paradoxes of the infinite kline xxv

Paradoxes of the Paradoxes of the InfiniteInfinite

Kline XXVKline XXV

Pre-May SeminarPre-May Seminar

March 14, 2011March 14, 2011

Galileo Galilei (1564-1642)Galileo Galilei (1564-1642)

Galileo: Galileo: Dialogue on Two Dialogue on Two New SciencesNew Sciences, 1638, 1638

SimplicioSimplicio: Here a difficulty presents itself which : Here a difficulty presents itself which appears to me insoluble. Since it is clear that we appears to me insoluble. Since it is clear that we may have one line segment longer than may have one line segment longer than another, each containing an infinite number of another, each containing an infinite number of points, we are forced to admit that, within one points, we are forced to admit that, within one and the same class, we may have something and the same class, we may have something greater than infinity, because the infinity of greater than infinity, because the infinity of points in the long line segment is greater than points in the long line segment is greater than the infinity of points in the short line segment. the infinity of points in the short line segment. This assigning to an infinite quantity a value This assigning to an infinite quantity a value greater than infinity is quite beyond my greater than infinity is quite beyond my comprehension.comprehension.

Galileo’s Galileo’s DialogoDialogo

SalviatiSalviati: This is one of the difficulties which : This is one of the difficulties which arise when we attempt, with our finite minds, arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those to discuss the infinite, assigning to it those properties which we give to the finite and properties which we give to the finite and limited; but this I think is wrong, for we limited; but this I think is wrong, for we cannot speak of infinite quantities as being cannot speak of infinite quantities as being the one greater or less than or equal to the one greater or less than or equal to another. To prove this I have in mind an another. To prove this I have in mind an argument, which, for the sake of clearness, I argument, which, for the sake of clearness, I shall put in the form of questions to Simplicio shall put in the form of questions to Simplicio who raised this difficulty.who raised this difficulty.


SalviatiSalviati: If I should ask further how : If I should ask further how many squares there are, one might many squares there are, one might reply truly that there are as many as reply truly that there are as many as the corresponding number of roots, the corresponding number of roots, since every square has its own root since every square has its own root and every root its own square, while and every root its own square, while no square has more than one root no square has more than one root and no root more than one square. and no root more than one square.

SimplicioSimplicio: Precisely so. : Precisely so.


SalviatiSalviati: But if I inquire how many roots : But if I inquire how many roots there are, it cannot be denied that there are there are, it cannot be denied that there are as many as there are numbers because every as many as there are numbers because every number is a root of some square. This being number is a root of some square. This being granted we must say that there are as many granted we must say that there are as many squares as there are numbers because they squares as there are numbers because they are just as numerous as their roots, and all are just as numerous as their roots, and all the numbers are roots. Yet at the outset we the numbers are roots. Yet at the outset we said there are many more numbers than said there are many more numbers than squares, since the larger portion of them are squares, since the larger portion of them are not squares.not squares.


SagredoSagredo: What then must one conclude : What then must one conclude under these circumstances? under these circumstances?

SalviatiSalviati: So far as I see we can only infer that : So far as I see we can only infer that the totality of all numbers is infinite, that the the totality of all numbers is infinite, that the number of squares is infinite, and that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of their roots is infinite; neither is the number of squares less than the totality of all number of squares less than the totality of all numbers, nor the latter greater than the numbers, nor the latter greater than the former; and finally the attributes "equal," former; and finally the attributes "equal," "greater," and "less" are not applicable to "greater," and "less" are not applicable to infinite, but only to finite, quantities.infinite, but only to finite, quantities.

Bernard Bolzano (1781-Bernard Bolzano (1781-1848)1848)


Czech PriestCzech Priest


Czech PriestCzech Priest [0,1]~[0,2][0,1]~[0,2]

CardinalityCardinality


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



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



Card(S)=|S|Card(S)=|S| |{1,2,3}|=|{a,b,c}||{1,2,3}|=|{a,b,c}|



Card(S)=|S|Card(S)=|S| |{1,2,3}|=|{a,b,c}||{1,2,3}|=|{a,b,c}| c=|[0,1]|c=|[0,1]|



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.



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 (1831-Richard Dedekind (1831-1916)1916)

Richard Dedekind (1831-Richard Dedekind (1831-1916)1916)

Definition of Definition of infinite sets:infinite sets:

Georg Cantor (1845-1918)Georg Cantor (1845-1918)

00אא

00אא = |{… ,3 ,2 ,1}| = |{… ,3 ,2 ,1}|

00אא

00אא = |{… ,3 ,2 ,1}| = |{… ,3 ,2 ,1}|

00אא = |{… , = |{… ,322 ,3 ,222 ,2 ,122}|1}|

00אא

00אא = |{… ,3 ,2 ,1}| = |{… ,3 ,2 ,1}|

00אא = |{… , = |{… ,322 ,3 ,222 ,2 ,122}|1}|

|{ rationals in (0,1) }| = |{ rationals in (0,1) }| = 00אא

00אא

00אא = |{… ,3 ,2 ,1}| = |{… ,3 ,2 ,1}|

00אא = |{… , = |{… ,322 ,3 ,222 ,2 ,122}|1}|


|{ rationals }| = |{ rationals }| = 00אא

00אא

00אא = |{… ,3 ,2 ,1}| = |{… ,3 ,2 ,1}|

00אא = |{… , = |{… ,322 ,3 ,222 ,2 ,122}|1}|



|{ algebraic numbers }| = |{ algebraic numbers }| = 00אא

00אא

00אא = |{… ,3 ,2 ,1}| = |{… ,3 ,2 ,1}|

00אא = |{… , = |{… ,322 ,3 ,222 ,2 ,122}|1}|




Arithmetic: Arithmetic: 00א א + + 00אא

00אא

00אא = |{… ,3 ,2 ,1}| = |{… ,3 ,2 ,1}|

00אא = |{… , = |{… ,322 ,3 ,222 ,2 ,122}|1}|




Arithmetic: Arithmetic: 00א א + + 00אא

Cardinality and DimensionalityCardinality and Dimensionality

Cantor’s Diagonal ArgumentCantor’s Diagonal Argument


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


|(0,1)|=c|(0,1)|=c c > c > 00אא

AttacksAttacks

AttacksAttacks

Leopold KroneckerLeopold Kronecker

AttacksAttacks

Leopold KroneckerLeopold Kronecker Henri PoincareHenri Poincare

AttacksAttacks


SupportSupport

AttacksAttacks


SupportSupport

David HilbertDavid Hilbert

Georg CantorGeorg Cantor““My theory stands as firm as a rock; My theory stands as firm as a rock; every arrow directed against it will every arrow directed against it will return quickly to its archer. How do I return quickly to its archer. How do I know this? Because I have studied it know this? Because I have studied it from all sides for many years; because I from all sides for many years; because I have examined all objections which have examined all objections which have ever been made against the have ever been made against the infinite numbers; and above all because infinite numbers; and above all because I have followed its roots, so to speak, to I have followed its roots, so to speak, to the first infallible cause of all created the first infallible cause of all created things.”things.”

Felix HausdorffFelix Hausdorff

Set theory is “a field in which Set theory is “a field in which nothing is self-evident, whose nothing is self-evident, whose true statements are often true statements are often paradoxical, and whose paradoxical, and whose plausible ones are false.” plausible ones are false.”

Foundations of Set TheoryFoundations of Set Theory (1914)(1914)

Math May Seminar: Math May Seminar: InterlakenInterlaken

Fun with Fun with 00אא


Hilbert’s HotelHilbert’s Hotel


Hilbert’s HotelHilbert’s Hotel Bottles of BeerBottles of Beer

The Power Set of The Power Set of SS

The Power Set of The Power Set of S S

S={1}S={1}


S={1}S={1} S={1, 2}S={1, 2}


S={1}S={1} S={1, 2}S={1, 2} S={1, 2, 3}S={1, 2, 3}


S={1}S={1} S={1, 2}S={1, 2} S={1, 2, 3}S={1, 2, 3} |S|=2|S|=2SS


c=2c=2 00אא

Axiom of ChoiceAxiom of Choice

If If pp is any collection of nonempty is any collection of nonempty sets {sets {A,B,…A,B,…}, then there exists a set }, then there exists a set ZZ consisting of precisely one element consisting of precisely one element each from each from AA, from , from BB, and so on for all , and so on for all sets in sets in pp..

Continuum HypothesisContinuum Hypothesis

1877 Cantor: “There is no set whose 1877 Cantor: “There is no set whose cardinality is strictly between that of the cardinality is strictly between that of the integers and that of the real numbers.”integers and that of the real numbers.”

1900 Hilbert’s 11900 Hilbert’s 1stst problem problem 1908 Ernst Zermelo: axiomatic set 1908 Ernst Zermelo: axiomatic set

theorytheory 1922 Abraham Fraenkel1922 Abraham Fraenkel 1940 Kurt Godel1940 Kurt Godel 1963 Paul Cohen1963 Paul Cohen

paradoxes of the infinite kline xxv

Documents

number of squares

infinite number of points

corresponding number

infinite quantities

infinite quantity

infinity of points

value greater

galileos dialogosalviati