On Ackermann's set theory

Let l(u)⊃ |G|. A central problem in higher non-linear graph theoryis the construction of projective numbers. We show that Recent developments in axiomatic set theory [6] have raised the questionof whetherEis not dominated byl. On the other hand, the work in [6, 24] did not consider the hyper-real case.

A system of axiomatic set theory. Part IV. General set theory

Journal of Symbolic Logic ◽

10.2307/2268110 ◽

1942 ◽

Vol 7 (4) ◽

pp. 133-145 ◽

Cited By ~ 12

Author(s):

Paul Bernays

Keyword(s):

Set Theory ◽

Axiom Of Choice ◽

Point Of View ◽

The Other ◽

Von Neumann ◽

Neumann System ◽

Other Hand ◽

The Axiom Of Choice ◽

Axiomatic Set Theory

Our task in the treatment of general set theory will be to give a survey for the purpose of characterizing the different stages and the principal theorems with respect to their axiomatic requirements from the point of view of our system of axioms. The delimitation of “general set theory” which we have in view differs from that of Fraenkel's general set theory, and also from that of “standard logic” as understood by most logicians. It is adapted rather to the tendency of von Neumann's system of set theory—the von Neumann system having been the first in which the possibility appeared of separating the assumptions which are required for the conceptual formations from those which lead to the Cantor hierarchy of powers. Thus our intention is to obtain general set theory without use of the axioms V d, V c, VI.It will also be desirable to separate those proofs which can be made without the axiom of choice, and in doing this we shall have to use the axiom V*—i.e., the theorem of replacement taken as an axiom. From V*, as we saw in §4, we can immediately derive V a and V b as theorems, and also the theorem that a function whose domain is represented by a set is itself represented by a functional set; and on the other hand V* was found to be derivable from V a and V b in combination with the axiom of choice. (These statements on deducibility are of course all on the basis of the axioms I–III.)

Everything, more or less

10.1093/oso/9780198719649.001.0001 ◽

2019 ◽

Cited By ~ 2

Author(s):

J. P. Studd

Keyword(s):

Set Theory ◽

Philosophy Of Language ◽

Ad Hoc ◽

The Other ◽

Foundations Of Mathematics ◽

Russell’S Paradox ◽

Other Hand ◽

Absolute Generality ◽

Russell's Paradox ◽

The Way

Almost no systematic theorizing is generality-free. Scientists test general hypotheses; set theorists prove theorems about every set; metaphysicians espouse theses about all things regardless of their kind. But how general can we be? Do we ever succeed in theorizing about ABSOLUTELY EVERYTHING in some interestingly final, all-caps-worthy sense of ‘absolutely everything’? Not according to generality relativism. In its most promising form, this kind of relativism maintains that what ‘everything’ and other quantifiers encompass is always open to expansion: no matter how broadly we may generalize, a more inclusive ‘everything’ is always available. The importance of the issue comes out, in part, in relation to the foundations of mathematics. Generality relativism opens the way to avoid Russell’s paradox without imposing ad hoc limitations on which pluralities of items may be encoded as a set. On the other hand, generality relativism faces numerous challenges: What are we to make of seemingly absolutely general theories? What prevents our achieving absolute generality simply by using ‘everything’ unrestrictedly? How are we to characterize relativism without making use of exactly the kind of generality this view foreswears? This book offers a sustained defence of generality relativism that seeks to answer these challenges. Along the way, the contemporary absolute generality debate is traced through diverse issues in metaphysics, logic, and the philosophy of language; some of the key works that lie behind the debate are reassessed; an accessible introduction is given to the relevant mathematics; and a relativist-friendly motivation for Zermelo–Fraenkel set theory is developed.

The κ-closed unbounded filter and supercpmpact cardinals

Journal of Symbolic Logic ◽

10.2307/2273253 ◽

1981 ◽

Vol 46 (1) ◽

pp. 31-40

Author(s):

Mitchell Spector

Keyword(s):

Set Theory ◽

A Priori ◽

Large Cardinals ◽

Large Cardinal ◽

Convincing Evidence ◽

The Other ◽

Other Hand ◽

Time Period ◽

The One ◽

The Universe

The consistency of the Axiom of Determinateness (AD) poses a somewhat problematic question for set theorists. On the one hand, many mathematicians have studied AD, and none has yet derived a contradiction. Moreover, the consequences of AD which have been proven form an extensive and beautiful theory. (See [5] and [6], for example.) On the other hand, many extremely weird propositions follow from AD; these results indicate that AD is not an axiom which we can justify as intuitively true, a priori or by reason of its consequences, and we thus cannot add it to our set theory (as an accepted axiom, evidently true in the cumulative hierarchy of sets). Moreover, these results place doubt on the very consistency of AD. The failure of set theorists to show AD inconsistent over as short a time period as fifteen years can only be regarded as inconclusive, although encouraging, evidence.On the contrary, there is a great deal of rather convincing evidence that the existence of various large cardinals is not only consistent but actually true in the universe of all sets. Thus it becomes of interest to see which consequences of AD can be proven consistent relative to the consistency of ZFC + the existence of some large cardinal. Earlier theorems with this motivation are those of Bull and Kleinberg [2] and Spector ([14]; see also [12], [13]).

Some proofs of independence in axiomatic set theory

Journal of Symbolic Logic ◽

10.2307/2269104 ◽

1956 ◽

Vol 21 (3) ◽

pp. 291-303 ◽

Cited By ~ 3

Author(s):

Elliott Mendelson

Keyword(s):

Set Theory ◽

The Other ◽

Formal Systems ◽

System A ◽

Similar Proof ◽

Independence Results ◽

The One ◽

Axiomatic Set Theory ◽

Gödel’S Theorem

1. Gödel's theorem that sufficiently strong formal systems cannot prove their own consistency and Tarski's method for constructing truth-definitions can be combined to give several independence results in axiomatic set theory. In substance, the following theorems can be obtained: (a) The existence of inaccessible ordinals is not provable from the axioms of set theory, if these axioms are consistent, (b) The axiom of infinity is independent of the other axioms, if these other axioms are consistent, (c) The axiom of replacement is independent of the other axioms, if these other axioms are consistent. In all cases, V = L will be included as an axiom.The result (a) concerning inaccessible ordinals already has been proved in Shepherdson [10] and Mostowski [6], but their proofs are somewhat different from the one given here. According to Mostowski [6], Kuratowski essentially had a proof of (a) in 1924. Propositions (b) and (c) have been proved, for axiomatic set theory without the axiom V = L, by Bernays [1] pp. 65–69. The method of proof used in this paper is due to Firestone and Rosser [2].An outline of a similar proof along these lines is given in Rosser [7] pp. 60–62.As our system G of set theory, we choose Gödel's system A, B, C, as given in [4], except for the following changes.

On the paradox of grounded classes

Journal of Symbolic Logic ◽

10.2307/2266899 ◽

1955 ◽

Vol 20 (2) ◽

pp. 140-140 ◽

Cited By ~ 5

Author(s):

Richard Montague

Keyword(s):

Natural Number ◽

Set Theory ◽

Infinite Sequence ◽

Axiom Of Choice ◽

The Other ◽

Other Hand ◽

The Family ◽

The One ◽

The Axiom Of Choice ◽

Regular Classes

Mr. Shen Yuting, in this Journal, vol. 18, no. 2 (June, 1953), stated a new paradox of intuitive set-theory. This paradox involves what Mr. Yuting calls the class of all grounded classes, that is, the family of all classes a for which there is no infinite sequence b such that … ϵ bn ϵ … ϵ b2ϵb1 ϵ a.Now it is possible to state this paradox without employing any complex set-theoretical notions (like those of a natural number or an infinite sequence). For let a class x be called regular if and only if (k)(x ϵ k ⊃ (∃y)(y ϵ k · ~(∃z)(z ϵ k · z ϵ y))). Let Reg be the class of all regular classes. I shall show that Reg is neither regular nor non-regular.Suppose, on the one hand, that Reg is regular. Then Reg ϵ Reg. Now Reg ϵ ẑ(z = Reg). Therefore, since Reg is regular, there is a y such that y ϵ ẑ(z = Reg) · ~(∃z)(z ϵ z(z = Reg) · z ϵ y). Hence ~(∃z)(z ϵ ẑ(z = Reg) · z ϵ Reg). But there is a z (namely Reg) such that z ϵ ẑ(z = Reg) · z ϵ Reg.On the other hand, suppose that Reg is not regular. Then, for some k, Reg ϵ k · [1] (y)(y ϵ k ⊃ (∃z)(z ϵ k · z ϵ y)). It follows that, for some z, z ϵ k · z ϵ Reg. But this implies that (ϵy)(y ϵ k · ~(ϵw)(w ϵ k · w ϵ y)), which contradicts [1].It can easily be shown, with the aid of the axiom of choice, that the regular classes are just Mr. Yuting's grounded classes.

Set theory, philosophy of

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y092-1 ◽

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.

The General Idea Behind Gödel’s Proof

Gödel's Incompleteness Theorems ◽

10.1093/oso/9780195046724.003.0004 ◽

1992 ◽

Author(s):

Raymond M. Smullyan

Keyword(s):

Number Theory ◽

Set Theory ◽

The Other ◽

General Idea ◽

Original Proof ◽

Extensive Class ◽

The One ◽

Axiomatic Set Theory ◽

Gödel’S Theorem ◽

Mathematical Systems

In the next several chapters we will be studying incompleteness proofs for various axiomatizations of arithmetic. Gödel, 1931, carried out his original proof for axiomatic set theory, but the method is equally applicable to axiomatic number theory. The incompleteness of axiomatic number theory is actually a stronger result since it easily yields the incompleteness of axiomatic set theory. Gödel begins his memorable paper with the following startling words. . . . “The development of mathematics in the direction of greater precision has led to large areas of it being formalized, so that proofs can be carried out according to a few mechanical rules. The most comprehensive formal systems to date are, on the one hand, the Principia Mathematica of Whitehead and Russell and, on the other hand, the Zermelo-Fraenkel system of axiomatic set theory. Both systems are so extensive that all methods of proof used in mathematics today can be formalized in them—i.e. can be reduced to a few axioms and rules of inference. It would seem reasonable, therefore, to surmise that these axioms and rules of inference are sufficient to decide all mathematical questions which can be formulated in the system concerned. In what follows it will be shown that this is not the case, but rather that, in both of the cited systems, there exist relatively simple problems of the theory of ordinary whole numbers which cannot be decided on the basis of the axioms.” . . . Gödel then goes on to explain that the situation does not depend on the special nature of the two systems under consideration but holds for an extensive class of mathematical systems. Just what is this “extensive class” of mathematical systems? Various interpretations of this phrase have been given, and Gödel’s theorem has accordingly been generalized in several ways. We will consider many such generalizations in the course of this volume. Curiously enough, one of the generalizations that is most direct and most easily accessible to the general reader is also the one that appears to be the least well known.

A study of cometary eruptions through OH radio-lines

International Astronomical Union Colloquium ◽

10.1017/s025292110003150x ◽

1999 ◽

Vol 173 ◽

pp. 249-254

Author(s):

A.M. Silva ◽

R.D. Miró

Keyword(s):

Short Duration ◽

Growth Curves ◽

The Other ◽

Initial Mass ◽

Radio Lines ◽

Other Hand ◽

Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100061938 ◽

1969 ◽

Vol 27 ◽

pp. 6-7

Author(s):

A. V. Crewe

Keyword(s):

Normal Form ◽

The Other ◽

Resolving Power ◽

Scanning Microscope ◽

Conventional Microscope ◽

Other Hand ◽

Closing The Gap ◽

Thermal Sources ◽

Better Than ◽

Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.