Seminário remoto "Lógicos em Quarentena" 12/11/2020 (quinta-feira) 16:00h

[Bruno Lopes]

Numa iniciativa conjunta da Sociedade Brasileira de Lógica e do Grupo de Interesse em Lógica da Sociedade Brasileira de Computação, gostaríamos de convidar a todos a participarem do Seminário "Lógicos em Quarentena". Trata-se de um seminário remoto com apresentações informais por membros da comunidade e espaço para perguntas no fim. As apresentações usualmente são gravadas e disponibilizadas na página do evento http://lq.sbl.org.br (com a agenda completa).

Data: 12 de novembro de 2020 (quinta-feira)
Horário: 16:00h GMT-3
Apresentador: Samuel Gomes da Silva (DM/UFBA)
Título: On striking, counterintuitive partitions - or: The Axiom of Choice is not to be blamed of anything

Resumo: One of the more counterintuitive consequences of the Axiom of Choice (perhaps the most celebrated among them all) is the well-known Banach-Tarski Paradox,  which is a theorem ({\it not a paradox}) stating that any closed, ``solid"\, ball of the three-dimensional Euclidean space may be decomposed into a finite number of subsets (``pieces", so to say) which, after rearranged using only rigid motions, turn out to form two identical copies of the original ball. Variations of this very same theorem (which heavily relies on the Axiom of Choice) can be spelled out even more strikingly   (``one could cut an orange into finite pieces and then reassemble those pieces in order to get a sphere of the size of the Sun").  Of course, the orange pieces referred to would be {\bf non-measurable} -- thus, Banach-Tarski Paradox could be understood as a fancy alternative proof of the well-known fact that the Axiom of Choice easily produces non-measurable subsets of any given Euclidean space.  Due to these undeniably counterintuitive aspects, Banach-Tarski Paradox is usually presented as an argument against the acceptance of the Axiom of Choice. In this talk, we will see that the possibly implicit desire of all those anti-Axiom of Choice researchers (which, apparently, would be to discard the Axiom of Choice and then freely work with models of Mathematics on which all subsets of any given Euclidean space are Lebesgue-measurable) would also yeld some highly counterintuitive results regarding partitions of sets -- so, the Axiom of Choice should not be considered the sole culprit when it comes to counterintuitive situations involving partitions ! For instance, we will show in the talk that: if all subsets of the real line are Lebesgue-measurable, then there is a partition of $\mathbb{R}$ into strictly more than $2^{\aleph_0}$ non-empty subsets -- that is, there would be a partition of a set (and not some obscure set -- arguably the most important set of all Mathematics, which is the real line $\mathbb{R}$) into {\bf more pieces than elements} (!!!). Having appeared as a common feature of a number of constructions, we will take the opportunity to discuss the so-called {\it Partition Principle} -- which is an immediate consequence of the Axiom of Choice for which the natural question in the context (``Is this principle, in fact, an equivalent of the Axiom of Choice ?") constitutes itself as one of the oldest (and still open) problems of this kind in the literature.

A apresentação ocorrerá pelo Google Meet através do link público https://meet.google.com/uwn-tyjb-rbr .

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Bruno Lopes

Professor Adjunto

Instituto de Computação

Universidade Federal Fluminense

http://www.ic.uff.br/~bruno