|
|
|
Mathematics
Seminar Series New
York City College of Technology(CUNY) Abstract of the Talk by: Dr. Victoria Gitman Standard
systems of non-standard models of Peano Arithmetic |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
There is a rich collection of
structures satisfying the Peano Axioms, which are
viewed as capturing the essential properties of number theory. The natural
numbers with the operations of plus and times, (N,+,x), is
called the standard model of Peano Arithmetic (PA),
while other structures are referred to as non-standard models. Non-standard models
contain the natural numbers as an initial segment of their linear order and
have very complex operations of plus and times on their non-standard part.
One of the most fundamental concepts in the field is the standard system of a
model of PA. The standard system is a particular collection of subsets of the
natural numbers associated to a model of PA. Intuitively, standard systems
are intended to capture the traces of information the non-standard model
leaves on its standard part –the natural numbers. One of the most important
open questions in the field of models of PA has been characterizing
collections of subsets of the natural numbers that arise as standard systems.
There is a proposed characterizing due to Scott from the 1960’s which holds
for standard systems of sizes countable and omega_1. It remains an open
question, known as Scott’s Problem, whether the characterization holds true
for standard systems of all cardinalities. In this talk, I will give a brief introduction
to non-standard models of PA, followed by a discussion of standard systems,
Scott’s problem and my own recent contribution to it. |
|
|
|
http://websupport1.citytech.cuny.edu/faculty/dkahrobaei/NYCCT_Math_Seminar.htm |
|
|
|
Address: © 2007
(since August 2007) |