BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:The unreasonable effectiveness of Nonstandard Anal
ysis - Sanders\, S (Ludwig-Maximilians-Universitt
Mnchen)
DTSTART;TZID=Europe/London:20151214T150000
DTEND;TZID=Europe/London:20151214T153000
UID:TALK62893AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/62893
DESCRIPTION:The aim of my talk is to highlight a hitherto unkn
own computational aspect of Nonstandard Analysis.
In particular\, we provide an algorithm which take
s as input the proof of a mathematical theorem fro
m pure Nonstandard Analysis\, i.e. formulated sole
ly with the nonstandard definitions (of continuity
\, integration\, dif- ferentiability\, convergence
\, compactness\, et cetera)\, and outputs a proof
of the as- sociated effective version of the theor
em. Intuitively speaking\, the effective version o
f a mathematical theorem is obtained by replacing
all its existential quantifiers by functionals com
puting (in a specific technical sense) the objects
claimed to exist. Our algorithm often produces th
eorems of Bishops Constructive Analysis ([2]). \n
The framework for our algorithm is Nelsons syntact
ic approach to Nonstandard Analysis\, called inter
nal set theory ([4])\, and its fragments based on
Goedels T as introduced in [1]. Finally\, we estab
lish that a theorem of Nonstandard Analysis has th
e same computational content as its highly constru
ctive Herbrandisation. Thus\, we establish an algo
rithmic two-way street between so-called hard and
soft analysis\, i.e. between the worlds of numeric
al and qualitative results.\n \nReferences: \n [1]
Benno van den Berg\, Eyvind Briseid\, and Pavol S
afarik\, A functional interpretation for non- stan
dard arithmetic\, Ann. Pure Appl. Logic 163 (2012)
\, no. 12\, 19621994. \n \n[2] Errett Bishop and D
ouglas S. Bridges\, Constructive analysis\, Grundl
ehren der Mathematis- chen Wissenschaften\, vol. 2
79\, Springer-Verlag\, Berlin\, 1985. \n \n[3] Fer
nando Ferreira and Jaime Gaspar\, Nonstandardness
and the bounded functional interpre- tation\, Ann.
Pure Appl. Logic 166 (2015)\, no. 6\, 701712. \n
\n[4] Edward Nelson\, Internal set theory: a new a
pproach to nonstandard analysis\, Bull. Amer. Math
. Soc. 83 (1977)\, no. 6\, 11651198. \n \n[5] Step
hen G. Simpson\, Subsystems of second order arithm
etic\, 2nd ed.\, Perspectives in Logic\, CUP\, 200
9. \n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
END:VEVENT
END:VCALENDAR