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Paradoxes of the Infinite
Kline XXV
Pre-May Seminar
March 14, 2011
Galileo Galilei (1564-1642)
Galileo: Dialogue on Two New
Sciences, 1638

Simplicio: Here a difficulty presents itself which
appears to me insoluble. Since it is clear that we
may have one line segment longer than another,
each containing an infinite number of points, we
are forced to admit that, within one and the
same class, we may have something greater
than infinity, because the infinity of points in the
long line segment is greater than the infinity of
points in the short line segment. This assigning
to an infinite quantity a value greater than
infinity is quite beyond my comprehension.
Galileo’s Dialogo

Salviati: This is one of the difficulties which
arise when we attempt, with our finite minds, to
discuss the infinite, assigning to it those
properties which we give to the finite and
limited; but this I think is wrong, for we cannot
speak of infinite quantities as being the one
greater or less than or equal to another. To
prove this I have in mind an argument, which,
for the sake of clearness, I shall put in the form
of questions to Simplicio who raised this
difficulty.
Galileo’s Dialogo
Salviati: If I should ask further how
many squares there are, one might reply
truly that there are as many as the
corresponding number of roots, since
every square has its own root and every
root its own square, while no square has
more than one root and no root more than
one square.
 Simplicio: Precisely so.

Galileo’s Dialogo

Salviati: But if I inquire how many roots there
are, it cannot be denied that there are as many
as there are numbers because every number is a
root of some square. This being granted we
must say that there are as many squares as
there are numbers because they are just as
numerous as their roots, and all the numbers
are roots. Yet at the outset we said there are
many more numbers than squares, since the
larger portion of them are not squares.
Galileo’s Dialogo
Sagredo: What then must one conclude under
these circumstances?
 Salviati: So far as I see we can only infer that
the totality of all numbers is infinite, that the
number of squares is infinite, and that the
number of their roots is infinite; neither is the
number of squares less than the totality of all
numbers, nor the latter greater than the former;
and finally the attributes "equal," "greater," and
"less" are not applicable to infinite, but only to
finite, quantities.

Bernard Bolzano (1781-1848)
Bernard Bolzano (1781-1848)

Czech Priest
Bernard Bolzano (1781-1848)
Czech Priest
 [0,1]~[0,2]

Cardinality
Cardinality

The number of elements in a set is the
cardinality of the set.
Cardinality
The number of elements in a set is the
cardinality of the set.
 Card(S)=|S|

Cardinality
The number of elements in a set is the
cardinality of the set.
 Card(S)=|S|
 |{1,2,3}|=|{a,b,c}|

Cardinality
The number of elements in a set is the
cardinality of the set.
 Card(S)=|S|
 |{1,2,3}|=|{a,b,c}|
 c=|[0,1]|

Cardinality
The number of elements in a set is the
cardinality of the set.
 Card(S)=|S|
 |{1,2,3}|=|{a,b,c}|
 c=|[0,1]|
 Lemma: c=|[a,b]| for any real a<b.

Cardinality
The number of elements in a set is the
cardinality of the set.
 Card(S)=|S|
 |{1,2,3}|=|{a,b,c}|
 c=|[0,1]|.
 Lemma: c=|[a,b]| for any real a<b.
 Lemma: |Reals|=c.

Richard Dedekind (1831-1916)
Richard Dedekind (1831-1916)

Definition of
infinite sets:
Georg Cantor (1845-1918)
‫‪0‬א‬
‫‪0‬א = |}… ‪|{1, 2, 3,‬‬
‫‪‬‬
‫‪0‬א‬
‫‪0‬א = |}… ‪|{1, 2, 3,‬‬
‫‪0‬א = |}… ‪ |{12, 22, 32,‬‬
‫‪‬‬
‫א‬0
|{1, 2, 3, …}| = ‫א‬0
 |{12, 22, 32, …}| = ‫א‬0
 |{ rationals in (0,1) }| = ‫א‬0

‫א‬0
|{1, 2, 3, …}| = ‫א‬0
 |{12, 22, 32, …}| = ‫א‬0
 |{ rationals in (0,1) }| = ‫א‬0
 |{ rationals }| = ‫א‬0

‫א‬0
|{1, 2, 3, …}| = ‫א‬0
 |{12, 22, 32, …}| = ‫א‬0
 |{ rationals in (0,1) }| = ‫א‬0
 |{ rationals }| = ‫א‬0
 |{ algebraic numbers }| = ‫א‬0

‫א‬0
|{1, 2, 3, …}| = ‫א‬0
 |{12, 22, 32, …}| = ‫א‬0
 |{ rationals in (0,1) }| = ‫א‬0
 |{ rationals }| = ‫א‬0
 |{ algebraic numbers }| = ‫א‬0
 Arithmetic: ‫א‬0 + ‫ א‬0

‫א‬0
|{1, 2, 3, …}| = ‫א‬0
 |{12, 22, 32, …}| = ‫א‬0
 |{ rationals in (0,1) }| = ‫א‬0
 |{ rationals }| = ‫א‬0
 |{ algebraic numbers }| = ‫א‬0
 Arithmetic: ‫א‬0 + ‫ א‬0
 Cardinality and Dimensionality

Cantor’s Diagonal Argument
Cantor’s Diagonal Argument

|(0,1)|=c
Cantor’s Diagonal Argument
|(0,1)|=c
 c > ‫א‬0

Attacks
Attacks

Leopold Kronecker
Attacks
Leopold Kronecker
 Henri Poincare

Attacks
Leopold Kronecker
 Henri Poincare

Support
Attacks
Leopold Kronecker
 Henri Poincare

Support
David Hilbert
Georg Cantor
“My theory stands as firm as a rock; every
arrow directed against it will return quickly
to its archer. How do I know this? Because
I have studied it from all sides for many
years; because I have examined all
objections which have ever been made
against the infinite numbers; and above all
because I have followed its roots, so to
speak, to the first infallible cause of all
created things.”
Felix Hausdorff
Set theory is “a field in which
nothing is self-evident, whose
true statements are often
paradoxical, and whose plausible
ones are false.”
Foundations of Set Theory (1914)
Math May Seminar: Interlaken
Math May Seminar: Interlaken
Math May Seminar: Interlaken
Math May Seminar: Interlaken
Fun with ‫א‬0
Fun with ‫א‬0

Hilbert’s Hotel
Fun with ‫א‬0
Hilbert’s Hotel
 Bottles of Beer

The Power Set of S
The Power Set of S

S={1}
The Power Set of S
S={1}
 S={1, 2}

The Power Set of S
S={1}
 S={1, 2}
 S={1, 2, 3}

The Power Set of S
S={1}
 S={1, 2}
 S={1, 2, 3}
 |S|=2S

The Power Set of S

c=2 ‫א‬0
Axiom of Choice

If p is any collection of nonempty sets
{A,B,…}, then there exists a set Z
consisting of precisely one element each
from A, from B, and so on for all sets in p.
Continuum Hypothesis
1877 Cantor: “There is no set whose
cardinality is strictly between that of the
integers and that of the real numbers.”
 1900 Hilbert’s 1st problem
 1908 Ernst Zermelo: axiomatic set theory
 1922 Abraham Fraenkel
 1940 Kurt Godel
 1963 Paul Cohen
