scholarly journals Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1429
Author(s):  
Torsten Asselmeyer-Maluga ◽  
Jerzy Król
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Category Theory ◽  
Internal Symmetry ◽  
Smooth Manifolds ◽  
Natural Numbers ◽  
Local Invariance ◽  
Geometric Morphisms

Category theory allows one to treat logic and set theory as internal to certain categories. What is internal to SET is 2-valued logic with classical Zermelo–Fraenkel set theory, while for general toposes it is typically intuitionistic logic and set theory. We extend symmetries of smooth manifolds with atlases defined in Set towards atlases with some of their local maps in a topos T . In the case of the Basel topos and R 4 , the local invariance with respect to the corresponding atlases implies exotic smoothness on R 4 . The smoothness structures do not refer directly to Casson handless or handle decompositions, which may be potentially useful for describing the so far merely putative exotic R 4 underlying an exotic S 4 (should it exist). The tovariance principle claims that (physical) theories should be invariant with respect to the choice of topos with natural numbers object and geometric morphisms changing the toposes. We show that the local T -invariance breaks tovariance even in the weaker sense.

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].

Categories: From Zero to Infinity

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Pierre Schapira
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
And Topology

Set theory, category theory, and topology. Pierre Schapira explores the concept of identity within category theory, and what it means for the properties to be satisfied only up to homotopy.

Category theory, introduction to

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Category Theory ◽  
Standard Set ◽  
Universe Of Sets ◽  
The Universe

A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets. This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard 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.

Some properties of epistemic set theory with collection

1986 ◽  
Vol 51 (3) ◽  
pp. 748-754 ◽  
Author(s):  
Andre Scedrov
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Existential Quantifier ◽  
Modal Interpretation ◽  
Conservative Extension ◽  
Disjunction Property ◽  
First Order ◽  
Slight Extension ◽  
Order Arithmetic ◽  
Epistemic Systems

Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal S4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (∃x) □ …. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of Gödel's modal interpretation [6] of intuitionistic logic.Myhill showed that whenever a sentence □A ∨ □B is provable in IST, then A is provable in IST or B is provable in IST (the disjunction property), and that whenever a sentence ∃x.□A(x) is provable in IST, then so is A(t) for some closed term t (the existence property). He adapted the Friedman slash [4] to epistemic systems.Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFER, cf. §1) suffices to interpret intuitionistic ZF set theory with Replacement (ZFIR, [10]). This is obtained by extending Gödel's modal interpretation [6] of intuitionistic logic. ZFER still had the properties of Goodman's system mentioned above.

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