MacLane's Postulates
In the 1960's Saunders MacLane proposed a new set of axioms that are more like Birkhoff's than Euclid's or Hilbert's axioms.
Like Birkhoff's axioms, MacLane's axioms also use the real numbers; and so has fewer axioms than Hilbert's axioms did.
MacLane's approach also does not generate any new theorems in Euclidean geometry but rather simplifies previous attempts at erecting the foundations of Euclidean geometry.
The main differences are that MacLane uses a function to measure distance between points (called a metric function) and the so called Continuity Axiom, which incorporates the Crossbar Proposition into Euclidean geometry as an axiom.
This topic states MacLane's axioms, from which all of Euclidean geometry can be proven.
Comment (Undefined Terms) Point, distance, line, and angle measure.
Axiom (Distance Axioms) (i) There are at least two points. (ii) If
and
are points,
is a nonnegative number. (iii) For points
and
if and only if
![the maclane postulates _gr_7.gif]](pages/the-maclane-postulates/Images/the-maclane-postulates_gr_7.gif) (iv) If
and
are points,
Axiom (Line Axioms) (i) A line is a set of points containing more than one point. (ii) Through two distinct points there is one and only one line. (iii) Three distinct points lie on a line if one of them is between the other two. (iv) On each ray from a point
and to each positive real number
there is a point
with
Axiom (Angle Axioms) (i) If
and
are rays from the same point,
is a real number modulo
![the maclane postulates _gr_18.gif]](pages/the-maclane-postulates/Images/the-maclane-postulates_gr_18.gif) (ii) If
and
are three rays from the same point, then
![the maclane postulates _gr_21.gif]](pages/the-maclane-postulates/Images/the-maclane-postulates_gr_21.gif) (iii) If
is a ray from
and
is a real number (modulo 360), then there is a ray from
such that
![the maclane postulates _gr_26.gif]](pages/the-maclane-postulates/Images/the-maclane-postulates_gr_26.gif) (iv) If
then
if and only if
Axiom (Similiarity Axiom) If two ordered triangles
and
have
and
(for
positive) they are similiar.
Axiom (Continuity Axiom) Let
be proper.
If
is between
and
then
Conversely, if
then the ray
meets the interval
![the maclane postulates _gr_43.gif]](pages/the-maclane-postulates/Images/the-maclane-postulates_gr_43.gif)
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Cite this as: The Maclane Postulates Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/the-maclane-postulates.html
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