If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable (...) without using the axiom of choice. (shrink)

We investigate whether classical combinatorial theorems are provable in ZF. Some statements are not provable in ZF, but they are equivalent within ZF. For example, the following statements (i)–(iii) are equivalent: $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1, $${\omega }_1\rightarrow ({\omega }_1,{\omega }+1)^2$$ ω 1 → ( ω 1, ω + 1 ) 2, any family $$\mathcal {A}\subset [{On}]^{<{\omega }}$$ A ⊂ [ On ] < ω of size $${\omega }_1$$ ω 1 contains a $$\Delta (...) $$ Δ -system of size $${\omega }_1$$ ω 1. Some classical results cannot be proven in ZF alone; however, we can establish weaker versions of these statements within the framework of ZF, such as $${{\omega }_2}\rightarrow ({\omega }_1,{\omega }+1)$$ ω 2 → ( ω 1, ω + 1 ), any family $$\mathcal {A}\subset [{On}]^{<{\omega }}$$ A ⊂ [ On ] < ω of size $${\omega }_2$$ ω 2 contains a $$\Delta $$ Δ -system of size $${\omega }_1$$ ω 1. Some statements can be proven in ZF using purely combinatorial arguments, such as: given a set mapping $$F:{\omega }_1\rightarrow {[{\omega }_1]}^{<{\omega }}$$ F : ω 1 → [ ω 1 ] < ω, the set $${\omega }_1$$ ω 1 has a partition into $${\omega }$$ ω -many F-free sets. Other statements can be proven in ZF by employing certain methods of absoluteness, for example: given a set mapping $$F:{\omega }_1\rightarrow {[{\omega }_1]}^{<{\omega }}$$ F : ω 1 → [ ω 1 ] < ω, there is an F-free set of size $${\omega }_1$$ ω 1, for each $$n\in {\omega }$$ n ∈ ω, every family $$\mathcal {A}\subset {[{\omega }_1]}^{{\omega }}$$ A ⊂ [ ω 1 ] ω with $$|A\cap B|\le n$$ | A ∩ B | ≤ n for $$\{A,B\}\in {[\mathcal {A}]}^{2}$$ { A, B } ∈ [ A ] 2 has property B. In contrast to statement (5), we show that the following ZFC theorem of Komjáth is not provable from ZF + $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1 : (6$$ ^*$$ ∗ ) every family $$\mathcal {A}\subset {[{\omega }_1]}^{{\omega }}$$ A ⊂ [ ω 1 ] ω with $$|A\cap B|\le 1$$ | A ∩ B | ≤ 1 for $$\{A,B\}\in {[\mathcal {A}]}^{2}$$ { A, B } ∈ [ A ] 2 is essentially disjoint. A function f is a uniform denumeration on$${\omega }_1$$ ω 1 iff $${\text {dom}}(f)={\omega }_1$$ dom ( f ) = ω 1, and for every $$1\le {\alpha }<{\omega }_1$$ 1 ≤ α < ω 1, $$f({\alpha })$$ f ( α ) is a function from $${\omega }$$ ω onto $${\alpha }$$ α. It is easy to see that the existence of a uniform denumeration of $${\omega }_1$$ ω 1 implies $$cf({\omega }_1)={\omega }_1$$ c f ( ω 1 ) = ω 1. We prove that the failure of the reverse implication is equiconsistent with the existence of an inaccessible cardinal. (shrink)

We now move to the demonstration of the left-to-right direction of the equivalence result. Let us assume that there is a winning $$\mathbf {P}$$ P -strategy in the dialogical game for $$\varphi $$ φ.

We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation.

Independence is the condition that, if X is preferred to Y, then a lottery between X and Z is preferred to a lottery between Y and Z given the same probability of Z. Is it rationally required that one’s preferences conform to Independence? The main objection to this requirement is that it would rule out the alleged rationality of Allais and Ellsberg Preferences. In this paper, I put forward a sequential dominance argument with fairly weak assumptions for a variant of (...) Independence (called Independence for Constant Prospects), which shows that Allais and Ellsberg Preferences are irrational. Hence this influential objection (that is, the alleged rationality of Allais and Ellsberg Preferences) can be rebutted. I also put forward a number of sequential dominance arguments that various versions of Independence are requirements of rationality. One of these arguments is based on very minimal assumptions, but the arguments for the versions of Independence which are strong enough to serve in the standard axiomatization of Expected Utility Theory need notably stronger assumptions. (shrink)

For each intermediate propositional logicJ, J * denotes the least predicate extension ofJ. By the method of canonical models, the strongly Kripke completeness ofJ *+D(=x(p(x)q)xp(x)q) is shown in some cases including:1. J is tabular, 2. J is a subframe logic. A variant of Zakharyashchev's canonical formulas for intermediate logics is introduced to prove the second case.

Let NBG be von Neumann-Bernays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.

An agent who violates independence can avoid dynamic inconsistency in sequential choice if he is sophisticated enough to make use of backward induction in planning. However, Seidenfeld has demonstrated that such a sophisticated agent with dependent preferences is bound to violate the principle of dynamic substitution, according to which admissibility of a plan is preserved under substitution of indifferent options at various choice nodes in the decision tree. Since Seidenfeld considers dynamic substitution to be a coherence condition on dynamic choice, (...) he concludes that sophistication cannot save a violator of independence from incoherence. In response to McClennen’s objection that relying on dynamic substitution when independence is at stake must be question-begging, Seidenfeld undertakes to prove that dynamic substitution follows from the principle of backward induction alone, provided we assume that the agent’s admissible choices from different sets of feasible plans are all based on a fixed underlying preference ordering of plans. This paper shows that Seidenfeld's proof fails: depending on the interpretation, it is either invalid or based on an unacceptable assumption. (shrink)

We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces.

N. B. Prof Williamson is now based at the Faculty of Philosophy, University of Oxford.

Concerning the notorious Sleeping Beauty problem, philosophers have debated whether 1/2 or 1/3 is rational as Beauty’s credence in the coin’s landing heads. According to Kierland and Monton, the answer depends on whether her goal is to minimize average or total inaccuracy because, while the expected average inaccuracy of Halfing is smaller than that of Thirding, the expected total inaccuracy of Thirding is lower than that of Halfing. In this paper, I argue that Halfing is average accuracy dominated but Thirding (...) is not; and that each of the standard forms of Halfing and Thirding regards a different credence assignment as better than itself in terms of total accuracy. Therefore, Halfing is irrational, and Thirding is likely to be rational, for the goal of minimizing average inaccuracy; but both Halfing and Thirding, at least in their standard forms, are irrational for the goal of minimizing total inaccuracy. (shrink)

Western thought since the seventeenth century has been dominated by methodological solipsism (Krieger, 1991). The famous sound-bite of René Descartes 'cogito, ergo sum': 'I think, therefore I am', became the starting point for most discourse on the nature of things. This dictum does not advocate idealism. It does not assert that everything is necessarily a construct of the human mind. But it assumes that the world of things and beings is surveyed and interpreted from the point of view of a (...) single individual. Nowhere is this stance more entrenched than in the philosophy of science. To a remarkable degree the scientist is represented as studying the natural world as if alone in it, served only by mindless assistants who might as well be replaced by machines. Scientific theories are presented as systems of thought conjured up and tested by that same individual in a further series of single-headed operations. Research results are formulated and treated philosophically as the independent findings of lone explorers, each reporting the evidence of their own eyes and their rational inferences concerning the hidden mechanisms by which these personal percepts might be generated. Our epistemological role models are Robinson Crusoe and Sherlock Holmes, self-sufficient intellectuals to whom their human companions, Friday and Watson, are mere stooges. (shrink)

This paper offers a particular intuitionistic negation completion of Urquhart's system C resulting in a super-intuitionistic contractionless propositional logic equivalent to Dummett's LC without contraction.

With the inclusion of an e ective methodology, this article answers in detail a question that, for a quarter of a century, remained open despite intense study by various researchers. Is the formula XCB = e(x e(e(e(x y) e(z y)) z)) a single axiom for the classical equivalential calculus when the rules of inference consist of detachment (modus ponens) and substitution? Where the function e represents equivalence, this calculus can be axiomatized quite naturally with the formulas (x x), e(e(x (...) y) e(y x)), and e(e(x y) e(e(y z) e(x z))), which correspond to reexivity, symmetry, and transitivity, respectively. (We note that e(x x) is dependent on the other two axioms.) Heretofore, thirteen shortest single axioms for classical equivalence of length eleven had been discovered, and XCB was the only remaining formula of that length whose status was undetermined. To show that XCB is indeed such a single axiom, we focus on the rule of condensed detachment, a rule that captures detachment together with an appropriately general, but restricted, form of substitution. The proof we present in this paper consists of twenty- ve applications of condensed detachment, completing with the deduction of transitivity followed by a deduction of symmetry. We also discuss some factors that may explain in part why XCB resisted relinquishing its treasure for so long. Our approach relied on diverse strategies applied by the automated reasoning program OTTER. Thus ends the search for shortest single axioms for the equivalential calculus. (shrink)

Henrich et al. describe an innovative research program investigating cross-cultural differences in the selfishness axiom (in economic games) in humans, yet humans are not the only species to show such variation. Chimpanzees and capuchin monkeys show signs of deviating from the standard self-interest paradigm in experimental settings by refusing to take foods that are less valuable than those earned by conspecifics, indicating that they, too, may pay attention to relative gains. However, it is less clear whether these species also (...) show the other-regarding preferences seen in humans. (shrink)

No detailed description available for "From Axiom to Dialogue".