MARCI  MAthematical Recreational CIrcle
 The main goal of the MARCI meeting is to enjoy mathematics.
 All attendees are free to talk/listen/eat as much as they like.
 One attendee is the meeting speaker. The speaker decides the topic.
 One attendee is the meeting host. It is the host's care to organize the
meeting.
 Speaker and host may be the same person.
MARCI meetings
 The MARCI met for the first time on 030318 (18th march 2003) with
three attendees:
 Maurizio Bruglieri
 Chiara De Santi (host)
 Leo Liberti (speaker)
The topic chosen by the speaker was on a theorem about generating prime
numbers, see the relevant paper and the
picture gallery.
 MARCI met virtually by email on 030326 with a problem proposed by
Maurizio, attendees Maurizio, Chiara, Leo; and a guest "speaker",
Franco Raimondi. Read the problem
statement with some proposed solutions, and Franco's
reply.

The 030401 MARCI meeting, with 4 attendees:
 Maurizio Bruglieri
 Marco Bruzzone
 Chiara De Santi (host)
 Leo Liberti
focused on more than one subject. First, we analysed the
paper Guy
et al., Primes at a Glance, and found many answers to questions
left pending from the first meeting. Secondly, Maurizio gave us an account
of the Genoa
"Chaos and Complexity" 2002 show, with some information
about fractal taste. Thirdly, Leo talked
about quantum computing. See the
notes, the idea for quantum
optimization, Steane's
survey paper, and further qubits
of information. Or just look at the pictures.

The 030429 MARCI meeting, with 5 attendees:
 Maurizio Bruglieri
 Marco Bruzzone
 Chiara De Santi (host, as usual)
 Leo Liberti
 Fabio Roda
started to look into Goedel's theorems and all that tours around it.
We looked at a short paper about
how easy it is to mistake provability for truth, and then we heard
Fabio talk about consistency, completeness and correctness of a
mathematical theory. As we all seem to be quite interested in this
topic, we all hope to have other sessions about Goedel's theorem
and logic / axiomatic set theory in general.
Some puzzles were also discussed, the most controversial being
the "missing area" problem illustrated
here. For hints (in Italian) about its solution, see
this mail.

The 030527 MARCI meeting, with 4 attendees:
 Marco Bruzzone
 Chiara De Santi (host, obviously)
 Leo Liberti
 Fabio Roda
went on with the investigation in the details of Goedel's theorem.
We started to look into the godel number of mathematical formulae,
based on Mendelson's book's numbering. Unfortunately, we were left
rather confused by Mendelson's choice of "constants", "predicates",
"functions": we seemed to remember that all these concepts could be
captured with just variables, symbols like ( ) , "implies" "not" and
quantifiers.
 The 030610 MARCI meeting, with 7 attendees:
 Pietro Belotti
 Maurizio Bruglieri
 Marco Bruzzone
 Chiara De Santi (host, again; and I shan't say it any more)
 Sonia Ferretti
 Leo Liberti
 Fabio Roda
again went on studying Goedel's theorem, and in particular most of the
concepts that come before the understanding of the theorem proper. We
clarified some concepts concerning the differences between
propositional logic, firstorder logic without proper axioms,
firstorder logic with proper axioms, and higher order logics.
Propositional logic does not involve the use of quantifiers; it is
the "least powerful" expressively speaking, but the "most powerful"
deductively speaking. In propositional logic every tautology is
provable (it is complete), and its deduction theorem does not have
any special hypothesis. In firstorder logic the quantifiers act on
individual variables. Without proper axioms (i.e. only counting the
logic axioms and the two inference rules quoted by Mendelson),
firstorder logic is complete (Goedel, 1930). However, its deduction
theorem is limited by some assumptions. Firstorder logic with proper
axioms (i.e. other axioms beyond the logic axioms, which might be
phrases describing the properties of some added relation, or function)
might not be complete, and its deduction theorems may be severely
limited in scope. We seemed to understand the fact that Goedel's
incompleteness theorem refers to a firstorder logic with enough
relation and functional letters and enough proper axioms so that a
description of the natural number is possible. In higherorder logics
the quantifiers may act on relation and functional letters as well
individual variables. We also talked about the difference between a
"structure" and a "model". Fabio Roda came up with the following
explanation. Every sentence in these logic systems we are talking
about contains names: individual variable names like x,y,
relation names, functional names, and so on. We can then assign a
"thing", an object, to each name: like x means "my mum" and so
on. The class of all these objects is a structure. The "union" of a
structure and its describing sentences in logic is a model.
Franco Raimondi, our correspondent from London, sent in
this mail.