23 October 2021
A Hitchhiker's Guide to Infinity - How Big is Infinity?online course
This workshop uses a variety of intuitive visual proofs, interspersed with a tantalising sprinkling of mathematical analysis and technology to examine the beautiful and contradictory world of “the infinite” from a variety of perspectives, and highlights key developments in the history of mathematics centred on the concept of “infinity”. In so doing, this workshop exposes the student to the branches of analysis, number theory, set theory and transfinite arithmetic.
This workshop is broken down into the following sections:
MOTIVATION: Paradoxes of Infinity
SECTION 1: Infinite Classes
SECTION 2: Hierarchies of Infinity
SECTION 3: Hilbert's Grand Hotel paradox
CONCLUSION: Continuum Hypothesis
This workshop begins with a seduction of mind-bending curiosities and paradoxes involving the mathematical notion of “infinity”. Specifically, Zeno's paradox of Achilles and the Tortoise is used as a motivation to introduce an analytical treatment of oscillating, divergent and convergent infinite series. This analytically focused workshop then proceeds to utilizes subtle mathematical tools to closely examine the nature of the different infinities of the natural numbers, integers, rational numbers and the real numbers systems. These different hierarchies of infinity are then further investigated by using such concepts as cardinality, countable v's uncountable, the number continuum and the bounding of irrational numbers by rational numbers, and transfinite arithmetic. Set Theory methods are then used to consider Hilbert's thought experiment of allocating an infinite number of guests into a hotel with an infinite number of rooms. The workshop culminates with Cantors concept of 1-1 correspondence of sets with proper subsets of themselves. The workshop concludes by examining the implications of the Continuum Hypothesis, and an update on its status!
Ian Tame is a Master postgraduate in Mathematics (with specialism stochastic processes). He has published mathematics articles (in English and German), possesses a postgraduate teaching qualification, and has over 25 years’ experience in education.
Students studying, or about to study mathematics, physics or philosophy at A-level or International Baccalaureate Diploma and other pre-university courses - especially those who aspire to study mathematics, science, computing or engineering-based disciplines at university; and current Undergraduate students studying Mathematics.
By the end of the course students will have achieved an understanding of how key developments in the history of mathematics have centred on the concept of “infinity”. Specifically, students will have learnt a selection of fundamental proofs from the fields of analysis and set theory about the properties of countable and uncountable infinite sets.
EUR 99: plus 19% VAT