In the conventional approach to quantum mechanics, indeterminism is an axiom and nonlocality is a theorem. We consider inverting the logical order, making nonlocality an axiom and indeterminism a theorem. Nonlocal “superquantum” correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock (...) Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. (shrink)

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the (...) bounded proper forcing axiom is exactly the existence of a Σ 1 -reflecting cardinal (which is less than the existence of a Mahlo cardinal). We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure. (shrink)

Summary of the talk given to the 22nd Conference on the History of Logic, Cracow (Poland), July 5–9, 1976.

Paul Hovda’s excellent paper ‘What Is Classical Mereology?' has fruitfully reshaped the debate concerning the axiomatic foundations of classical mereology. Precisely because of the importance of Hovda’s work and its usefulness as a reference tool, we note here that one of the five axiom systems presented therein, corresponding the ‘Third Way’ to classical mereology, is defective and must be amended. In addition, we note that two other axiom systems, corresponding to the ‘First Way’ and to the ‘Fifth Way’, (...) involve redundancies. (shrink)

Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face (...) complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology. (shrink)

We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of (...) their Hilbert-style deductive proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the "Tangibility Reflection Principle". We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further. (shrink)

If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at N₂ with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC-model. This yields that if Woodin's ℙ max axiom (*) holds, then BPFA implies that V is closed under the "Woodin-in-the-next-ZFC-model" operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus "NS ω1 is precipitous" (...) and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor. [2, Question 6.11]. (shrink)

This paper advances the study of so-called “iterative” paraconsistent set theories. Unlike a naive set theory, which validates both Unrestricted Comprehension and Extensionality, an iterative paraconsistent set theory uphold the axioms of Zermelo–Fraenkel set theory. The principal advantage of the iterative approach is that it yields set theories that are highly mathematically expressive. It has recently been conjectured that the mathematical expressiveness of certain iterative paraconsistent set theories may even rival that of classical set theory. However, until now the status (...) of the Axiom of Choice in this setting has remained open. We show that iterative paraconsistent set theories are compatible with the Axiom of Choice. Finally, in the conclusion, we contrast this result with the failure of Choice in the intuitionistic setting. (shrink)

In the Zermelo–Fraenkel set theory (ZF), |$|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$| for any infinite set |$A$|⁠, where |$\textrm {fin}(A)$| is the set of finite subsets of |$A$|⁠, |$2^{|A|}$| is the cardinality of the power set of |$A$| and |$\textrm {Part}(A)$| is the set of partitions of |$A$|⁠. In this paper, we show in ZF that |$|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$| for any set |$A$| with |$|A|\geq 5$|⁠, where |$\textrm {Part}_{\textrm {fin}}(A)$| is the set of partitions of |$A$| whose members are finite. We (...) also show that, without the Axiom of Choice, any relationship between |$|\textrm {Part}_{\textrm {fin}}(A)|$| and |$2^{|A|}$| for an arbitrary infinite set |$A$| cannot be concluded. (shrink)

In this paper, I shall discuss several topics related to Frege's paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege's notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege's attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik Frege hardly could have construed Hume's Principle as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck's arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

In this paper we introduce the Wholeness Axiom, which asserts that there is a nontrivial elementary embedding from V to itself. We formalize the axiom in the language {∈, j }, adding to the usual axioms of ZFC all instances of Separation, but no instance of Replacement, for j -formulas, as well as axioms that ensure that j is a nontrivial elementary embedding from the universe to itself. We show that WA has consistency strength strictly between I 3 (...) and the existence of a cardinal that is super- n -huge for every n. ZFC + WA is used as a background theory for studying generalizations of Laver sequences. We define the notion of Laver sequence for general classes E consisting of elementary embeddings of the form i :V β → M, where M is transitive, and use five globally defined large cardinal notions – strong, supercompact, extendible, super-almost-huge, superhuge – for examples and special cases of the main results. Assuming WA at the beginning, and eventually refining the hypothesis as far as possible, we prove the existence of a strong form of Laver sequence for a broad range of classes E that include the five large cardinal types mentioned. We show that if κ is globally superstrong, if E is Laver-closed at beth fixed points, and if there are superstrong embeddings i with critical point κ and arbitrarily large targets such that E is weakly compatible with i, then our standard constructions are E -Laver at κ., there is an extendible Laver sequence at κ.) In addition, in most cases our Laver sequences can be made special if E is upward λ -closed for sufficiently many λ. (shrink)

We give a new characterization of the class of completely representable cylindric algebras of dimension 2 #lt; n w via special neat embeddings. We prove an independence result connecting cylindric algebra to Martin''s axiom. Finally we apply our results to finite-variable first order logic showing that Henkin and Orey''s omitting types theorem fails for L n, the first order logic restricted to the first n variables when 2 #lt; n#lt;w. L n has been recently (and quite extensively) studied as (...) a many-dimensional modal logic. (shrink)

Start with an extension of Q (Robinson arithmetic) that internalizes an axiom predicate, and has an axiom that denies axiom-status to a formula using a constant $\alpha$. Then, whether the system is consistent depends on which number is assigned to $\alpha$. Contradiction is provable if $\alpha$ is ``directly'' self-referential as per recent work by Kripke. The contradiction is structurally akin to the liar paradox but arises without the usual semantic or modal vocabulary. Several solutions are noted. Yet (...) it remains that axiomhood exhibits liar-like fragility under naive internalization. (shrink)

The axiom of recovery, while capturing a central intuition regarding belief change, has been the source of much controversy. We argue briefly against putative counterexamples to the axiom—while agreeing that some of their insight deserves to be preserved—and present additional recovery-like axioms in a framework that uses epistemic states, which encode preferences, as the object of revisions. This makes iterated revision possible and renders explicit the connection between iterated belief change and the axiom of recovery. We provide (...) a representation theorem that connects the semantic conditions we impose on iterated revision and our additional syntactical properties. We show interesting similarities between our framework and that of Darwiche–Pearl (Artificial Intelligence 89:1–29 1997). In particular, we show that intuitions underlying the controversial (C2) postulate are captured by the recovery axiom and our recovery-like postulates (the latter can be seen as weakenings of (C2)). We present postulates for contraction, in the same spirit as the Darwiche–Pearl postulates for revision, and provide a theorem that connects our syntactic postulates with a set of semantic conditions. Lastly, we show a connection between the contraction postulates and a generalisation of the recovery axiom. (shrink)

The McKinsey axiom $$(\textrm{M})\ \Box \Diamond p\rightarrow \Diamond \Box p$$ has a local first-order correspondent on the class of all weakly transitive frames $${{\mathcal {W}}}{{\mathcal {T}}}$$. It globally corresponds to Lemmon’s condition $$({\textsf{m}}^\infty )$$ on $${{\mathcal {W}}}{{\mathcal {T}}}$$. The formula $$(\textrm{M})$$ is canonical over the weakly transitive modal logic $$\textsf{wK4}={\textsf{K}}\oplus p\wedge \Box p\rightarrow \Box \Box p$$. The modal logic $$\mathsf {wK4.1}=\textsf{wK4}\oplus \textrm{M}$$ has the finite model property. The modal logics $$\mathsf {wK4.1T}_0^n$$ ( $$ n>0$$ ) form an infinite descending (...) chain in the interval $$[\mathsf {wK4.1},\mathsf {K4.1}]$$ and each of them has the finite model property. Thus all the modal logics $$\mathsf {wK4.1}$$ and $$\mathsf {wK4.1T}_0^n$$ ( $$n>0$$ ) are decidable. (shrink)

We prove that MM (Martin maximum) is equivalent (in ZFC) to the older axiom SPFA (semiproper forcing axiom). We also prove that SPFA does not imply SPFA + or even PFA + (using the consistency of a large cardinal).

We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.

The Ultrapower Axiom is an abstract combinatorial principle inspired by the fine structure of canonical inner models of large cardinal axioms. In this paper, it is established that the Ultrapower A...

We strengthen a result of Harrington and Shelah by showing that, unless ω1 is an inaccessible cardinal in L, a relatively weak fragment of Martin's axiom implies that there exists a δ13 set of reals without the property of Baire.

We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].

A method of constructing Hilbert-type axiom systems for standard many-valued propositional logics was offered by Rosser and Turquette. Although this method is considered to be a solution of the problem of axiomatisability of a wide class of many-valued logics, the article demonstrates that it fails to produce adequate axiom systems. The article concerns finitely many-valued propositional logics of Łukasiewicz. It proves that if standard propositional connectives of the Rosser–Turquette axiom systems are definable in terms of the propositional (...) connectives of Łukasiewicz’s logics, and thus, they are normal ones, then every Rosser–Turquette axiom system for a finite-valued Łukasiewicz’s logic is semantically incomplete. (shrink)

We introduce a new simple way of defining the forcing method that works well in the usual setting under FA, the Foundation Axiom, and moreover works even under Aczel's AFA, the Anti-Foundation Axiom. This new way allows us to have an intuition about what happens in defining the forcing relation. The main tool is H. Friedman's method of defining the extensional membership relation ∈ by means of the intensional membership relation ε .Analogously to the usual forcing and the (...) usual generic extension for FA-models, we can justify the existence of generic filters and can obtain the Forcing Theorem and the Minimal Model Theorem with some modifications. These results are on the line of works to investigate whether model theory for AFA-set theory can be developed in a similar way to that for FA-set theory.Aczel pointed out that the quotient of transition systems by the largest bisimulation and transition relations have the essentially same theory as the set theory with AFA. Therefore, we could hope that, by using our new method, some open problems about transition systems turn out to be consistent or independent. (shrink)

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability (...) in the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry. (shrink)

We provide a quantifier-free axiom system for plane hyperbolic geometry in a language containing only absolute geometrically meaningful ternary operations (in the sense that they have the same interpretation in Euclidean geometry as well). Each axiom contains at most 4 variables. It is known that there is no axiom system for plane hyperbolic consisting of only prenex 3-variable axioms. Changing one of the axioms, one obtains an axiom system for plane Euclidean geometry, expressed in the same (...) language, all of whose axioms are also at most 4-variable universal sentences. We also provide an axiom system for plane hyperbolic geometry in Tarski's language L B which might be the simplest possible one in that language. (shrink)

The title of this paper is in the past tense to indicate that the question it will address is whether the Axiom of Reducibility is a principle of logic according to the view of logic that Russell had when writing the first edition of Principia Mathematica.'It is often said that Logicism was a failure because when it avoided the Scylla of contradiction in Frege's system it fell into the Charybdis of requiring'obviously non-logical principles at Russell's hands. The axiom~non-logical (...) principle which Russell had to add to his system in order to be able to develop mathematics. I want to consider this criticism of the axiom from several points of view.Why is it thought that the axiom of reducibility is not a principle of logic?What reasons does Russell actually give for doubting its logical status? Are they good reasons? (shrink)

This paper seeks to question the position of ZF as the dominant system of set theory, and in particular to examine whether there is any philosophical justification for the axiom of foundation. After some historical observations regarding Poincare and Russell, and the notions of circularity and hierarchy, the iterative conception of set is argued to be a semi-constructvist hybrid without philosophical coherence. ZF cannot be justified as necessary to avoid paradoxes, as axiomatizing a coherent notion of set, nor on (...) pragmatic grounds, and should be replaced by a system allowing non-well-founded sets. (shrink)

In this chapter, we will examine some problems concerning axiomatization of finitely many-valued propositional logics. It has been recently demonstrated that both a particular axiom system for the functionally complete three-valued logic and a certain general method of constructing axiom systems for finitely many-valued logics do not satisfy some salient metalogical requirements. Firstly, we will examine an axiom system for the functionally complete three-valued logic based on the well-known Mordchaj Wajsberg axiom system for the three-valued logic (...) of Jan Łukasiewicz. The examined axiom system was introduced in 1936, by Jerzy Słupecki. As we will see, this axiom system is not semantically complete. Then, we will investigate a method of constructing axiom systems for standard many-valued propositional logics introduced by John B. Rosser and Atwell R. Turquette. The Rosser-Turquette method is considered to be a solution of the problem of axiomatizability of a particular class of many-valued logics, i.e., standard many-valued propositional logics, including finitely many-valued propositional logics of Łukasiewicz and Emil Post. However, we will demonstrate that the Rosser-Turquette method fails to produce adequate axiom systems for a particular class of many-valued propositional logics, i.e., for finitely many-valued propositional logics of Łukasiewicz. The presented investigations concern the mutual definability of the Rosser-Turquette standard connectives and the connectives of finitely many-valued propositional logics of Łukasiewicz. The conclusion is that every Rosser-Turquette axiom system for finitely many-valued propositional logics of Łukasiewicz that satisfies a necessary condition of being a sound system, is semantically incomplete. (shrink)

In this paper, we present set theories based upon the paraconsistent logic Pac. We describe two different techniques to construct models of such set theories. The first of these is an adaptation of one used to construct classical models of positive comprehension. The properties of the models obtained in that way give rise to a natural paraconsistent set theory which is presented here. The status of the axiom of choice in that theory is also discussed. The second leads to (...) show that any classical universe of set theory (e.g. a model of ZF) can be extended to a paraconsistent one, via a term model construction using an adapted bisimulation technique. (shrink)

The minimal ordinal-connection axiom $$MOC$$ was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that $$MOC$$ is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, $$MOC$$ is in fact equivalent to the $${{\,\mathrm{GCH}\,}}$$. Our main results then show that $$MOC$$ corresponds to a weak version of global choice in models of the $${{\,\mathrm{GCH}\,}}$$ : it can fail in models of (...) the $${{\,\mathrm{GCH}\,}}$$ without global choice, but also global choice can fail in models of $$MOC$$. (shrink)

Most accounts of our knowledge of the successor axiom claim that this is based on the procedure of adding one. While they usually don’t claim to provide an account of how children actually acquire this knowledge, one may well think that this is how they get that knowledge. I argue that when we look at children’s responses in interviews, the time when they learn the successor axiom and the intermediate learning stages they find themselves in, that there is (...) an empirically viable alternative. I argue that they could also learn it on the basis of a method that has to do with the structure of the numeral system. Specifically, that they (1) use the syntactic structure of the numeral system and (2) attend to the leftmost digits, the one with the highest place-value. Children can learn that this is a reliable method of forming larger numbers by combining two elements. First, a grasp of the syntactic structure of the numeral system. That way they know that the leftmost digit receives the highest value. Second, an interpretation of numerals as designating cardinal values, so that they also realise that increasing or adding digits on the lefthand side of a numeral produces a larger number. There are thus two, currently equally well-supported, ways in which children might learn that there are infinitely many natural numbers. (shrink)

Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...) the Continuum Hypothesis, is a classic of modern mathematics. The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems, famously delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for this problem. In 1999, Time magazine ranked him higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. He is most renowned for his proof in 1931 of the 'incompleteness theorem,' in which he demonstrated that there are problems that cannot be solved by any set of rules or procedures. His proof wrought fruitful havoc in mathematics, logic, and beyond. (shrink)

The Covering Property Axiom, which attempts to capture some of the combinatorics of the Sacks model, the model obtained from by countable support iteration of length of the Sacks forcing, seems to miss a Suslin tree. We add a diamond polish to the axiom to remedy this.

This paper addresses the "algorithmic fine-tuning problem": why does our universe exhibit quantum mechanics if quantum mechanics is algorithmically improbable on a classical substrate? Building on our trilogy establishing the impossibility of deriving quantum structure (Axiom A1) from classical computation, we propose the Substrate Hypothesis: the universe's computational substrate is "quantum-native." We extend Chaitin's halting probability Ω from a real scalar to a state vector |Ω_Q⟩ in Hilbert space---the wavefunction of the algorithmic multiverse. We prove its normalizability (Theorem 1) (...) and establish a direct connection to classical Ω through the halting projection (Theorem 2). Using a minimal model language A1---named after the axiom it natively implements---we measure description length asymmetry: quantum operations require 25× fewer tokens on average (up to 40× at bit level) on a quantum substrate compared to classical NumPy implementations. This corresponds to an algorithmic probability gain of 2^878. We validate these theoretical predictions experimentally on AWS Braket, achieving >97% fidelity across all benchmarks with only 2--6 A1 tokens. This provides engineering validation of the Substrate Hypothesis: quantum mechanics is algorithmically natural on a quantum substrate. (shrink)

We first start by clarifying what axiomatizing everything can mean. We then study a famous case of axiomatization, the axiomatization of natural numbers, where two different aspects of axiomatization show up, the model-theoretical one and the proof-theoretical one. After that we discuss a case of axiomatization in a sense opposed to the one of arithmetic, the axiomatization of the notion of order, where the idea is not to catch a specific structure, but a notion. A third mathematical case is then (...) examined, the one of identity, a simple and obvious notion, but that cannot be axiomatized in first-order logic. We then move on to more general notions: the axiomatization of causality and the universe. To end with we deal with an even more tricky question: the axiomatization of reasoning itself. In conclusion we discuss in the light of our investigations the relation between axiomatization and understanding. (shrink)

It is shown that AC, the axiom of choice for families of non-empty subsets of the real line ℝ, does not imply the statement PW, the powerset of ℝ can be well ordered. It is also shown that the statement “the set of all denumerable subsets of ℝ has size 2math image” is strictly weaker than AC and each of the statements “if every member of an infinite set of cardinality 2math image has power 2math image, then the union (...) has power 2math image” and “ℵ ≠ ℵω” is Hartogs' aleph, the least ℵ not ≤ 2math image), is strictly weaker than the full axiom of choice AC. (shrink)

We show that the Axiom of Choice is equivalent to each of the following statements: A product of closures of subsets of topological spaces is equal to the closure of their product ; A product of complete uniform spaces is complete.

We show that the axiom of choice AC is equivalent to the Vector Space Kinna-Wagner Principle, i.e., the assertion: “For every family [MATHEMATICAL SCRIPT CAPITAL V]= {Vi : i ∈ k} of non trivial vector spaces there is a family ℱ = {Fi : i ∈ k} such that for each i ∈ k, Fiis a non empty independent subset of Vi”. We also show that the statement “every vector space over ℚ has a basis” implies that every infinite (...) well ordered set of pairs has an infinite subset with a choice set, a fact which is known not to be a consequence of the axiom of multiple choice MC. (shrink)

It is known that – assuming the axiom of choice – for subsets A of ℝ the following hold: (a) A is compact iff it is sequentially compact, (b) A is complete iff it is closed in ℝ, (c) ℝ is a sequential space. We will show that these assertions are not provable in the absence of the axiom of choice, and that they are equivalent to each.

Gödel's first steps in set theory, from the summer of 1935 to the end of his stay in Princeton half a year later, are described in the light of his shorthand notebooks. The notes end with an English manuscript titled ‘The freedom from contradiction of the axiom of choice’ that is analyzed in detail. Gödel works out a logical hierarchical construction that systematically incorporates well-orderings, thereby affirming the title of his paper. He also sees an avenue to having his (...) construction affirm the relative consistency of the Continuum Hypothesis, even to a key lemma about condensation along the hierarchy. However, he could not see a way to establishing the lemma until two years later. (shrink)

We show that some well known theorems in topology may not be true without the axiom of choice.

Suppose that T^∗ is an ω_1-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA(T^∗) for proper forcings which preserve these properties of T^∗. We prove that PFA(T^∗) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than ω_1, and the P-ideal dichotomy. On the other hand, PFA(T^∗) implies some of the consequences of diamond principles, such as the (...) existence of Knaster forcings which are not stationarily Knaster. (shrink)

In their logical analysis of theorems about disjoint rays in graphs, Barnes, Shore, and the author (hereafter BGS) introduced a weak choice scheme in second-order arithmetic, called the $\Sigma ^1_1$ axiom of finite choice (hereafter finite choice). This is a special case of the $\Sigma ^1_1$ axiom of choice ( $\Sigma ^1_1\text {-}\mathsf {AC}_0$ ) introduced by Kreisel. BGS showed that $\Sigma ^1_1\text {-}\mathsf {AC}_0$ suffices for proving many of the aforementioned theorems in graph theory. While it is (...) not known if these implications reverse, BGS also showed that those theorems imply finite choice (in some cases, with additional induction assumptions). This motivated us to study the proof-theoretic strength of finite choice. Using a variant of Steel forcing with tagged trees, we show that finite choice is not provable from the $\Delta ^1_1$ -comprehension scheme (even over $\omega $ -models). We also show that finite choice is a consequence of the arithmetic Bolzano–Weierstrass theorem (introduced by Friedman and studied by Conidis), assuming $\Sigma ^1_1$ -induction. Our results were used by BGS to show that several theorems in graph theory cannot be proved using $\Delta ^1_1$ -comprehension. Our results also strengthen results of Conidis. (shrink)

A new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice (...) fails; this is arranged by means of an application of the omitting types theorem. The general theory of pseudo-ultrapowers is developed. Following that, we study supercompactness in the absence of choice, and we analyze pseudo-ultrapowers of models of the axiom of determinateness and various infinite exponent partition relations. Relationships between pseudo-ultrapowers and forcing are also discussed. (shrink)

Journal of Mathematical Logic, Volume 25, Issue 03, December 2025. We prove that the forcing axiom [math] implies [math]. Using this implication, we show that the forcing axiom [math] is inconsistent. We also derive weak Chang’s Conjecture from [math] and use this second implication to give another proof of the inconsistency of [math].

The present work is an attempt to break ground in mathematics proper, armed with the accepting view just described. Specifically, we shall examine various versions of antinomic set theory, in particular the axiom of choice, keeping the presentation as intuitive as possible, more in the manner of a nineteenth century paper than as a thoroughly formalized system. The reason for such a presentation is the conviction that at this point it should be the mathematics that eventually determines the logic, (...) rather than the other way around. (shrink)

After presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we consider the bounded forcing axiom for a class of partially ordered sets Γ 1 such that, letting Γ 0 be the class of all (...) stationary-set-preserving partially ordered sets, one can prove the following: (a) $\Gamma_0 \subseteq \Gamma_1$, (b) Γ 0 = Γ 1 if and only if NS ω 1 is ℵ 1 -dense. (c) If P $\notin \Gamma_1$, then BFA({P}) fails. We call the bounded forcing axiom for Γ 1 Maximal Bounded Forcing Axiom (MBFA). Finally we prove MBFA consistent relative to the consistency of an inaccessible Σ 2 -correct cardinal which is a limit of strongly compact cardinals. (shrink)

We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of 1-dense real order types, Ann. Pure Appl. Logic 29 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal (...) invariants in Cichoń’s diagram are at least 2. (shrink)