The Center "Leo Apostel"
                         ************************
 
in collaboration with the Doctoral Programme of the VUB, invites 
everybody to the 21st of its interdisciplinary seminars in the series 
"Foundations"; a colloqium on Thursday, 22 May 1997:

   "Everything you always wanted to know about Godel,
   ************************************** but were afraid to ask."
                                          ***********************

In this series CLEA invites scholars that are actively engaged in the 
research on the foundations of a  particular discipline. 
Their lectures will always be directed to an interdisciplinary 
audience, and the discussions aim at confronting the foundations of the 
different disciplines.


            1)  Godel's theorem on elementary arithmetic
                ****************************************

                        by Prof. E. Colebunders
 Thursday, 22 May 1997 from 15.00 till 16.00 p.m., room B301 (building B)
              Vrije Universiteit Brussel, Campus Oefenplein .

                                 &

            2)     How to Godel and how not to Godel
                   *********************************

                          by Prof. J.P. Van Bendegem

 Thursday, 22 May 1997 from 16.15 till 17.15 p.m., room B301 (building B)
              Vrije Universiteit Brussel, Campus Oefenplein .


               About the lecture of Prof. Colebunders :
               ----------------------------------------
By elementary arithmetic we mean the entire collection of truths and 
untruths on the  natural numbers, addition, multiplication and
exponentiation. We consider an axiomatic system for this arithmetic in a
language strong enough to express all these truths and untruths. We show 
that this arithmetic calculus will never be reducible to such an 
axiomatic system; this means that there will always be a formula (called 
Godel's formula) which is true in arithmetic calculus and which is not a 
theorem in the formal theory, or which is not true in arithmetic calculus 
and is a theorem in the axiomatic theory. This fundamental discrepancy 
will be shown to be independent from our, by accident badly chosen, 
axioms.  The problems arise because the axiomatic systems under 
consideration are capable to make self-referring assertions.

               About the lecture of Prof. van Bendegem :
               -----------------------------------------
Godel's results - the famous incompleteness theorems - have given rise to 
an enormous literature on incompleteness, undecidability and non-
computability. Many of these applications are well-founded and clearly 
demonstrate limitations of human calculating powers. One specific example 
- one of the simplest - will be discussed, namely the busy beaver 
problem. There are an impressive number of results in mathematics and, 
recently, in physics. Some of these results will be mentioned. But as 
many of these applications are quite "shaky". Probably the most famous 
example is John Lucas' argument, based on Godel's theorems, to show that 
the human mind cannot be equated to a machine. The argument, dating back 
to the 60s, was almost dead and buried, however, recently, Roger Penrose, 
has taken up the argument once again. An attempt will be made to present 
an evalutation of this argument.

About the speakers :
--------------------
Prof. Eva Colebunders is head of the research group TOPO (Dept. of 
Mathematics, Free University of Brussels) and active in the field of 
Topology. She is author of articles on Convergence Theory and Categorical 
Topology and of the book : "Function classes of Cauchy continuous maps", 
published in the series "Monographs and textbooks in pure and 
applied Mathematics", Marcel Dekker,New York, 1989. Several research 
results, published in the book mentioned above, make explicit use of set 
theoretical assumptions. So although these results belong to the field of 
Topology, they depend on a specific choice concerning the axioms of Set 
Theory. This phenomenon is not unusual. These days many problems in 
various mathematical disciplines such as Topology, Measure Theory and 
Analysis and that were open for decades, are now solved by making 
specific assumptions about sets. In fact they depend on the foundation of 
Mathematics. This phenomenon is nowadays responsible for the development 
of important research fields, such as "Set theoretic Topology" in which 
topologists and logicians find each other.  This explains the interest of 
the speaker in the foundations of Mathematics in general and in Godel's 
Theorem in particular.

Prof. Jean Paul Van Bendegem teaches logic and philosophy of science at 
the Vrije Universiteit Brussel. His research concerns foundations of 
mathematics - more specifically, the possibility of strict finitism - and 
philosophy of space and time. Recently he has been working on a book 
about science and religion, entitled "Tot in der eindigheid" to appear in 
October.

Both presentations with questions will last about an hour. Afterwards, an 
hour or more is reserved for an in-depth, group discussion of the topic. 

More info at the CLEA office: phone 02-644 26 77 or via the Web-page: 
http://pespmc1.vub.ac.be/CLEA/