Download Lecture Notes 2: Infinity

Philosophy 024: Big Ideas
Prof. Robert DiSalle ([email protected])
Talbot College 408, 519-661-2111 x85763
Office Hours: Monday and Wednesday 12-2 PM
Course Website:
http://instruct.uwo.ca/philosophy/024/
Lucretius (Roman, ca. 94 BCE- 51 BCE)
Background: Ancient Greek atomism
Leucippus (ca. 435 BCE), Democritus (ca. 410 BCE):
-space is an infinite void;
-bodies are composed of atoms (indivisible particles)
moving freely in the void, colliding and combining;
-qualitative properties that our senses apprehend
arise from quantitative properties of insensible atoms, their
motions and combinations
Epicurus (341-270 BCE): atoms naturally tend to “fall”
through the void, and collisions and combinations result from
spontaneous and random “swerves”.
The random “swerves” allow the atomistic, materialistic view
to be reconciled with freedom of the will.
Understanding the material origins of human life frees human
beings from ignorance and fear of death. At death the body
and the soul dissolve into their atomic parts.
“Therefore death is nothing to us; for that which is dissolved
is without sensation; and that which lacks sensation is nothing
to us.”
The infinite turns out to be the contrary of what it is said to be.
It is not what has nothing outside it that is infinite, but what
always has something outside it…
A quantity is infinite if it is such that we can always take a part
outside what has been already taken. On the other hand, what
has nothing outside it is complete and whole. For thus we define
the whole-that from which nothing is wanting, as a whole man
or a whole box. What is true of each particular is true of the
whole as such-the whole is that of which nothing is outside. On
the other hand that from which something is absent and outside,
however small that may be, is not 'all'. 'Whole' and 'complete'
are either quite identical or closely akin. Nothing is complete
(teleion) which has no end (telos); and the end is a limit.
(Aristotle, Physics)
Euclid’s Postulate 2: To produce a finite straight line
continuously in a straight line.
Aristotle: “My argument does not even rob mathematicians of
their study, although it denies the existence of the infinite in the
sense of actual existence as something increased to such an
extent that it cannot be gone through; for, as it is, they do not
even need the infinite or use it, but only require that the finite
straight line shall be as long as they please….”
Lucretius: the case for the infinitude of space
• If the universe has a boundary, there must be
something outside of it to limit it. But nothing can
be outside the universe.
• Suppose there is an edge, and you can stand there
and throw a missile. If it goes forward, then there
is no boundary and you are not at the edge. If it is
blocked, then there is something outside and you
are not at the edge.
• If the void were finite, then all matter would
eventually collapse by its weight into the center.
The “elemental” view of nature (Empedocles, 440 BCE):
Material things are composed of four elements (earth, air,
fire, water, based on four fundamental qualities (hot, cold,
moist, dry):
(Isidore of Seville, 1472)
The geocentric universe (from the Cosmographia of Apianus, 1524)
Copernicus’s heliocentric model (1543)
The heliocentric system in infinite space (Thomas Digges, 1576)
Understanding infinite sets
1-to-1 correspondence: For two sets, F and G: the number of F’s
is equal to the number of G’s if and only if the F’s and the G’s
are in 1-to-1 correspondence. (Frege, 1884)
Subset of a set S: A set that has some of the members of S.
Proper subset: subset that does not have all the members of S.
For finite sets, “the whole is greater than the part.”
An infinite set can have proper subsets that are in 1-to-1
correspondence with the whole set. (“Hilbert’s Hotel”)
E.g. for every natural number 1,2,3,…. there is an even number.
The table of all the rational numbers
1
2
3
4
5
6
7
8
9 ….
1/2 2/2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 ….
1/3 2/3 3/3 4/3 5/3 6/3 7/3 8/3 9/3 ….
1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 ….
1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5 ….
1/6 2/6 3/6 4/6 5/6 6/6 7/6 8/6 9/6 ….
1/7 2/7 3/7 4/7 5/7 6/7 7/7 8/7 9/7 ….
1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 9/8 ….
1/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 9/9 ….
Counting the rational numbers: a set is countable if it can be
placed in 1-1 correspondence with the natural numbers.
1
2
3
4
5
6
7
8
9 ….
1/2 2/2 3/2 4/2 5/2 6/2 7/2 8/2 9/2 ….
1/3 2/3 3/3 4/3 5/3 6/3 7/3 8/3 9/3 ….
1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 ….
1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5 ….
1/6 2/6 3/6 4/6 5/6 6/6 7/6 8/6 9/6 ….
1/7 2/7 3/7 4/7 5/7 6/7 7/7 8/7 9/7 ….
1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 9/8 ….
1/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 9/9 ….
Can we count up all the real numbers?
1st number:
N1.a1a2a3a4a5…
2nd number: N2.b1b2b3b4b5…
3rd number:
N3.c1c2c3c4c5… ….and so on.
First, suppose that all of the real numbers are listed in this way.
Then, choose aa1, bb2, cc3, etc. (Assume in each case that the
digit is neither 0 nor 9, to avoid the case 0.999…=1.000.)
Now, consider the number z = 0.abc….
Evidently z is not on the list!
(It differs from the first number in the first decimal place, the
second number in the second decimal place, etc. )
The continuum of real numbers is equivalent to (in 1-1
correspondence with) any part of it.
A1
A2
A∞
B1
B2
B∞
The dome is bigger than the disc, but the points are in 1-1
correspondence; each point (x,y z) on the dome can be mapped
(“projected”) onto the point (x,y) beneath it.
(x,y,z)
(x,y)
A paradox of the infinite (“Russell’s paradox”):
Ordinary set: does not contain itself as an element.
E.g.: The set of all natural numbers is not itself a natural number,
and so it is not an element of itself.
Extraordinary set: does contain itself as an element.
E.g.: The set of all sets not mentioned in the Bible is itself not
mentioned in the Bible, so it is an element of itself.
Question: Since extraordinary sets are obviously quite bizarre,
can we confine ourselves to talking about all and only the
ordinary sets?
I.e., can we talk about the set of all ordinary sets?
The set S: the set of all and only ordinary sets, i.e. the set of
all sets that are not elements of themselves.
Question: Is S an ordinary set?
If S is an ordinary set, then it contains itself as an element,
since it is supposed to contain all ordinary sets.
But that means that S is an extraordinary set, since
extraordinary sets contain themselves as elements.
So S is not the set of ordinary sets, since it contains an
extraordinary set (itself).
Cf. the “Barber of Alcalà”: he shaves all men who do not
shave themselves.
So, does he shave himself or not?