The Problem of Conceptual Understanding

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Real Number ◽  
Set Theory ◽  
Conceptual Understanding ◽  
Philosophy Of Mathematics ◽  
Axiom System ◽  
The Real ◽  
Iterative Conception ◽  
Universe Of Sets ◽  
The Universe ◽  
The Web

This chapter considers one of the most intriguing questions that philosophy of mathematics in practice must, sooner or later, confront: how understanding of mathematics is obtained. In particular, it examines how issues of meaning and understanding in relation to practice and use relate to the question of the acceptability of “classical” or postulational mathematics, a question usually formulated in terms of consistency. The chapter begins with a discussion of the iterative conception of the universe of sets and its presuppositions, analyzing it from the standpoint of the web of practices. It then addresses the issue of conceptual understanding in mathematics, as exemplifid by the theory Zermelo–Fraenkel axiom system (ZFC). Finally, it looks at arguments based on the idea of the real-number continuum as a source of justification for the axioms of set theory.

scholarly journals A New Set Theory for Analysis

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 31
Author(s):  
Juan Ramírez
Keyword(s):  
Real Number ◽  
Number System ◽  
Order Structure ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Power Set ◽  
Universe Of Sets ◽  
The Universe ◽  
Real Number System

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

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.

Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Set Theory ◽  
Mathematical Practice ◽  
Axiom Of Choice ◽  
Small Fragment ◽  
Real Numbers ◽  
The Real ◽  
Classical Construction

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

Natural internal forcing schemata extending ZFC: Truth in the universe?

1994 ◽  
Vol 59 (2) ◽  
pp. 461-472
Author(s):  
Garvin Melles
Keyword(s):  
Set Theory ◽  
Mathematical Truth ◽  
Unified Theory ◽  
Rule Of Thumb ◽  
Guiding Principle ◽  
Survey Article ◽  
Inner Models ◽  
Independence Results ◽  
Universe Of Sets ◽  
The Universe

Mathematicians have one over on the physicists in that they already have a unified theory of mathematics, namely, set theory. Unfortunately, the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, ZFC? This author does not think so, he feels that set theorists' intuition about the universe of sets is stronger than ZFC. Here in this paper, using part of this intuition, we introduce some axiom schemata which we feel are very natural candidates for being considered as part of the axioms of set theory. These schemata assert the existence of many generics over simple inner models. The main purpose of this article is to present arguments for why the assertion of the existence of such generics belongs to the axioms of set theory.Our central guiding principle in justifying the axioms is what Maddy called the rule of thumb maximize in her survey article on the axioms of set theory, [8] and [9]. More specifically, our intuition conforms with that expressed by Mathias in his article What is Maclane Missing? challenging Mac Lane's view of set theory.

Inner models for set theory – Part III

1953 ◽  
Vol 18 (2) ◽  
pp. 145-167 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Original System ◽  
Axiom System ◽  
Complete Model ◽  
Universal Class ◽  
Inner Models ◽  
Consistency Results ◽  
Image Position ◽  
The Universe ◽  
New Hypothesis

In this third and last paper on inner models we consider some of the inherent limitations of the method of using inner models of the type defined in 1.2 for the proof of consistency results for the particular system of set theory under consideration. Roughly speaking this limitation may be described by saying that practically no further consistency results can be obtained by the construction of models satisfying the conditions of theorem 1.5, i.e., conditions 1.31, 1.32, 1.33, 1.51, viz.:This applies in particular to the ‘complete models’ defined in 1.4. Before going on to a precise statement of these limitations we shall consider now the theorem on which they depend. This is concerned with a particular type of complete model examples of which we call “proper complete models”; they are those complete models which are essentially interior to the universe, those whose classes are sets of the universe constituting a class thereof, i.e., those for which the following proposition is true:The main theorem of this paper is that the statement that there are no models of this kind can be expressed formally in the same way as the axioms A, B, C and furthermore it can be proved that if the axiom system A, B, C is consistent then so is the system consisting of axioms A, B, C, plus this new hypothesis that there exist no proper complete models. When combined with the axiom ‘V = L’ introduced by Gödel in (1) this new hypothesis yields a system in which any normal complete model which exists has for its universal class V, the universal class of the original system.

scholarly journals AN ORDINAL ANALYSIS OF ADMISSIBLE SET THEORY USING RECURSION ON ORDINAL NOTATIONS

2002 ◽  
Vol 02 (01) ◽  
pp. 91-112 ◽  
Author(s):  
JEREMY AVIGAD
Keyword(s):  
Set Theory ◽  
Proof Theory ◽  
Admissible Set ◽  
Ordinal Analysis ◽  
Rate Of Growth ◽  
Ordinal Notations ◽  
Universe Of Sets ◽  
The Universe

The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal analysis.

A quasi-intumonistic set theory

1971 ◽  
Vol 36 (3) ◽  
pp. 456-460 ◽  
Author(s):  
Leslie H. Tharp
Keyword(s):  
Set Theory ◽  
Obvious Candidate ◽  
Universe Of Sets ◽  
Basic Philosophy ◽  
The Universe

It is natural, given the usual iterative description of the universe of sets, to investigate set theories which in some way take account of the unfinished character of the universe. We do not here consider any arguments aimed at justifying one system over another, or at clarifying the basic philosophy. Rather, we look at an obvious candidate which is similar to a system discussed by L. Pozsgay in [1]. Pozsgay sketched the development of the ordinary theorems in such a system and attempted to show it equiconsistent with ZF. In this paper we show that the consistency of the system we call IZF can be proved in the usual ZF set theory.

Gödel and Set Theory

2007 ◽  
Vol 13 (2) ◽  
pp. 153-188 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Order Logic ◽  
Point Of View ◽  
First Order ◽  
Kurt Gödel ◽  
The Axiom Of Choice ◽  
Universe Of Sets ◽  
The Universe ◽  
The Continuum

Kurt Gödel (1906–1978) with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was.

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.

Platonism and Mathematical Intuition in Kurt Gödel's Thought

1995 ◽  
Vol 1 (1) ◽  
pp. 44-74 ◽  
Author(s):  
Charles Parsons
Keyword(s):  
Set Theory ◽  
Real World ◽  
Mathematical Knowledge ◽  
Philosophy Of Mathematics ◽  
Foundations Of Mathematics ◽  
Mathematical Objects ◽  
The Real ◽  
Robust Realism ◽  
Mathematical Intuition

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish.

Sign in / Sign up

Export Citation Format

Share Document