scholarly journals FRAGMENTS OF FREGE’SGRUNDGESETZEAND GÖDEL’S CONSTRUCTIBLE UNIVERSE

2016 ◽  
Vol 81 (2) ◽  
pp. 605-628 ◽  
Author(s):  
SEAN WALSH
Keyword(s):  
Set Theory ◽  
The Other ◽  
Structure Theory ◽  
Sufficient Condition ◽  
Weak Version ◽  
Power Set ◽  
Consistency Problem ◽  
Universe Of Sets ◽  
The 19Th Century ◽  
Abstraction Principles

AbstractFrege’sGrundgesetzewas one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of theGrundgesetzeformed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of theGrundgesetze, and our main theorem (Theorem 2.9) shows that there is a model of a fragment of theGrundgesetzewhich defines a model of all the axioms of Zermelo–Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to Gödel’s constructible universe of sets and to Kripke and Platek’s idea of the projectum, as well as to a weak version of uniformization (which does not involve knowledge of Jensen’s fine structure theory). The axioms of theGrundgesetzeare examples ofabstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension (Theorem 3.5). As an application, we resolve an analogue of the joint consistency problem in the predicative setting.

A proof-theoretic characterization of the primitive recursive set functions

1992 ◽  
Vol 57 (3) ◽  
pp. 954-969 ◽  
Author(s):  
Michael Rathjen
Keyword(s):  
Set Theory ◽  
Elementary Proof ◽  
Weak Version ◽  
Definable Set ◽  
Set Functions ◽  
Theoretic Characterization ◽  
Universe Of Sets ◽  
Set Function ◽  
Primitive Recursive

AbstractLet KP− be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V ≔ universe of sets) be a Δ0-definable set function, i.e. there is a Δ0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and V ⊨ ∀x∃!yφ(x, y). In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with the collection of those functions which are Σ1-definable in KP− + Σ1-Foundation + ∀x∃!yφ(x, y). Moreover, we show that this is still true if one adds Π1-Foundation or a weak version of Δ0-Dependent Choices to the latter theory.

Varieties of involution monoids with extreme properties

2019 ◽  
Vol 70 (4) ◽  
pp. 1157-1180
Author(s):  
Edmond W H Lee
Keyword(s):  
The Other ◽  
Sufficient Condition ◽  
Finitely Based ◽  
Finitely Based Variety ◽  
Lattice Of Subvarieties ◽  
Power Set ◽  
Cross Variety ◽  
Positive Integers

Abstract A variety that contains continuum many subvarieties is said to be huge. A sufficient condition is established under which an involution monoid generates a variety that is huge by virtue of its lattice of subvarieties order-embedding the power set lattice of the positive integers. Based on this result, several examples of finite involution monoids with extreme varietal properties are exhibited. These examples—all first of their kinds—include the following: finite involution monoids that generate huge varieties but whose reduct monoids generate Cross varieties; two finite involution monoids sharing a common reduct monoid such that one generates a huge, non-finitely based variety while the other generates a Cross variety; and two finite involution monoids that generate Cross varieties, the join of which is huge.

COMPUTABLE POLISH GROUP ACTIONS

2018 ◽  
Vol 83 (2) ◽  
pp. 443-460
Author(s):  
ALEXANDER MELNIKOV ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Set Theory ◽  
Group Actions ◽  
General Setting ◽  
Structure Theory ◽  
Computable Structure ◽  
Descriptive Set Theory ◽  
Sufficient Condition ◽  
Computable Structure Theory ◽  
Image Position ◽  
Descriptive Set

AbstractUsing methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. As we will see, the usual action of ${S_\infty }$ on the space of structures in a given language is effective in a certain algorithmic sense that we need, and ${S_\infty }$ itself carries a natural computability structure (to be defined). Among other results, we give a sufficient condition for an orbit under effective ${\cal G}$-action of a computable Polish ${\cal G}$ to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.

The model of set theory generated by countably many generic reals

1981 ◽  
Vol 46 (4) ◽  
pp. 732-752 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Minimal Model ◽  
The Other ◽  
Power Set ◽  
Standard Models ◽  
Weak Axioms Of Choice

AbstractAdjoin, 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.

The equivalence of NF-style set theories with “tangled” type theories; the construction of ω-models of predicative NF (and more)

1995 ◽  
Vol 60 (1) ◽  
pp. 178-190 ◽  
Author(s):  
M. Randall Holmes
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
The Other ◽  
Natural Numbers ◽  
Consistency Proof ◽  
Natural Behaviour ◽  
New Foundations ◽  
Consistency Problem

AbstractAn ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory “New Foundations” (NF) is constructed. Marcel Crabbé has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for any ordinal α. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such “tangled type theories” in general, exhibiting a “tangled type theory” (and also an extension of Zermelo set theory with Δ0 comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that “tangled type theory with urelements” has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU.

End-extensions preserving power set

1991 ◽  
Vol 56 (1) ◽  
pp. 323-328 ◽  
Author(s):  
Thomas Forster ◽  
Richard Kaye
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Power Set ◽  
Consistency Problem

AbstractWe consider the quantifier hierarchy of Takahashi [1972] and show how it gives rise to reflection theorems for some large cardinals in ZF, a new natural subtheory of Zermelo's set theory, a potentially useful new reduction of the consistency problem for Quine's NF, and a sharpening of another reduction of this problem due to Boffa.

Set theory, philosophy of

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Set Theory ◽  
The Other ◽  
Second Stage ◽  
Other Hand ◽  
Iterative Conception ◽  
Power Set ◽  
After Effect ◽  
Almost All ◽  
The Universe ◽  
Philosophy Of Set Theory

The various attitudes that have been taken to mathematics can be split into two camps according to whether they take mathematical theorems to be true or not. Mathematicians themselves often label the former camp realist and the latter formalist. (Philosophers, on the other hand, use both these labels for more specific positions within the two camps.) Formalists have no special difficulty with set theory as opposed to any other branch of mathematics; for that reason we shall not consider their view further here. For realists, on the other hand, set theory is peculiarly intractable: it is very difficult to give an unproblematic explanation of its subject matter. The reason this difficulty is not of purely local interest is an after effect of logicism. Logicism, in the form in which Frege and Russell tried to implement it, was a two-stage project. The first stage was to embed arithmetic (Frege) or, more ambitiously, the whole of mathematics (Russell) in the theory of sets; the second was to embed this in turn in logic. The hope was that this would palm off all the philosophical problems of mathematics onto logic. The second stage is generally agreed to have failed: set theory is not part of logic. But the first stage succeeded: almost all of mathematics can be embedded in set theory. So the logicist aim of explaining mathematics in terms of logic metamorphoses into one of explaining it in terms of set theory. Various systems of set theory are available, and for most of mathematics the method of embedding is fairly insensitive to the exact system that we choose. The main exceptions to this are category theory, whose embedding is awkward if the theory chosen does not distinguish between sets and proper classes; and the theory of sets of real numbers, where there are a few arguments that depend on very strong axioms of infinity (also known as large cardinal axioms) not present in some of the standard axiomatizations of set theory. All the systems agree that sets are extensional entities, so that they satisfy the axiom of extensionality: ∀x(xЄa ≡ xЄb) → a=b. What differs between the systems is which sets they take to exist. A property F is said to be set-forming if {x:Fx} exists: the issue to be settled is which properties are set-forming and which are not. What the philosophy of set theory has to do is to provide an illuminating explanation for the various cases of existence. The most popular explanation nowadays is the so-called iterative conception of set. This conceives of sets as arranged in a hierarchy of stages (sometimes known as levels). The bottom level is a set whose members are the non-set-theoretic entities (sometimes known as Urelemente) to which the theory is intended to be applicable. (This set is often taken by mathematicians to be empty, thus restricting attention to what are known as pure sets, although this runs the danger of cutting set theory off from its intended application.) Each succeeding level is then obtained by forming the power set of the preceding one. For this conception three questions are salient: Why should there not be any sets other than these? How rich is the power-set operation? How many levels are there? An alternative explanation which was for a time popular among mathematicians is limitation of size. This is the idea that a property is set-forming provided that there are not too many objects satisfying it. How many is too many is open to debate. In order to prevent the system from being contradictory, we need only insist that the universe is too large to form a set, but this is not very informative in itself: we also need to be told how large the universe is.

On the nature of chemistry

2018 ◽  
Vol 10 (2) ◽  
pp. 74 ◽  
Author(s):  
Eric R. Scerri
Keyword(s):  
19Th Century ◽  
The Other ◽  
Recent History ◽  
History Of Chemistry ◽  
History Of ◽  
The One ◽  
Fundamental Unity ◽  
The 19Th Century

<span>The very nature of chemistry presents us with a tension. A tension between the exhilaration of diversity of substances and forms on the one hand and the safety of fundamental unity on the other. Even just the recent history of chemistry has been al1 about this tension, from the debates about Prout's hypothesis as to whether there is a primary matter in the 19th century to the more recent speculations as to whether computers will enable us to virtually dispense with experimental chemistry.</span>

La Nueva gramática de la lengua castellana de martínez de noboa

1997 ◽  
Vol 24 (1-2) ◽  
pp. 115-138
Author(s):  
Marina Maquieira
Keyword(s):  
19Th Century ◽  
The Other ◽  
The 19Th Century ◽  
Grammatical Tradition ◽  
Philosophical Grammar

Summary This paper examines a treatise on Spanish grammar, i.e., a particular grammar which follows the tradition of French philosophical grammar. Bachiller D. Antonio Martínez de Noboa’s work, published in 1839, appears in a century when the Spanish grammatical tradition is at its best. Texts like Vicente Salvá’s (1786–1849) and of course Andrés Bello’s (1781–1865) have in recent years attracted the attention of researchers. However, Martínez de Noboa’s work is much less known, although Gómez Asencio (1981, 1985) did highlight its importance in his two indispensable studies of the period between 1771 and 1847. The Nueva Gramática de la lengua Castellana is indebted to the framework set by José Gómez de Hermosilla (1835) and Jacobo Saqueniza (1828), although it does include some original observations. This paper examines the structure of the work in question and aims to show how it is in global terms a unified text combining different aspects, of which the most striking is without doubt the syntactic one. With this aim in mind certain specific examples of the analogy pertaining to syntax have been studied. First those he himself highlighted, e.g., the article/pronoun and verb and then those comments on syntax which are logically pertinent, e.g., conjunctions. Noboa himself was cited as was Saqueniza as having been responsible for the introduction of distinction between coordinate and subordinate conjunctions in Spanish grammar, along with the distinction between simple and complex clauses. On the purely syntactic level, it was also Noboa who refined the whole notion of verbal government. Finally, there is a brief summary of the section dedicated to pronunciation and spelling which are also considered by the author to be in some way related to the other parts of the grammar. In sum, what makes this work particularly interesting is undoubtedly the emphasis on syntax as more studies had been carried out on morphology than in any other area up until the 19th century and continued after Noboa to monopolise questions concerning grammar throughout this century.

The consistency problem for positive comprehension principles

1989 ◽  
Vol 54 (4) ◽  
pp. 1401-1418 ◽  
Author(s):  
M. Forti ◽  
R. Hinnion
Keyword(s):  
Set Theory ◽  
Difficult Problem ◽  
Large Cardinal ◽  
Positive Theory ◽  
Construction Principle ◽  
Short Summary ◽  
Free Construction ◽  
Unpublished Thesis ◽  
Consistency Problem ◽  
Original Construction

Since Gilmore showed that some theory with a positive comprehension scheme is consistent when the axiom of extensionality is dropped and inconsistent with it (see [1] and [2]), the problem of the consistency of various positive comprehension schemes has been investigated. We give here a short classification, which shows clearly the importance of the axiom of extensionality and of the abstraction operator in these consistency problems. The most difficult problem was to show the consistency of the comprehension scheme for positive formulas, with extensionality but without abstraction operator. In his unpublished thesis, Set theory in which the axiom of foundation fails [3], Malitz solved partially this problem but he needed to assume the existence of some unusual kind of large cardinal; as his original construction is very interesting and his thesis is unpublished, we give a short summary of it. M. Forti solved the problem completely by working in ZF with a free-construction principle (sometimes called an anti-foundation axiom), instead of ZF with the axiom of foundation, as Malitz did.This permits one to obtain the consistency of this positive theory, relative to ZF. In his general investigations about “topological set theories” (to be published), E. Weydert has independently proved the same result. The authors are grateful to the Mathematisches Forshungsinstitut Oberwolfach for giving them the opportunity of discussing these subjects and meeting E. Weydert during the meeting “New Foundations”, March 1–7, 1987.

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