Achilles and a tortoise take part in a race in which
the tortoise is given a head start. Consider Achilles’
circumstance. To overtake the tortoise, he must first
reach the tortoise’s starting position. By then, the tortoise will, of course, have moved on to a new position. To overtake the tortoise, Achilles must reach
this second position, and, again, by the time he does
so the tortoise will have moved farther along. Continuing this way, we see that it is impossible for Achilles
to ever overtake the tortoise, no matter how swiftly
he moves.
3. The Arrow Paradox: Assume that time and
space each are composed of discrete fundamental units.
Consider an arrow in flight. At any one fundamental
unit of time, the arrow occupies a particular instantaneous position in space. At this moment, it is indistinguishable from a motionless arrow occupying the same
location in space during that same unit of time. How
then is it that motion is perceived?
4. The Stadium Paradox: Assume that time
and space each are composed of discrete
fundamental units.
Imagine two runners in a stadium moving in opposite
directions from the same starting position. Suppose the
runners each move at a speed equivalent to one fundamental distance of space for each fundamental unit of
time. Ask now, “What does a runner see when he looks
behind his shoulder?” The answer: his opponent moving 2 units of space in 1 unit of time. We can only conclude that there must be a unit smaller than the
supposedly indivisible unit of time that corresponds to
just 1 unit of this motion.
Some comments are in order. It is worth noting
that many texts have attempted to resolve Zeno’s
dichotomy paradox by arguing that the fact that we
can walk across a room proves that the infinite sum
(SERIES)
+
+
+ … converges to the finite value
1. This, however, is not relevant to the argument proposed in this first paradox. Zeno is suggesting instead
that it is impossible to build such an infinite sum by
starting at the “wrong end,” as it were. (The construction of an infinite sum, however, is appropriate to analyze the second paradox.)
Some scholars suggest that Zeno’s third paradox is
an important first step in the study of INFINITESIMALs
and issues of DIFFERENTIAL CALCULUS. The standard
formula for velocity is v = , where d is the distance
traveled and t the time taken to do so. At a particular
instant, this leads to the meaningless equation v = .
Here, it is said, Zeno is pointing out the mathematical
difficulty of infinitesimals, the same problem that was
to haunt the inventors of calculus two millennia later.
Zermelo, Ernst Friedrich Ferdinand (1871–1953)
German Set theory, Physics Born on July 27, 1871, in
Berlin, Germany, mathematician Ernst Zermelo is
remembered for founding axiomatic SET THEORY. In
1908 he published a set of seven axioms designed to
overcome the paradoxes posed by BERTRAND ARTHUR
WILLIAM RUSSELL (1872–1970) in set theory, and to
provide, for the first time, a rigorous and logical foundation for the developing subject. Zermelo’s axioms,
although later modified by other mathematicians, have
remained the starting point on which much continued
work on this topic has been based.
Zermelo received a doctorate in mathematics from
the University of Berlin in 1894 after completing a thesis exploring issues in calculus, the calculus of variations, and their applications to physics. He continued
work in this area at the university for an additional 3
years before moving to Göttingen to study hydrodynamics. He was awarded a lectureship at Göttingen for
his outstanding research in this field.
At the time elsewhere in Europe, mathematicians
were attempting to make sense of the controversial work
put forward by German mathematician GEORG CANTOR
(1845–1918). The German mathematician DAVID
HILBERT (1862–1943), in his famous Paris lecture of
1900, cited the resolution of Cantor’s CONTINUUM
HYPOTHESIS as the most challenging and important question of the 20th century. Zermelo was captivated by the
problem when he learned of it, and began devoting all
his time and efforts to solving it. He soon realized that a
first step toward solving the general conjecture would be
to establish a second of Cantor’s claims, namely, that
any set can be a WELL-ORDERED SET.
Zermelo published his first collection of results on
this topic in 1901 with the paper “Über die Addition
transfiniter Cardinalzahlen” (On the addition of
0
–
0
d
–
t
1
–
8
1
–
4
1
–
2
Zermelo, Ernst Friedrich Ferdinand 533
the tortoise is given a head start. Consider Achilles’
circumstance. To overtake the tortoise, he must first
reach the tortoise’s starting position. By then, the tortoise will, of course, have moved on to a new position. To overtake the tortoise, Achilles must reach
this second position, and, again, by the time he does
so the tortoise will have moved farther along. Continuing this way, we see that it is impossible for Achilles
to ever overtake the tortoise, no matter how swiftly
he moves.
3. The Arrow Paradox: Assume that time and
space each are composed of discrete fundamental units.
Consider an arrow in flight. At any one fundamental
unit of time, the arrow occupies a particular instantaneous position in space. At this moment, it is indistinguishable from a motionless arrow occupying the same
location in space during that same unit of time. How
then is it that motion is perceived?
4. The Stadium Paradox: Assume that time
and space each are composed of discrete
fundamental units.
Imagine two runners in a stadium moving in opposite
directions from the same starting position. Suppose the
runners each move at a speed equivalent to one fundamental distance of space for each fundamental unit of
time. Ask now, “What does a runner see when he looks
behind his shoulder?” The answer: his opponent moving 2 units of space in 1 unit of time. We can only conclude that there must be a unit smaller than the
supposedly indivisible unit of time that corresponds to
just 1 unit of this motion.
Some comments are in order. It is worth noting
that many texts have attempted to resolve Zeno’s
dichotomy paradox by arguing that the fact that we
can walk across a room proves that the infinite sum
(SERIES)
+
+
+ … converges to the finite value
1. This, however, is not relevant to the argument proposed in this first paradox. Zeno is suggesting instead
that it is impossible to build such an infinite sum by
starting at the “wrong end,” as it were. (The construction of an infinite sum, however, is appropriate to analyze the second paradox.)
Some scholars suggest that Zeno’s third paradox is
an important first step in the study of INFINITESIMALs
and issues of DIFFERENTIAL CALCULUS. The standard
formula for velocity is v = , where d is the distance
traveled and t the time taken to do so. At a particular
instant, this leads to the meaningless equation v = .
Here, it is said, Zeno is pointing out the mathematical
difficulty of infinitesimals, the same problem that was
to haunt the inventors of calculus two millennia later.
Zermelo, Ernst Friedrich Ferdinand (1871–1953)
German Set theory, Physics Born on July 27, 1871, in
Berlin, Germany, mathematician Ernst Zermelo is
remembered for founding axiomatic SET THEORY. In
1908 he published a set of seven axioms designed to
overcome the paradoxes posed by BERTRAND ARTHUR
WILLIAM RUSSELL (1872–1970) in set theory, and to
provide, for the first time, a rigorous and logical foundation for the developing subject. Zermelo’s axioms,
although later modified by other mathematicians, have
remained the starting point on which much continued
work on this topic has been based.
Zermelo received a doctorate in mathematics from
the University of Berlin in 1894 after completing a thesis exploring issues in calculus, the calculus of variations, and their applications to physics. He continued
work in this area at the university for an additional 3
years before moving to Göttingen to study hydrodynamics. He was awarded a lectureship at Göttingen for
his outstanding research in this field.
At the time elsewhere in Europe, mathematicians
were attempting to make sense of the controversial work
put forward by German mathematician GEORG CANTOR
(1845–1918). The German mathematician DAVID
HILBERT (1862–1943), in his famous Paris lecture of
1900, cited the resolution of Cantor’s CONTINUUM
HYPOTHESIS as the most challenging and important question of the 20th century. Zermelo was captivated by the
problem when he learned of it, and began devoting all
his time and efforts to solving it. He soon realized that a
first step toward solving the general conjecture would be
to establish a second of Cantor’s claims, namely, that
any set can be a WELL-ORDERED SET.
Zermelo published his first collection of results on
this topic in 1901 with the paper “Über die Addition
transfiniter Cardinalzahlen” (On the addition of
0
–
0
d
–
t
1
–
8
1
–
4
1
–
2
Zermelo, Ernst Friedrich Ferdinand 533