Paradoxes of the Infinite Kline XXV Pre-May Seminar March 14, 2011

Galileo Galilei ( ) Galileo Galilei ( )

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. 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. 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. 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. 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. 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? 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. 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 ( )

Czech Priest Czech Priest

Bernard Bolzano ( ) Czech Priest Czech Priest [0,1]~[0,2] [0,1]~[0,2]

Cardinality

Cardinality The number of elements in a set is the cardinality of the set. The number of elements in a set is the cardinality of the set.

Cardinality Card(S)=|S| Card(S)=|S|

Cardinality The number of elements in a set is the cardinality of the set. The number of elements in a set is the cardinality of the set. Card(S)=|S| Card(S)=|S| |{1,2,3}|=|{a,b,c}| |{1,2,3}|=|{a,b,c}|

Cardinality The number of elements in a set is the cardinality of the set. The number of elements in a set is the cardinality of the set. Card(S)=|S| Card(S)=|S| |{1,2,3}|=|{a,b,c}| |{1,2,3}|=|{a,b,c}| c=|[0,1]| c=|[0,1]|

Cardinality The number of elements in a set is the cardinality of the set. The number of elements in a set is the cardinality of the set. Card(S)=|S| Card(S)=|S| |{1,2,3}|=|{a,b,c}| |{1,2,3}|=|{a,b,c}| c=|[0,1]| c=|[0,1]| Lemma: c=|[a,b]| for any real a<b. Lemma: c=|[a,b]| for any real a<b.

Cardinality The number of elements in a set is the cardinality of the set. The number of elements in a set is the cardinality of the set. Card(S)=|S| Card(S)=|S| |{1,2,3}|=|{a,b,c}| |{1,2,3}|=|{a,b,c}| c=|[0,1]|. c=|[0,1]|. Lemma: c=|[a,b]| for any real a<b. Lemma: c=|[a,b]| for any real a<b. Lemma: |Reals|=c. Lemma: |Reals|=c.

Richard Dedekind ( ) Richard Dedekind ( )

Definition of infinite sets: Definition of infinite sets:

Georg Cantor ( )

א0א0א0א0 |{1, 2, 3, …}| = א 0 |{1, 2, 3, …}| = א 0

א0א0א0א0 |{1 2, 2 2, 3 2, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0

א0א0א0א0 |{1, 2, 3, …}| = א 0 |{1, 2, 3, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals in (0,1) }| = א 0

א0א0א0א0 |{1, 2, 3, …}| = א 0 |{1, 2, 3, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals }| = א 0 |{ rationals }| = א 0

א0א0א0א0 |{1, 2, 3, …}| = א 0 |{1, 2, 3, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals }| = א 0 |{ rationals }| = א 0 |{ algebraic numbers }| = א 0 |{ algebraic numbers }| = א 0

א0א0א0א0 |{1, 2, 3, …}| = א 0 |{1, 2, 3, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals }| = א 0 |{ rationals }| = א 0 |{ algebraic numbers }| = א 0 |{ algebraic numbers }| = א 0 Arithmetic: א 0 + א 0 Arithmetic: א 0 + א 0

א0א0א0א0 |{1, 2, 3, …}| = א 0 |{1, 2, 3, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{1 2, 2 2, 3 2, …}| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals in (0,1) }| = א 0 |{ rationals }| = א 0 |{ rationals }| = א 0 |{ algebraic numbers }| = א 0 |{ algebraic numbers }| = א 0 Arithmetic: א 0 + א 0 Arithmetic: א 0 + א 0 Cardinality and Dimensionality Cardinality and Dimensionality

Cantor’s Diagonal Argument

|(0,1)|=c |(0,1)|=c

Cantor’s Diagonal Argument |(0,1)|=c |(0,1)|=c c > א 0 c > א 0

Attacks

Attacks Leopold Kronecker Leopold Kronecker

Attacks Henri Poincare Henri Poincare

Attacks Leopold Kronecker Leopold Kronecker Henri Poincare Henri Poincare Support

Attacks Leopold Kronecker Leopold Kronecker Henri Poincare Henri Poincare Support David Hilbert 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

Fun with א 0

Hilbert’s Hotel Hilbert’s Hotel

Fun with א 0 Hilbert’s Hotel Hilbert’s Hotel Bottles of Beer Bottles of Beer

The Power Set of S

S={1} S={1}

The Power Set of S S={1} S={1} S={1, 2} S={1, 2}

The Power Set of S S={1} S={1} S={1, 2} S={1, 2} S={1, 2, 3} S={1, 2, 3}

The Power Set of S S={1} S={1} S={1, 2} S={1, 2} S={1, 2, 3} S={1, 2, 3} |S|=2 S |S|=2 S

The Power Set of S c=2 א0 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. 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.” 1877 Cantor: “There is no set whose cardinality is strictly between that of the integers and that of the real numbers.” 1900 Hilbert’s 1 st problem 1900 Hilbert’s 1 st problem 1908 Ernst Zermelo: axiomatic set theory 1908 Ernst Zermelo: axiomatic set theory 1922 Abraham Fraenkel 1922 Abraham Fraenkel 1940 Kurt Godel 1940 Kurt Godel 1963 Paul Cohen 1963 Paul Cohen