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 .
--
Bruno Lopes
Professor Adjunto
Instituto de Computação
Universidade Federal Fluminense
http://www.ic.uff.br/~bruno
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Para ver esta discussão na web, acesse
https://groups.google.com/a/dimap.ufrn.br/d/msgid/logica-l/CAD-Wq0_9CMSss6iLN0AzDMTUsxctF34Radswu0XrWP8Na%3D-vwg%40mail.gmail.com.
