Adjoin, to a countable standard model M of Zermelo-Fraenkel set theory (ZF), a countable set A of independent Cohen generic reals. If one attempts to construct the model generated over M by these reals (not necessarily containing A as an element) as the intersection of all standard models that include M ∪ A, the resulting model fails to satisfy the power set axiom, although it does satisfy all the other ZF axioms. Thus, there is no smallest ZF model (...) including M ∪ A, but there are minimal such models. These are classified by their sets of reals, and there is one minimal model whose set of reals is the smallest possible. We give several characterizations of this model, we determine which weak axioms of choice it satisfies, and we show that some better known models are forcing extensions of it. (shrink)

A model \ of ZF is said to be condensable if \\prec _{\mathbb {L}_{{\mathcal {M}}}} {\mathcal {M}}\) for some “ordinal” \, where \:=,\in )^{{\mathcal {M}}}\) and \ is the set of formulae of the infinitary logic \ that appear in the well-founded part of \. The work of Barwise and Schlipf in the 1970s revealed the fact that every countable recursively saturated model of ZF is cofinally condensable \prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}\) for an unbounded collection of \). Moreover, it (...) can be readily shown that any \-nonstandard condensable model of \ is recursively saturated. These considerations provide the context for the following result that answers a question posed to the author by Paul Kindvall Gorbow.Theorem A. Assuming a modest set-theoretic hypothesis, there is a countable model \ of ZFC that is both definably well-founded is in the well-founded part of \}\) and cofinally condensable. We also provide various equivalents of the notion of condensability, including the result below.Theorem B. The following are equivalent for a countable model \ of \: \ is condensable. \ is cofinally condensable. \ is nonstandard and \\prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}\) for an unbounded collection of \. (shrink)

A DO model (here also referred to a Paris model) is a model of set theory all of whose ordinals are first order definable in . Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our (...) results include the following:1. If T is a consistent completion of ZF+V≠OD, then T has continuum-many countable nonisomorphic Paris models.2. Every countable model of ZFC has a Paris generic extension.3. If there is an uncountable well-founded model of ZFC, then for every infinite cardinal κ there is a Paris model of ZF of cardinality κ which has a nontrivial automorphism.4. For a model ZF, is a prime model ⇒ is a Paris model and satisfies AC⇒ is a minimal model. Moreover, Neither implication reverses assuming Con(ZF). (shrink)

By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, (...) as defined by Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given. (shrink)

A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the Leibniz-Mycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF, T has a Leibnizian model if and only if T proves LM. Here we prove: THEOREM A. Every complete theory T extending ZF + LM has $2^{\aleph_{0}}$ nonisomorphic countable Leibnizian models. THEOREM B. If $\kappa$ is aprescribed definable infinite (...) cardinal of a complete theory T extending ZF + V = OD. then there are $2^{\aleph_{1}}$ nonisomorphic Leibnizian models $\mathfrak{M}$ of T of power $\aleph_{1}$ such that $(\kappa^{+})^\mathfrak{M}$ is $\aleph_{1}-like$ . THEOREM C. Every complete theory T extending ZF + V = OD has $2^{\aleph_{1}}$ nonisomorphic \aleph_{1}-like$ Leibnizian models. (shrink)

We study the complexity of the classification problem for countable models of set theory (). We prove that the classification of arbitrary countable models of is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well‐founded models of.

We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.

We construct two recursive models of fragments of set theory. We also show that the fragments of Kripke-Platek set theory that prove -induction for -formulas have no recursive models but the standard model of the hereditarily finite sets.

A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that (...) is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. (shrink)

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic (...) to a submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations. (shrink)

A model M = (M, E,...) of Zermelo-Fraenkel set theory ZF is said to be θ-like, where E interprets ∈ and θ is an uncountable cardinal, if |M| = θ but $|\{b \in M: bEa\}| for each a ∈ M. An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ 1 -like model. Coupled with Chang's two cardinal (...) theorem this implies that if θ is a regular cardinal θ such that $2^{ then every consistent extension of ZF also has a θ + -like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ 2 -like model. Here we prove: THEOREM A. If θ has the tree property then the following are equivalent for any completion T of ZFC: (i) T has a θ-like model. (ii) $\Phi \subseteq T$ , where Φ is the recursive set of axioms {∃ κ(κ is n-Mahlo and "V κ is a Σ n -elementary submodel of the universe"): n ∈ ω}. (iii) T has a λ-like model for every uncountable cardinal λ. THEOREM B. The following are equiconsistent over ZFC: (i) "There exists an ω-Mahlo cardinal". (ii) "For every finite language L, all ℵ 2 -like models of ZFC(L) satisfy the scheme Φ(L). (shrink)

We prove the following extension of a result of Keisler and Morley. Suppose U is a countable model of ZFC and c is an uncountable regular cardinal in U. Then there exists an elementary extension of U which fixes all ordinals below c, enlarges c, and either (i) contains or (ii) does not contain a least new ordinal. Related results are discussed.

We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω1‐like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of many ω1‐like models of, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω1‐like model of that does (...) not embed into its own constructible universe; and there can be an ω1‐like model of whose structure of hereditarily finite sets is not universal for the ω1‐like models of set theory. (shrink)

By a classical theorem of Harvey Friedman, every countable nonstandard model $\mathcal {M}$ of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding j, i.e., j is a self-embedding of $\mathcal {M}$ such that $j[\mathcal {M}]\subsetneq \mathcal {M}$, and the ordinal rank of each member of $j[\mathcal {M}]$ is less than the ordinal rank of each element of $\mathcal {M}\setminus j[\mathcal {M}]$. Here, we investigate the larger family of proper initial-embeddings j of models $\mathcal {M}$ of fragments (...) of set theory, where the image of j is a transitive submodel of $\mathcal {M}$. Our results include the following three theorems. In what follows, $\mathrm {ZF}^-$ is $\mathrm {ZF}$ without the power set axiom; $\mathrm {WO}$ is the axiom stating that every set can be well-ordered; $\mathrm {WF}$ is the well-founded part of $\mathcal {M}$ ; and $\Pi ^1_\infty \text{-}\mathrm {DC}_\alpha $ is the full scheme of dependent choice of length $\alpha $.Theorem A.There is an $\omega $ -standard countable nonstandard model $\mathcal {M}$ of $\mathrm {ZF}^-+\mathrm {WO}$ that carries no initial self-embedding $j:\mathcal {M} \longrightarrow \mathcal {M}$ other than the identity embedding.Theorem B.Every countable $\omega $ -nonstandard model $\mathcal {M}$ of $\ \mathrm {ZF}$ is isomorphic to a transitive submodel of the hereditarily countable sets of its own constructible universe $L^{\mathcal {M}}$.Theorem C.The following three conditions are equivalent for a countable nonstandard model $\mathcal {M}$ of $\mathrm {ZF}^{-}+\mathrm {WO}+\forall \alpha \ \Pi ^1_\infty \text{-}\mathrm {DC}_\alpha $. There is a cardinal in $\mathcal {M}$ that is a strict upper bound for the cardinality of each member of $\mathrm {WF}$. $\mathrm {WF}$ satisfies the powerset axiom.For all $n \in \omega $ and for all $b \in M$, there exists a proper initial self-embedding $j: \mathcal {M} \longrightarrow \mathcal {M}$ such that $b \in \mathrm {rng}$ and $j[\mathcal {M}] \prec _n \mathcal {M}$. (shrink)

Aristotle is oftentimes viewed through a strictly philosophical lens as heir to Plato and has having introduced logical rigor where an emphasis on the theory of Forms formerly prevailed. It must be appreciated that Aristotle was the son of a physician, and that his inculcation of the thought of other Greek philosophers addressing health and the natural elements led to an extremely broad set of biologically- and medically-related writings. As this article proposes, Aristotle deepened the fourfold theory of (...) the elements with anatomic and physiologic observations. In books like History of Animals, Parts of Animals, and the lost Anatomai, he actively dissected organisms and recorded his findings. The corpus of medically-related literature Aristotle developed had a direct influence on subsequent Greek thinkers, including Galen, and on Medieval Islamic and modern Western practitioners such as William Harvey. Aristotle's ideas continue to influence modern medical thinkers in Europe and America through both his interpretation by European philosophers gaining the attention of medical humanists, and his writings' enduring impact on medical researchers balancing scientific with more personalistic approaches to medicine. (shrink)

A theorem of Enayat's concerning models of ZFC which had been proved using several different additional set-theoretical hypotheses is shown here to be absolute.

TheoremEvery model of ZFChas a conservative elementary extension which possesses a cofinal minimal elementary extension.An application of Boolean ultrapowers to models of full arithmetic is also presented.

This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST. The paper extends the results in (...) [2] by introducing a new and perhaps more natural notion of ideal, and in the class theory of part three. (shrink)

Given a model \ of set theory, and a nontrivial automorphism j of \, let \\) be the submodel of \ whose universe consists of elements m of \ such that \=x\) for every x in the transitive closure of m ). Here we study the class \ of structures of the form \\), where the ambient model \ satisfies a frugal yet robust fragment of \ known as \, and \=m\) whenever m is a finite ordinal in the (...) sense of \ Our main achievement is the calculation of the theory of \ as precisely \-\. The following theorems encapsulate our principal results:Theorem A. Every structure in \ satisfies \-\.Theorem B. Each of the following three conditions is sufficient for a countable structure \ to be in \: \ is a transitive model of \-\. \ is a recursively saturated model of \-\. \ is a model of \.Theorem C. Suppose \ is a countable recursively saturated model of \ and I is a proper initial segment of \ that is closed under exponentiation and contains \. There is a group embedding \ from \\) into \\) such that I is the longest initial segment of \ that is pointwise fixed by \ for every nontrivial \.\)In Theorem C, \\) is the group of automorphisms of the structure X, and \ is the ordered set of rationals. (shrink)

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non-standard (...) models of the fragment \ + \-Separation of \; and Gaifman’s technique of iterated ultrapowers is employed to show that any countable model of \ can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of “strong rank-cut” is characterized in terms of the theory \, and in terms of fixed-point sets of self-embeddings. (shrink)

Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (`unfoldable cardinals') lie in the boundary of the propositions consistent with `V = L' and the existence of 0 ♯ . We also provide an `embedding characterisation' of the unfoldable cardinals and study their preservation and destruction by various forcing constructions.

An ordinal in a model of set theory is truly countable if its set of predecessors is countable in the real world. We classify the order types of the sets of truly countable ordinals. Models with indiscernibles and other related results are discussed.

We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ (...) set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly 1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry. (shrink)

We introduce a new sheaf-theoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produce models of various elementary set theories. These results are then used to prove the conservativity of different set theories over various classical and constructive type theories.

Let U be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that U has a Σ n end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If U is a well-founded model of ZFC, such that U has (...) uncountable cofinality and U has a topless Σ 3 end extension, then U has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in U) highly hyperinaccessible. In § 3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σ n end extension for each n ∈ ω. The case κ = ω 1 is investigated more deeply in § 4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω 1 -like models of ZFC are constructed which have a Σ n end extension for all n ∈ ω but have no elementary end extension. ω 1 -like models of ZFC which have no Σ 3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions. (shrink)

We characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).

If a˜cardinal κ1, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ0 and its new cofinality, ρ, is less than κ0, then, under some additional assumptions, each cardinal λ>κ1 less than cc/[κ1]<κ1) is collapsed to κ0 as well. If in addition N=M[f], where f : ρ→κ1 is an unbounded mapping, then N is a˜λ=κ0-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba.