Transcendence of cardinals

1965 ◽  
Vol 30 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Power Set ◽  
Recursive Predicate

In this paper, by a function of ordinals we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined recursive functions and predicates of ordinals, following Kleene's definition on natural numbers. A predicate will be called arithmetical, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called arithmetical, if its representing predicate is arithmetical.The cardinals are identified with those ordinals a which have larger power than all smaller ordinals than a. For any given ordinal a, we denote by the cardinal of a and by 2a the cardinal which is of the same power as the power set of a. Let χ be the function such that χ(a) is the least cardinal which is greater than a.Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; χ is such a function. If we define χ in making use of only the language on ordinals, it seems necessary to use the notion of all the functions from ordinals, e.g., as in [6].

Frege's theorem in a constructive setting

1999 ◽  
Vol 64 (2) ◽  
pp. 486-488 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
The Family ◽  
Power Set ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Image Position ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the conditionthen E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the followingTheorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:(i) ø ϵdom(υ)(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.Then we can define a subset N of E which is the domain of a model of Peano's axioms.

Continuum hypothesis

2018 ◽  
Author(s):  
Mary Tiles
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Natural Numbers ◽  
Generalized Continuum ◽  
Power Set ◽  
Independence Results ◽  
Infinite Set ◽  
Special Case ◽  
Insight Into ◽  
The Continuum

The ‘continuum hypothesis’ (CH) asserts that there is no set intermediate in cardinality (‘size’) between the set of real numbers (the ‘continuum’) and the set of natural numbers. Since the continuum can be shown to have the same cardinality as the power set (that is, the set of subsets) of the natural numbers, CH is a special case of the ‘generalized continuum hypothesis’ (GCH), which says that for any infinite set, there is no set intermediate in cardinality between it and its power set. Cantor first proposed CH believing it to be true, but, despite persistent efforts, failed to prove it. König proved that the cardinality of the continuum cannot be the sum of denumerably many smaller cardinals, and it has been shown that this is the only restriction the accepted axioms of set theory place on its cardinality. Gödel showed that CH was consistent with these axioms and Cohen that its negation was. Together these results prove the independence of CH from the accepted axioms. Cantor proposed CH in the context of seeking to answer the question ‘What is the identifying nature of continuity?’. These independence results show that, whatever else has been gained from the introduction of transfinite set theory – including greater insight into the import of CH – it has not provided a basis for finally answering this question. This remains the case even when the axioms are supplemented in various plausible ways.

PRIMITIVE INDEPENDENCE RESULTS

2003 ◽  
Vol 03 (01) ◽  
pp. 67-83
Author(s):  
HARVEY M. FRIEDMAN
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Power Set ◽  
Independence Results

We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality (see Secs. 4 and 5). They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC (see Sec. 6). It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without power set (see [4]).

Universal recursion theoretic properties of r.e. preordered structures

1985 ◽  
Vol 50 (2) ◽  
pp. 397-406 ◽  
Author(s):  
Franco Montagna ◽  
Andrea Sorbi
Keyword(s):  
Equivalence Relation ◽  
Point Of View ◽  
Main Concern ◽  
Natural Numbers ◽  
Axiomatic Theory ◽  
Recursive Functions ◽  
Theoretic Point

When dealing with axiomatic theories from a recursion-theoretic point of view, the notion of r.e. preordering naturally arises. We agree that an r.e. preorder is a pair = 〈P, ≤P〉 such that P is an r.e. subset of the set of natural numbers (denoted by ω), ≤P is a preordering on P and the set {〈;x, y〉: x ≤Py} is r.e.. Indeed, if is an axiomatic theory, the provable implication of yields a preordering on the class of (Gödel numbers of) formulas of .Of course, if ≤P is a preordering on P, then it yields an equivalence relation ~P on P, by simply letting x ~Py iff x ≤Py and y ≤Px. Hence, in the case of P = ω, any preordering yields an equivalence relation on ω and consequently a numeration in the sense of [4]. It is also clear that any equivalence relation on ω (hence any numeration) can be regarded as a preordering on ω. In view of this connection, we sometimes apply to the theory of preorders some of the concepts from the theory of numerations (see also Eršov [6]).Our main concern will be in applications of these concepts to logic, in particular as regards sufficiently strong axiomatic theories (essentially the ones in which recursive functions are representable). From this point of view it seems to be of some interest to study some remarkable prelattices and Boolean prealgebras which arise from such theories. It turns out that these structures enjoy some rather surprising lattice-theoretic and universal recursion-theoretic properties.After making our main definitions in §1, we examine universal recursion-theoretic properties of some r.e. prelattices in §2.

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

Epistemic set theory is a conservative extension of intuitionistic set theory

1985 ◽  
Vol 50 (4) ◽  
pp. 895-902 ◽  
Author(s):  
R. C. Flagg
Keyword(s):  
Set Theory ◽  
Affirmative Answer ◽  
Conservative Extension ◽  
Successful Program ◽  
Power Set ◽  
Epistemic Notion ◽  
Intuitionistic Mathematics ◽  
Topological Boolean Algebra ◽  
Faithful Interpretation ◽  
Intuitionistic Set Theory

In [6] Gödel observed that intuitionistic propositional logic can be interpreted in Lewis's modal logic (S4). The idea behind this interpretation is to regard the modal operator □ as expressing the epistemic notion of “informal provability”. With the work of Shapiro [12], Myhill [10], Goodman [7], [8], and Ščedrov [11] this simple idea has developed into a successful program of integrating classical and intuitionistic mathematics.There is one question quite central to the above program that has remained open. Namely:Does Ščedrov's extension of the Gödel translation to set theory provide a faithful interpretation of intuitionistic set theory into epistemic set theory?In the present paper we give an affirmative answer to this question.The main ingredient in our proof is the construction of an interpretation of epistemic set theory into intuitionistic set theory which is inverse to the Gödel translation. This is accomplished in two steps. First we observe that Funayama's theorem is constructively provable and apply it to the power set of 1. This provides an embedding of the set of propositions into a complete topological Boolean algebra . Second, in a fashion completely analogous to the construction of Boolean-valued models of classical set theory, we define the -valued universe V(). V() gives a model of epistemic set theory and, since we use a constructive metatheory, this provides an interpretation of epistemic set theory into intuitionistic set theory.

Certain predicates defined by induction schemata

1953 ◽  
Vol 18 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Formal System ◽  
General Notion ◽  
Constant Number ◽  
Natural Numbers ◽  
System L ◽  
Truth Definition ◽  
Consistency Proofs ◽  
Special Case

It is known that we can introduce in number theory (for example, the system Z of Hilbert-Bernays) by induction schemata certain predicates of natural numbers which cannot be expressed explicitly within the framework of number theory. The question arises how we can define these predicates in some richer system, without employing induction schemata. In this paper a general notion of definability by induction (relative to number theory), which seems to apply to all the known predicates of this kind, is introduced; and it is proved that in a system L1 which forms an extension of number theory all predicates which are definable by induction (hereafter to be abbreviated d.i.) according to the definition are explicitly expressible.In order to define such predicates and prove theorems answering to their induction schemata, we have to allow certain impredicative classes in L1. However, if we want merely to prove that for each constant number the special case of the induction schema for a predicate d.i. is provable, we do not have to assume the existence of impredicative classes. A certain weaker system L2, in which only predicative classes of natural numbers are allowed, is sufficient for the purpose. It is noted that a truth definition for number theory can be obtained in L2. Consistency proofs for number theory do not seem to be formalizable in L2, although they can, it is observed, be formalized in L1.In general, given any ordinary formal system (say Zermelo set theory), it is possible to define by induction schemata, in the same manner as in number theory, certain predicates which are not explicitly definable in the system. Here again, by extending the system in an analogous fashion, these predicates become expressible in the resulting system. The crucial predicate instrumental to obtaining a truth definition for a given system is taken as an example.

On undecidable statements in enlarged systems of logic and the concept of truth

1939 ◽  
Vol 4 (3) ◽  
pp. 105-112 ◽  
Author(s):  
Alfred Tarski
Keyword(s):  
Set Theory ◽  
Modus Ponens ◽  
Logical System ◽  
Natural Numbers ◽  
System L ◽  
Concept Of Truth ◽  
Made In

It is my intention in this paper to add somewhat to the observations already made in my earlier publications on the existence of undecidable statements in systems of logic possessing rules of inference of a “non-finitary” (“non-constructive”) character (§§1–4).I also wish to emphasize once more the part played by the concept of truth in relation to problems of this nature (§§5–8).At the end of this paper I shall give a result which was not touched upon in my earlier publications. It seems to be of interest for the reason (among others) that it is an example of a result obtained by a fruitful combination of the method of constructing undecidable statements (due to K. Gödel) with the results obtained in the theory of truth.1. Let us consider a formalized logical system L. Without giving a detailed description of the system we shall assume that it possesses the usual “finitary” (“constructive”) rules of inference, such as the rule of substitution and the rule of detachment (modus ponens), and that among the laws of the system are included all the postulates of the calculus of statements, and finally that the laws of the system suffice for the construction of the arithmetic of natural numbers. Moreover, the system L may be based upon the theory of types and so be the result of some formalization of Principia mathematica. It may also be a system which is independent of any theory of types and resembles Zermelo's set theory.

scholarly journals Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 86 ◽  
Author(s):  
Dmitri Shakhmatov ◽  
Víctor Yañez
Keyword(s):  
Topological Group ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Natural Numbers ◽  
Compactness Property ◽  
Free Ultrafilter ◽  
The Axiom Of Choice

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence ( U n ) of non-empty open subsets of G, one can choose a point x n ∈ U n for all n ∈ N in such a way that the resulting sequence ( x n ) has a p-limit in G; that is, { n ∈ N : x n ∈ V } ∈ p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo- ω -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⨁ i ∈ I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets.

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