Proefschrift--Universitaat Leiden.

Proefschrift--Universitaat Leiden.

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In this lecture we introduce a new transfinite dimension function for metric spaces which utilizes Henderson's topological D-dimension and ascribes to any metric space either an ordinal number or the symbol Ω. The construction of our function is motivated by that of Urbański's transfinite Hausdorff dimension, tHD. Henderson's dimension is a topological invariant, however, like Hausdorff dimension and tHD the function presented will be invariant under bi-Lipschitz continuous maps and generally not under homeomorphisms. We present some original results on D-dimension and build the general theory for the D-variant of transfinite Hausdorff dimension, \mathrm{t}_D\mathrm{HD}. In particular, we will show for any ordinal number α, existence of a metrizable space which has \mathrm{t}_D\mathrm{HD} greater than or equal to α and less than or equal to \omega_\tau, where τ is the least ordinal which satisfies α < \omega_\tau.

The purpose of this thesis is to study the equivalence relation between sets A and B: A o B if and only if there exists a one to one function f from A onto B. In Chapter I, some of the fundamental properties of the equivalence relation are derived. Certain basic results on countable and uncountable sets are given. In Chapter II, a number of theorems on equivalent sets are proved and Dedekind's definitions of finite and infinite are compared with the ordinary concepts of finite and infinite. The Bernstein Theorem is studied and three different proofs of it are given. In Chapter III, the concept of cardinal number is introduced by means of two axioms of A. Tarski, and some fundamental theorems on cardinal arithmetic are proved.

Thesis advisor: Patrick Byrne / In the late 19th century, Georg Cantor opened up the mathematical field of set theory with his development of transfinite numbers. In his radical departure from previous notions of infinity espoused by both mathematicians and philosophers, Cantor created new notions of transcendence in order to clearly described infinities of different sizes. Leading the opposition against Cantor's theory was Leopold Kronecker, Cantor's former mentor and the leading contemporary German mathematician. In their lifelong dispute over the transfinite numbers emerge philosophical disagreements over mathematical existence, consistency, and freedom. This thesis presents a short summary of Cantor's controversial theories, describes Cantor and Kronecker's philosophical ideas, and attempts to state clearly their differences of opinion. In the end, the author hopes to present the shock caused by Cantor's work and an appreciation of the two very different philosophies of mathematics represented by Cantor and Kronecker. / Thesis (BA) — Boston College, 2005. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Philosophy. / Discipline: College Honors Program.

Lie, Nga Sze. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 103-111). / Abstracts in English and Chinese. / Abstract --- p.i / Chapter 1 --- Introduction and Preliminary Discussions --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- Overview of the Thesis --- p.2 / Chapter 1.1.2 --- Background --- p.3 / Chapter 1.1.3 --- About Chapter 3: Details of the Theory --- p.4 / Chapter 1.1.4 --- About Chapter 4: Defects of the Theory --- p.7 / Chapter 1.2 --- Preliminary Discussions --- p.12 / Chapter 1.2.1 --- number --- p.12 / Chapter 1.2.2 --- mathematical existence and abstract reality --- p.12 / Chapter 1.2.3 --- finite/infinite --- p.12 / Chapter 1.2.4 --- actually/potentially infinite --- p.13 / Chapter 1.2.5 --- denumerability --- p.13 / Chapter 1.3 --- Concluding Remarks --- p.14 / Chapter 2 --- Mapping Mathematical Philosophies --- p.15 / Chapter 2.1 --- Preview --- p.15 / Chapter 2.1.1 --- Nominalism --- p.16 / Chapter 2.1.2 --- Conceptualism --- p.16 / Chapter 2.1.3 --- Intuitionism --- p.17 / Chapter 2.1.4 --- Realism --- p.18 / Chapter 2.1.5 --- Empiricism --- p.19 / Chapter 2.1.6 --- Logicism --- p.19 / Chapter 2.1.7 --- Neo-logicism --- p.21 / Chapter 2.1.8 --- Formalism --- p.21 / Chapter 2.1.9 --- Practicism --- p.23 / Chapter 2.2 --- Central Problem of Philosophy of Mathematics --- p.23 / Chapter 2.3 --- Metaphysics --- p.24 / Chapter 2.3.1 --- Abstractism --- p.24 / Chapter 2.3.2 --- Abstractist Schools --- p.25 / Chapter 2.3.3 --- Non-abstractism --- p.25 / Chapter 2.3.4 --- Non-abstractist Schools --- p.26 / Chapter 2.4 --- Semantics --- p.26 / Chapter 2.4.1 --- Literalism --- p.26 / Chapter 2.4.2 --- Literalistic schools --- p.27 / Chapter 2.4.3 --- Non-literalism --- p.27 / Chapter 2.4.4 --- Non-literalistic schools --- p.27 / Chapter 2.5 --- Epistemology --- p.28 / Chapter 2.5.1 --- Scepticism --- p.28 / Chapter 2.5.2 --- Scepticist Schools --- p.28 / Chapter 2.5.3 --- Non-scepticism --- p.29 / Chapter 2.5.4 --- Non-scepticist Schools --- p.29 / Chapter 2.6 --- Foundations of Mathematics --- p.30 / Chapter 2.6.1 --- Foundationalism --- p.31 / Chapter 2.6.2 --- Foundationalist Schools --- p.32 / Chapter 2.6.3 --- N on-foundationalism --- p.33 / Chapter 2.6.4 --- Non-foundationalist schools --- p.33 / Chapter 2.7 --- Finitistic Considerations --- p.33 / Chapter 2.7.1 --- Finitism --- p.41 / Chapter 2.7.2 --- Finitist Schools --- p.42 / Chapter 2.7.3 --- Non-finitism --- p.44 / Chapter 2.7.4 --- Non-finitist Schools --- p.44 / Chapter 2.8 --- Finitistic Reconsiderations --- p.44 / Chapter 2.8.1 --- C-finitism --- p.45 / Chapter 2.8.2 --- C-finitist Schools --- p.45 / Chapter 2.8.3 --- Non-C-finitism --- p.46 / Chapter 2.8.4 --- Non-C-finitist Schools --- p.46 / Chapter 2.9 --- Concluding Remarks --- p.47 / Chapter 3 --- Principles of Transfinite Theory --- p.48 / Chapter 3.0.1 --- Historical Notes on Infinity --- p.48 / Chapter 3.0.2 --- Cantor´ةs Proof --- p.49 / Chapter 3.1 --- The Domain Principle --- p.51 / Chapter 3.1.1 --- Variables and Domain --- p.53 / Chapter 3.1.2 --- Attack and Defense --- p.54 / Chapter 3.2 --- The Enumeral Principle --- p.56 / Chapter 3.2.1 --- Cantor´ةs Ordinal Theory of Numbers --- p.58 / Chapter 3.2.2 --- A Well-ordered Set --- p.59 / Chapter 3.2.3 --- An Enumeral --- p.59 / Chapter 3.2.4 --- An Ordinal Number --- p.60 / Chapter 3.2.5 --- Attack and Defense --- p.60 / Chapter 3.3 --- The Abstraction Principle --- p.63 / Chapter 3.3.1 --- Cantor´ةs Cardinal Theory of Numbers --- p.64 / Chapter 3.3.2 --- An Abstract One --- p.65 / Chapter 3.3.3 --- One-one Correspondence --- p.65 / Chapter 3.3.4 --- A Cardinal Number --- p.65 / Chapter 3.3.5 --- Attack and Defense --- p.65 / Chapter 3.4 --- Concluding Remarks --- p.68 / Chapter 4 --- Problems in Transfinite Theory --- p.70 / Chapter 4.1 --- Structure and Procedure --- p.70 / Chapter 4.1.1 --- Free Mathematics --- p.72 / Chapter 4.1.2 --- Non-constructive Proof --- p.75 / Chapter 4.2 --- Number and Numerosity --- p.85 / Chapter 4.2.1 --- Weak Reductionism --- p.85 / Chapter 4.2.2 --- Non-Cantorian Sets --- p.87 / Chapter 4.2.3 --- Intension in an Extensional Theory --- p.89 / Chapter 4.3 --- Conceivability and Comparability --- p.95 / Chapter 4.3.1 --- Tension with Absolute Infinity --- p.95 / Chapter 4.4 --- Conclusion --- p.100 / Bibliography --- p.103

Explains properties of mathematical sets, algebra of sets, and set order types.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIES. / Bibliography: leaves 205-206. / by Fredric M. Katz. / Ph.D.