!=x_8, x_8!=x_9} has exactly two solutions in positive integers x_1,...,x_9, namely (1,...,1) and (f(1),...,f(9)). No known system S⊆B with a finite (...) number of solutions in positive integers x_1,...,x_9 has a solution (x_1,...,x_9)∈(N\{0})^9 satisfying max(x_1,...,x_9)>f(9). For every known system S⊆B, if the finiteness/infiniteness of the set {(x_1,...,x_9)∈(N\{0})^9: (x_1,...,x_9) solves S} is unknown, then the statement ∃ x_1,...,x_9∈N\{0} ((x_1,...,x_9) solves S)∧(max(x_1,...,x_9)>f(9)) remains unproven. Let Λ denote the statement: if the system of equations {x_2!=x_3, x_3!=x_4, x_5!=x_6, x_8!=x_9, x_1 \cdot x_1=x_2, x_3 \cdot x_5=x_6, x_4 \cdot x_8=x_9, x_5 \cdot x_7=x_8} has at most finitely many solutions in positive integers x_1,...,x_9, then each such solution (x_1,...,x_9) satisfies x_1,...,x_9≤f(9). The statement Λ is equivalent to the statement Φ. It heuristically justifies the statement Φ . This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. Algorithms always terminate. We explain the distinction between existing algorithms (i.e. algorithms whose existence is provable in ZFC) and known algorithms (i.e. algorithms whose definition is constructive and currently known). Assuming that the infiniteness of a set X⊆N is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. *** (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) No known algorithm with no input returns the logical value of the statement card(X)=ω. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. *** Conditions (2)-(5) hold for X=P(n^2+1). The statement Φ implies Condition (1) for X=P(n^2+1). The set X={n∈N: the interval [-1,n] contains more than 29.5+\frac{11!}{3n+1}⋅sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. 501893∈X. Condition (1) holds with n=501893. card(X∩[0,501893])=159827. X∩[501894,∞)= {n∈N: the interval [-1,n] contains at least 30 primes of the form k!+1}. We present a table that shows satisfiable conjunctions of the form #(Condition 1) ∧ (Condition 2) ∧ #(Condition 3) ∧ (Condition 4) ∧ #(Condition 5), where # denotes the negation ¬ or the absence of any symbol. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove this assumption. (shrink)

Let G be an additive subgroup of ℂ, let Wn = {xi = 1, xi + xj = xk: i, j, k ∈ {1, …, n }}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk: i, j, k ∈ {1, …, n }}. We discuss two conjectures. If a system S ⊆ En is consistent over ℝ, then S has a real solution which consists of numbers whose absolute values belong to (...) [0, 22n –2]. If a system S ⊆ Wn is consistent over G, then S has a solution ∈ n in which |xj| ≤ 2n –1 for each j. (shrink)

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smoryński's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether (...) or not the equation has a solution in R. We prove that a positive solution to H_{10}(R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R \subseteq Q such that there exist computable functions \tau_1,\tau_2:N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). This implication for R=N guarantees that Smoryński's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R=Q. (shrink)

The height of a rational number p/q is defined by max(|p|,|q|) provided p/q is written in lowest terms. The height of a rational tuple (x_1,...,x_n) is defined as the maximum of n and the heights of the numbers x_1,...,x_n. Let h:\bigcup_{n=1}^\infty Q^n \to N\{0} denote the height function. We conjecture that \forall x_1,...,x_n \in Q \exists y_1,...,y_n \in Q (n=1 ==> h(y_1,...,y_n)=1) \wedge (n \geq 2 ==> h(y_1,...,y_n) \leq 2^(2^(n-2))) \wedge \forall i,j,k \in {1,...,n} ((x_i+1=x_k ==> y_i+1=y_k) \wedge (x_i \cdot (...) x_j=x_k ==> y_i \cdot y_j=y_k)). We prove that the conjecture implies that there is an algorithm which decides whether or not a Diophantine equation has a rational solution. (shrink)

Figner, Mackinlay, Wilkening, and Weber (2009) developed the Columbia Card Task (CCT) to measure risk-taking attitudes. This tool consists of two versions: in the COLD version the decision maker needs to state in advance how many cards (out of 32) they want to turn over (so called static risk taking), in the HOT version they have the possibility of turning over all 32 cards one-by-one until they decide to finish (dynamic risk taking). We argue that the HOT version confounds an (...) individual’s willingness to accept risk with their beliefs in trend continuation vs. trend reversal in a prognostic task. In two experimental studies we show that people believing in trend continuation (momentum subjects) turn over more cards than those believing in trend reversal (contrarians) in the HOT version of the task. However, this is not the case in the COLD version. Thus, we provide evidence that, when considered as a dynamic risk propensity measure, the number of turned over cards in the HOT version of the CCT is a contaminated measure and reflects two phenomena: (1) risk preference and (2) the decision-maker’s belief in trend continuation. We speculate that other dynamic risk taking measures can also be biased by a momentum strategy. (shrink)

The strength of emotions in moral judgment and decision-making under risk The focus of this paper is the role of emotions in judgments and choices associated with moral issues. Study 1 shows that depending on the strength of emotions when making a moral decision, people become sensitive to the severity and the probability of harm that their decisions can bring to others. A possible interpretation is that depending on the strength of emotions, people in their moral judgments choose to be (...) either utilitarian or deontologist. In Study 2, following the priority heuristic model, we found that in situations in which the violation of moral norms does not evoke strong negative emotions, people are sensitive to quantitative risk parameters, and the decision-making process requires a relatively longer time. In moral situations in which a violation of the moral norm evokes strong emotions, decision-making is based on arguments other than quantitative risk parameters, and the process takes a shorter time. (shrink)

Several years ago, Cohen, Dearnaley, and Hansel [1] demonstrated that under the influence of alcohol drivers became more risk prone, although their risk perception remained unchanged. Research shows that ambiguity aversion is to some extent positively correlated with risk aversion, though not very highly [2]. The question addressed by the present research is whether alcohol reduces ambiguity aversion. Our research was conducted in a natural setting (a restaurant bar), where customers with differing levels of alcohol intoxication were offered a choice (...) between a risky and an ambiguous lottery. We found that alcohol reduced ambiguity aversion and that the effect occurred in men but not women. We interpret these findings in terms of the risk-as-value hypothesis, according to which, people in Western culture tend to value risk, and suggest that alcohol consumption triggers adherence to socially and culturally valued patterns of conduct different for men and women. (shrink)

Data from surveys indicate that people, in general, do not trust others. On the other hand, in one-shot trust games, where the player decides whether to send money to an anonymous partner, the actual rate of trust is relatively high. In two experiments, we showed that although reciprocity expectations and profit maximization matter, they are not decisive for trusting behaviour. Crucial factors that motivate behaviour in trust games seem to be altruism and a type of moral obligation related to a (...) social norm encouraging cooperative behaviour. Finally, we were able to divide participants into specific profiles based on amount of money transferred to the partner, altruistic motivation, and belief in partners’ trustworthiness. This shows that the trust game is differently perceived and interpreted by different participants. (shrink)

Using randomly generated sequences of binary events we asked participants to make predictions about the next event. It turned out that while predicting uncertain events, people do not behave unsystematically. Our research identifies four types of relatively consistent strategies for predicting uncertain binary events: a strategy immune to short-run sequential dependencies consisting of the persistent prediction of long-run majority events, hereafter called the long-run momentum strategy ; a strategy immune to short-run sequential dependencies consisting of the persistent prediction of long-run (...) minority events, called the long-run contrarian strategy ; a strategy sensitive to short-run sequential dependencies consisting of the prediction of short-run majority events, called the short-run momentum strategy ; and a strategy sensitive to short-run sequential dependencies consisting of the prediction of short-run minority events, called the short-run contrarian strategy . When the character of events remains unknown, the most common strategy is the short-run momentum strategy. With the increase of a perceived randomness of the situation, people tend more often to use the short-run contrarian strategy. People differ in their general beliefs about the continuation or reversal of a trend in various natural and social processes. Trend believers, when facing sequences of binary events commonly perceived as random, tend to use momentum strategies, whereas those who believe in the trend's reversal tend to use contrarian strategies. (shrink)