Objectivity in Mathematical Knowledge

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Mathematical Knowledge ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Real Numbers ◽  
Basic Notion ◽  
Mathematical Practices ◽  
Ordinal Numbers ◽  
The Axiom Of Choice ◽  
The Web ◽  
The Continuum

This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new hypotheses are embedded in the web of mathematical practices, they become systematically linked with previous strata of mathematical knowledge, and this forces upon us agents (for example, research mathematicians or students of math) certain results, be they principles or conclusions. The chapter first considers a simple case that illustrates objective features in the introduction of basic mathematical hypotheses. It then discusses Georg Cantor's “purely arithmetical” proofs of his set-theoretic results, along with the notion of arbitrary set in relation to the Axiom of Choice that has strong roots in the theory of real numbers. It also explores Cantor's ordinal numbers and the Continuum Hypothesis.

scholarly journals On Arbitrary sets andZFC

2011 ◽  
Vol 17 (3) ◽  
pp. 361-393 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Real Numbers ◽  
The Real ◽  
Formal Systems ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
The Continuum

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

The constructible universe

2018 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Constructible Sets ◽  
Kurt Gödel ◽  
The Axiom Of Choice ◽  
The Universe ◽  
The Continuum

the ‘universe’ of constructible sets was introduced by Kurt Gödel in order to prove the consistency of the axiom of choice (AC) and the continuum hypothesis (CH) with the basic (ZF) axioms of set theory. The hypothesis that all sets are constructible is the axiom of constructibility (V = L). Gödel showed that if ZF is consistent, then ZF + V = L is consistent, and that AC and CH are provable in ZF + V = L.

Forcing

2018 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Forcing Axioms ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Axiom Of Choice ◽  
The Continuum

The method of forcing was introduced by Paul J. Cohen in order to prove the independence of the axiom of choice (AC) from the basic (ZF) axioms of set theory, and of the continuum hypothesis (CH) from the accepted axioms (ZFC = ZF + AC) of set theory (see set theory, axiom of choice, continuum hypothesis). Given a model M of ZF and a certain P∈M, it produces a ‘generic’ G⊆P and a model N of ZF with M⊆N and G∈N. By suitably choosing P, N can be ‘forced’ to be or not be a model of various hypotheses, which are thus shown to be consistent with or independent of the axioms. This method of proving undecidability has been very widely applied. The method has also motivated the proposal of new so-called forcing axioms to decide what is otherwise undecidable, the most important being that called Martin’s axiom (MA).

Some sets with measurable inverses

1969 ◽  
Vol 65 (2) ◽  
pp. 437-438
Author(s):  
Roy O. Davies
Keyword(s):  
Real Function ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Borel Set ◽  
Analytic Set ◽  
Linear Set ◽  
Measurable Set ◽  
Intermediate Results ◽  
The Axiom Of Choice ◽  
The Continuum

Goldman (4) conjectured that if Z is a linear set having the property that for every (Lebesgue) measurable real function f the set f−1[Z] is a measurable set, then Z must be a Borel set. I pointed out (2) that any analytic non-Borel set provides a counterexample, and Eggleston(3) showed that a set can have the property but be neither analytic nor even an analytic complement, for example, any Luzin set. As Eggleston mentions, in the construction of Luzin sets the continuum hypothesis is assumed (compare Sierpiński(6), Chapter II), and the question arises whether it can be dispensed with in his theorem. We shall show that a non-analytic set having Goldman's property can be constructed with the help of the axiom of choice alone, without the continuum hypothesis; the problem for analytic complements remains open. We shall also generalize one of Eggleston's intermediate results.

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