AXIOMATIC CONSTRUCTION OF THE TANGENT BUNDLE IN SYNTHETIC DIFFERENTIAL GEOMETRY
Synthetic differential geometry (SDG) is a reformulation of differential geometry based on an axiom that is synthetic in nature, in the sense that it relates to geometric objects and the relations between those objects, such that a rigorous calculus that incorporates nilpotent infinitesimals can...
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Format: | Final Project |
Language: | Indonesia |
Online Access: | https://digilib.itb.ac.id/gdl/view/71908 |
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Institution: | Institut Teknologi Bandung |
Language: | Indonesia |
Summary: | Synthetic differential geometry (SDG) is a reformulation of differential geometry
based on an axiom that is synthetic in nature, in the sense that it relates to geometric
objects and the relations between those objects, such that a rigorous calculus that
incorporates nilpotent infinitesimals can be done.
As a basic assumption, we assume the existence of a basic continuum object R
which obeys a principle we call the Microaffineness Principle, which states that
every curve can be seen as a composition of microstraight lines, which is somewhat
in line with the classical understanding of the continuum, an ideal object on which
true continuous variation is possible. To be exact, we assume that R obeys the
Kock-Lawvere axiom, which states that all functions from the domain D = {d ?
R : d2 = 0} to R can be uniquely described as an affine function. Having been
allowed to work with infinitesimal elements explicitly, we are now able to carry on
rigorous constructions of geometric objects in a more general manner that better
accomodates the geometric intuition we might naturally have of the object, through
arguing with infinitesimal elements.
However, to work with this principle, we have to sacrifice the universality of the
Law of Excluded Middle, a principle of logic which states that P ?¬P always holds
for all propositions P. A significant consequence of this is the loss of our ability to
use non-constructive arguments of proof, particularly the non-constructive version
of the reductio ad absurdum argument. In this Thesis, we will examine the benefits
we gain from this sacrifice. |
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