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Think about preparing a thesis statement that responds to ONE of the following essay questions, using appropriate course content as your evidence.
If you are preparing for this aspect of the final exam now, I would suggest first carefully read the essay questions. Determine – what is the question asking you? Then, go for a walk. Think about it, and determine your own point of view. Next, look back at our course materials. What evidence can you use to argue for your point of view? The evidence you use needs to demonstrate your knowledge of course content – your examples must come from the course content. And last but not least – you need to define your terms, so you know what you mean when you use those words. A good argument and tight logic requires a precise understanding of concepts.  
In your essay, you may refer to any course readings, lectures, or activities you feel would help you make the best historical argume nt to support your thesis claim. When you refer to course content in your essay, best practice would be to put in round brackets ( ) at the end of a sentence, the name of the author, or title of the lecture, or number of the activity, from which you draw that particular point, argument, idea or fact. This is a casual way of citing your sources, more casual than what I asked for in the historical essay assignment, but still requiring you to practice giving credit where appropriate.

In terms of the length of your essay. I am asking for a four-paragraph answer. An introduction and three body paragraphs. Each paragraph should contain a sub argument. The paragraph should identify what episode from history you have chosen to support your thesis statement, and then in the paragraph you need to make connections between that past event and the claim you have made. That means in each paragraph you are going to tell me some details about that past event but also why those details are relevant in terms of providing evidence to support your thesis claim.
Choose one of the following questions and answer it in the form of an essay. Your essay should have a clear and identifiable thesis statement and should be organized into several paragraphs. Three different episodes from the history of mathematics as drawn from our course materials should be used as historical evidence with which you support your thesis claim. At the end of your essay, a one to two sentence conclusion will be sufficient.

Q1. Mathematics and Society
Is mathematics separate from society? Have cultural beliefs and social values limited the ways in which mathematicians have practiced or invented mathematics? Or have mathematical developments had significant impact on society in modern history? Develop a thesis statement about the relationship between mathematics and society and draw upon three different examples from the course as evidence to support your claim.
Q2. Philosophy and the Limits of Mathematics
Does our philosophical belief system change what we might consider to be legitimate entities in mathematics? At times mathematical expressions once deemed inadmissible in mathematics have become foundational ideas for whole branches of the subject. Pick a philosophy of mathematics introduced in this course, defining your chosen position in the introduction to your essay. Taking as your examples complex numbers, Cantor’s transfinite set theory and non-Euclidean geometry, argue whether or not your philosophy admits or denies these mathematical entities and theories. Justify your claims using the criteria of the philosophical position you have chosen.THE HISTORY OF PROOF &
EMERGENCE OF PROOF BY COMPUTER
THIS WEEK
PEER SCHOLAR ACTIVITY “ASSESS” PHASE
Opens today during the activity period. This will close tonight at 11:59 p.m.
Tomorrow you will be able to login to Peer Scholar once again to receive
peer reviews on your historical essay. You will complete the “Reflect” phase
of the activity by briefly commenting on the helpfulness of each review.
You have until April 6 at 11:59 p.m. to complete the “Reflect” phase.
For completing all three phases of this activity you will receive 2% of your
course grade. If you do not complete all three phases, you won’t receive this
credit.
LOOKING AHEAD
REVISE DRAFT HISTORICAL ESSAY
Using the feedback you will get on your draft historical essay from course
staff and from peers in Peer Scholar, go ahead and revise your draft essay.
FINAL HISTORICAL ESSAY
Submit the final version by April 6 at 11:59 p.m. Your final historical essay
evaluation is 20% of your course grade.
FINAL EXAMINATION
Final examination is 30% of your grade. April 14 from 9-11 a.m.
Please check the exam schedule to know your correct room assignments.
IS PROOF NECESSARILY HUMAN?
?
TODAY’S LECTURE
Proof through the ages:
• Simple deductive proof
• Proof by contradiction
• Proof by complete mathematical induction
• Proofs of existence
A new type of proof:
• Proof by computer
• Four-colour theorem and its computer assisted proof
• Philosophical issues of using computers in mathematical proof
• Automated Theorem Provers (ATPs) and Interactive Theorem Provers
(ITPs) today
PROOF THROUGH THE AGES
TYPES OF PROOF
Simple deductive proof (c. 300 BCE)
Proof by contradiction (c. 300 BCE)
Proof by complete mathematical induction (c. ~1575 CE)
Proofs of existence (c. 300 BCE)
Proof by computer (c. ~1960 CE)
Will artificial intelligence tools be proving theorems in the future?
SIMPLE DEDUCTIVE PROOF
SIMPLE DEDUCTIVE PROOF
Simple deductive proof is given in
Euclid’s Elements, circa 300 BC
From these initial statements

23 definitions

5 postulates

5 common notions
constructions of plane and solid
geometric figures are given
Theorems about these figures and
their properties are proved using
the simple deductive proof
method
Euclid’s methods were typically
direct and constructive, with few
exceptions
SIMPLE DEDUCTIVE PROOF
An example of a simple deductive proof is Euclid’s Elements Book 1,
Proposition 15: If two straight lines cut one another, then they make the
vertical angles equal to one another.
PROOF
There are straight lines AB and CD which cut one another at point E.
[Proposition 13 on line CD]
Angle CEA + AED = 180o
Angle AED + DEB = 180o
Angle AED +
Proposition 13 If a straight line stands on a straight line,
then it makes either two right angles or angles whose
sum equals two right angles.
SIMPLE DEDUCTIVE PROOF
PROOF
There are straight lines AB and CD which cut one another at point E.
Angle CEA + AED = 180o
[Proposition 13 on line CD]
o
Angle AED + DEB = 180
[Proposition 13 on line AB]
Angle CEA + AED = angle AED + DEB [Postulate 4 and Common Notion 1]
Proposition 13 If a straight line stands on a straight line,
then it makes either two right angles or angles whose
sum equals two right angles.
Postulate 4 That all right angles are equal to one
another.
Common Notion 1 Things that are equal to the same
thing are equal to one another.
SIMPLE DEDUCTIVE PROOF
PROOF
There are straight lines AB and CD which cut one another at point E.
Angle CEA + AED = 180o
[Proposition 13 on line CD]
o
Angle AED + DEB = 180
[Proposition 13 on line AB]
Angle CEA + AED = angle AED + DEB [Postulate 4 and Common Notion 1]
Let the angle AED be subtracted from each.
Therefore remaining angle CEA = angle DEB
[Common Notion 3]
Proposition 13 If a straight line stands on a straight line,
then it makes either two right angles or angles whose
sum equals two right angles.
Postulate 4 That all right angles are equal to one
another.
Common Notion 1 Things that are equal to the same
thing are equal to one another.
Common Notion 3 If equals be subtracted from equals,
then the remainders are equal.
SIMPLE DEDUCTIVE PROOF
PROOF
There are straight lines AB and CD which cut one another at point E.
Angle CEA + AED = 180o
[Proposition 13 on line CD]
o
Angle AED + DEB = 180
[Proposition 13 on line AB]
Angle CEA + AED = angle AED + DEB [Postulate 4 and Common Notion 1]
Let the angle AED be subtracted from each.
Therefore remaining angle CEA = angle DEB
[Common Notion 3]
Similarly it can be proved that angles CEB, DEA are also equal.
Q.E.D.
Proposition 13 If a straight line stands on a straight line,
then it makes either two right angles or angles whose
sum equals two right angles.
Postulate 4 That all right angles are equal to one
another.
Common Notion 1 Things that are equal to the same
thing are equal to one another.
Common Notion 3 If equals be subtracted from equals,
then the remainders are equal.
SIMPLE DEDUCTIVE PROOF
Using simple deductive proof we
have demonstrated the truth of
Book 1, Proposition 15: If two
straight lines cut one another, then
they make the vertical angles
equal to one another.
Euclid’s Elements applies the
simple deductive proof method to
build up a system of theorems
from first principles. From initial
statements and prior theorems
more complex results are
deduced.
Euclid’s Elements is the model of
synthesis. From the axioms
everything that follows is put
together or synthesized.
SIMPLE DEDUCTIVE PROOF
Ian Mueller’s book Philosophy of
Mathematics and Deductive Structure
in Euclid’s Elements (1981) examines
Euclid’s deductive proof structure
from a philosophical and logical point
of view.
It shows how simple deductive proof
can invoke implicit assumptions (as
well as explicit assumptions, e.g.
Euclid’s stated postulates, common
notions, and definitions)
Euclid invoked implicit assumptions
like motion without deformation and
continuity of the continuum
PROOF BY CONTRADICTION
PROOF BY CONTRADICTION
Proof by contradiction is also called the reductio ad absurdum method.
In general, to prove a statement of the form P using proof by
contradiction, you begin with an assumption of not P and derive some
contradiction, Q and not Q. Because holding Q and not Q together
breaks the logical law of non-contradiction, conclude P must be true.
Proof by contradiction is an indirect proof method as the existence of P
is not necessarily shown by way of the proof of P.
An example of this is given in the Elements whereby Euclid employed
proof by contradiction to show the infinitude of primes without giving a
method by which one could produce an infinite progression of primes.
Cantor also used indirect proof to show the non-denumerability of the
continuum.
PROOF BY CONTRADICTION
Euclid proved that there are infinitely many prime numbers using proof
by contradiction. (à First assume not P)
PROOF
1. Suppose there are a finite number of primes (say n primes).
PROOF BY CONTRADICTION
PROOF
1. Suppose there are a finite number of primes (say n primes).
2. Multiply all n primes together and add 1 to form N.
N = p1 * p2 * p3 *…* pn + 1
PROOF BY CONTRADICTION
PROOF
1. Suppose there are a finite number of primes (say n primes).
2. Multiply all n primes together and add 1 to form N.
N = p1 * p2 * p3 *…* pn + 1
3. N is not divisible by p1 , p2 , p3 ,…, pn as diving by any of these gives a
remainder of 1.
PROOF BY CONTRADICTION
PROOF
1. Suppose there are a finite number of primes (say n primes).
2. Multiply all n primes together and add 1 to form N.
N = p1 * p2 * p3 *…* pn + 1
3. N is not divisible by p1 , p2 , p3 ,…, pn as diving by any of these gives a
remainder of 1.
4. Therefore, N must either be prime itself or be divisible by some other
prime not included on our original list.
PROOF BY CONTRADICTION
PROOF
1. Suppose there are a finite number of primes (say n primes).
2. Multiply all n primes together and add 1 to form N.
N = p1 * p2 * p3 *…* pn + 1
3. N is not divisible by p1 , p2 , p3 ,…, pn as diving by any of these gives a
remainder of 1.
4. Therefore, N must either be prime itself or be divisible by some other
prime not included on our original list.
5. Therefore, there must be at least n + 1 primes.
PROOF BY CONTRADICTION
PROOF
1. Suppose there are a finite number of primes (say n primes).
2. Multiply all n primes together and add 1 to form N.
N = p1 * p2 * p3 *…* pn + 1
3. N is not divisible by p1 , p2 , p3 ,…, pn as diving by any of these gives a
remainder of 1.
4. Therefore, N must either be prime or divisible by some other prime
not included on our original list.
5. Therefore, there must be at least n + 1 primes.
6. This is a contradiction (it was assumed that there are exactly n primes).
PROOF BY CONTRADICTION
PROOF
1. Suppose there are a finite number of primes (say n primes).
2. Multiply all n primes together and add 1 to form N.
N = p1 * p2 * p3 *…* pn + 1
3. N is not divisible by p1 , p2 , p3 ,…, pn as diving by any of these gives a
remainder of 1.
4. Therefore, N must either be prime or divisible by some other prime
not included on our original list.
5. Therefore, there must be at least n + 1 primes.
6. This is a contradiction (it was assumed that there are exactly n primes).
7. Therefore, the assumption that there is a finite number of primes is
false.
8. Therefore, there are an infinite number of primes.
PROOF BY CONTRADICTION
While Euclid has proved the infinitude of primes using proof by
contradiction, there is no method or procedure provided to construct
this infinite sequence of numbers. This is why proofs by contradiction
are sometimes thought of as non-constructive proof.
In the case of prime numbers a construction algorithm is known.
The Sieve of Eratosthenes is a well known algorithm for determining the
prime numbers up to a given number n. The Sieve of Eratosthenes is a
construction algorithm for the prime numbers.
PROOF BY CONTRADICTION
The Sieve of Eratosthenes is a well known algorithm for determining the
existence of prime numbers up to a given number n.
PROOF BY CONTRADICTION
Archimedes proves that A = π r 2 in Measurement of a Circle using the reductio
ad absurdum method of indirect proof.
PROOF STRATEGY
Show that the area of a circle is equivalent to a right-angled triangle in which the
height is equal to the radius and the base is the circumference of the circle.
In the book Measurement of a Circle proved that
He did this by showing that “The area of any circle is equal to a right-angled triangle in
which one of the sides about the right angle is equal to the radius, and the other to the
circumference of the circle”.
C=2π r
PROOF BY CONTRADICTION
PROOF STRATEGY
Show that the area of a circle is equivalent to a right-angled triangle in which the
height is equal to the radius and the base is the circumference of the circle.
Archimedes applied the reductio ad absurdum proof method on three cases
1. Circle area is greater than triangular area
In the book Measurement of a Circle proved that
2. Circle area is less than than triangular area
3. Circle area is equal to triangular area
He did this
by showing
that “The
area of any circle
is 1equal
a right-angled
triangle
in
Archimedes
determines
a contradiction
in cases
and 2to
thereby
proving case
3.
which one of the sides about the right angle is equal to the radius, and the other to the
circumference of the circle”.
C=2π r
PROOF BY CONTRADICTION
Archimedes determination of circular area is another classic example
proof by contradiction.
This proof was especially clever seeing as pi is not a constructible
number. Using Greek tools of compass and straightedge it is
impossible to produce a rectilinear figure of exact area π r2.
Archimedes had proved the circle was equal in area to an imaginary
triangle with A=π r2 without ever constructing such a triangle.
PROOF BY INDUCTION
PROOF BY COMPLETE MATHEMATICAL INDUCTION
Consider a statement P(n) involving the number n.
BASE CASE Show that P(1) holds.
INDUCTIVE STEP Show that if P(n) holds then P(n+1) holds.
Conclude P(n) holds for all n.
PROOF BY COMPLETE MATHEMATICAL INDUCTION
Francesco Maurolico in 1575 presented the first proof by complete
mathematical induction, showing 1+2+3+ …+n = n(n+1)/2.
1+2+3…..+100 = (100*101)/2 = 5050
PROOF BY INDUCTION
Show
PROOF BY COMPLETE
MATHEMATICAL INDUCTION
Pascal’s Traité du Triangle
Arithmétique (1654) shows the
relationship of the arithmetical triangle
to the binomial coefficients. This work
also contained the first explicit
formulation of the principle of proof
by induction.
PROOF BY COMPLETE MATHEMATICAL INDUCTION
Pascal used the triangle to perform calculations involving games of
chance. He studied patterns in the triangle and deduced nineteen
consequences for showing various number patterns in the triangle. In
the twelfth consequence, Pascal used complete mathematical induction
to prove how to generate the following property of the triangle.
If we designate the kth entry in the nth row of Pascal’s triangle as Cnk,
Pascal shows that Cn-1k+ Cnk-1 = Cnk is true by induction.
Showing this holds for the base case and then for any indexed cell, he
proved the triangle can be generated indefinitely.
PROOFS OF EXISTENCE
PROOFS OF EXISTENCE
One historical view is that the Greek emphasis on geometric
construction stemmed from their belief that to construct a geometric
object was to show that it existed.
Examples of some existence proofs:
• Gauss’s proof of the fundamental theorem of algebra (1797)
• Bolzano’s proof of the intermediate value theorem (1817)
• Liouville’s construction of a transcendental number (1844)
• Cantor’s proof of the existence of transcendental numbers (1874)
Existence proofs may be constructive or non-constructive.
.
PROOFS OF EXISTENCE
Euclid Book 13 There are only five convex regular polyhedra.
Turing’s Theorem: There exists no universal decision procedure (1936)
Every perfect number is even. Stated existentially: odd perfect numbers
do not exist (No proof yet. No perfect odd number has yet been
found.)
A perfect number is a positive integer that is equal to the sum of its
proper divisors.
.
Note: Perfect numbers 6, 28, 496, 8128…
PROOF BY COMPUTER
PROOF BY COMPUTER
1976: Appel, Haken, and Koch offered a proof of the Four-Colour
Theorem which used a computer result as part of their justification
procedure.
FOUR COLOUR THEOREM
Every planar map can be coloured with just four colours such that
neighbouring countries are always coloured differently.
The theoretical question of the legitimacy of computer proofs then
became tied to the practical question of whether the Four-Colour
Conjecture was solved.
PROOF BY COMPUTER
How many colors are necessary to color a map in such a way that no two
bordering countries have the same color?
Clearly at least four colors are required.
My dear Hamilton,…
A student of mine asked me to day to give him a reason for a fact
which I did not know was a fact – and do not yet. He says that if a figure be
any how divided and the compartments differently coloured so that figures
with any portion of common boundary line are differently coloured – four
colours may be wanted, but not more – the following is his case in which
four are wanted
Query cannot a necessity for five or more be invented…?
What do you say? And has it, if true been noticed? My pupil says
he guessed it in colouring a map of England… the more I think if it the
more evident it seems. If you retort with some very simple case which
makes me out a stupid animal, I think I must do as the Sphynx did….
[jokingly, leaping to his death]
…Augustus de Morgan’s letter to Hamilton from 1852
PROOF BY COMPUTER
Maps are equivalent to planar graphs
country ↔ vertex
boundary ↔ edge
PROOF BY COMPUTER
It is not too difficult to
prove that six colors will
color any map in such a
way that no two
bordering countries have
the same color.
It is more difficult but
possible to prove that
five colors will do the
job.
It is very difficult to prove
that four colors suffice to
color all maps.
PROOF BY COMPUTER
The idea was to prove the four-colour theorem by induction.
If all maps with n countries can be coloured with four colours, then so
can all maps with n+1 countries. But for 4CT a general method for
proceeding from maps with n countries to maps with n+1 countries was
difficult.
PROOF BY COMPUTER
The four-color conjecture was an outstanding problem that was finally
settled in 1976 with the aid of a mainframe computer at University of
Illinois.
PROOF BY COMPUTER
1976: Appel, Haken, and Koch offer a proof of the Four-Colour
conjecture using 1200 hours of computer time applying a procedure for
checking for whether a set of unavoidable configurations could be
reduced to show all were four-colourable configurations.
Their proof of the 4CT used an inductive method involving several
cases. The first case was trivial and the second case had several
subcases, and the third had over a thousand sub-cases, with 366 million
possible colourations to be checked. This checking cannot be handled
except by computers.
While computers being used in calculation is non-controversial, Appel,
Haken and Koch’s use of computers was significant because it lead to
an extension of knowledge in pure mathematics.
Computers à not just aids to calculation?
PROOF BY COMPUTER
Robin Wilson’s 2002 book gives a
detailed historical account of the
problem, and quite a detailed
exposition of attempts at a proof
before 1976 as well as details of the
proof by Appel, Haken, and Koch.
PHILOSOPHICAL IMPLICATIONS
OF COMPUTER PROOF
PHILOSOPHICAL IMPLICATIONS OF COMPUTER PROOF
What are the criteria for traditional mathematical proof?
1. Proofs are convincing.
2. Proofs are surveyable.
3. Proofs are formalizable.
Thomas Tymoczko’s 1979 article “The Four-Colour Problem and It’s
Philosophical Significance” says it’s the latter two criteria which are
really important.
PHILOSOPHICAL IMPLICATIONS OF COMPUTER PROOF
Are the criteria met for a proof using a computer?
1. The proof is convincing.
2. The proof is surveyable.
3. The proof is formalizable.
PHILOSOPHICAL IMPLICATIONS OF COMPUTER PROOF
Are the criteria met for a proof using a computer?
1. The proof is convincing YES
2. The proof is surveyable NO
3. The proof is formalizable YES and NO
The reliability of the proof rests upon reliability of the machine and of
the program. Most feel the reliability is sufficiently high to warrant
acceptance of the theorem. But Tymoczko says the reliability of the
4CT proof is lesser than traditional proof methods.
He concludes a computer proof is a “lesser” form of proof.
PHILOSOPHICAL IMPLICATIONS OF COMPUTER PROOF
Questions arise from accepting computers as a proof method:
What if computers become more clever at inventing proofs than
humans?
What if the computer is asked to prove the consistency of Peano
arithmetic and it delivered a result of inconsistency? If the result is
surprising will mathematicians accept it?
Do we accept the computer as an independent authority?
Do we accept a new vision of mathematics as an empirical science?
PHILOSOPHICAL IMPLICATIONS OF COMPUTER PROOF
Wolchover, “In Computers We Trust?”
In the 1970s computer-assisted proof
was met by feelings of mistrust
Isn’t computer code fallible?
Programs like Mathematica, Maple and
Magma are closed source.
Writing open-source theorem-proving
software and verifying previously
proved results is not highly rewarded.
In 1998 Thomas Hales proved the Kepler conjecture with a computer
checking possible cases to ensure no counter-examples existed.
ATPs and ITPs: PROOF BY COMPUTER TODAY
Automated Theorem Provers (ATPs) use brute-force methods to find
results. It’s hard for human mathematicians to follow the reasoning
behind why ATPs find a theorem correct.
Interactive Theorem Provers (ITPs) act as proof assistants and can verify
an argument or check existing proofs, the usefulness is guided by the
creativity and human ingenuity required in traditional proof.
The state of automated reasoning or theorem-proving using computers
is still developing. In 1998 the Kepler conjecture was proved using
computer techniques.
Is the use of computers in theorem-proving as controversial today (and
unpopular among mathematicians) as it was in 1977?
Will artificial intelligence tools be proving theorems in
the future?
Chat GPT — Open AI
Google – Bard
Will artificial intelligence tools put mathematicians out
of business?
END
CANTOR AND THE TRANSFINITE REALM
COURSE ANNOUNCEMENTS
COURSE ANNOUNCEMENTS
Term Test – grading is not yet complete, but I hope to publish your grades
and feedback this week.
Today’s activity will help focus and motivate you on starting your historical
essay assignment.
The graded activity for this week is the Thesis Statement assignment which
is due March 3. This is an individual assignment.
Ameer Sarwar will be here to lead your activity period. Regardless of your
group number attend the activity in AH400 today.
COURSE ANNOUNCEMENTS
Sylvia’s weekly office hours are cancelled for the time being.
Please email me if you wish to make a virtual meeting.
Tessa and Ameer have scheduled office hours to assist you in your course
work for the remainder of the semester:
Ameer – Feb 28, 2023 09:00 AM over Zoom
Tessa – Wednesday, March 8th @1:00-2:00pm JHB416
Ameer – Mar 10, 2023 09:00 AM over Zoom
Tessa – Wednesday, March 22nd @1:00-2:00pm JHB416
Ameer – Apr 3, 2023 09:00 AM over Zoom
Tessa – Wednesday, April 5th @1:00-2:00pm JHB416
We are happy to help you in OH with your course paper planning or term
test questions. Please take note of these upcoming opportunities to meet
with course staff. Come to OH rather than writing us emails.
THE TRANSFINITE REALM
ARE THERE INFINITIES OF DIFFERENT SIZES?
In the 1870s and 1880s, German mathematician Georg Cantor
developed set theory. Using this theory, he was able to demonstrate
that there are levels of infinity, or infinite sets of different sizes.
ARE THERE INFINITIES OF DIFFERENT SIZES?
Not everyone accepted or appreciated Cantor’s work.
Cantor’s teacher Leopold Kronecker called Cantor a corruptor of youth
Henri Poincaré allegedly described Cantor’s mathematics as a disease from
which one day mathematics would hopefully recover.
The essence of mathematics lies in its freedom – Georg Cantor
OVERVIEW OF TODAY’S LECTURE
THE MATH PART
• Review
• How can we order cardinal numbers?
• Power sets
• The continuum hypothesis
• Set theoretic paradoxes
• Zermelo Frankel axioms of set theory & independence result
THE HISTORY PART
• Cantor’s later life
REVIEW
REVIEW
What is Cantor’s definition of a set?
What does it mean for an infinite set to be countable/denumerable?
What does it mean for an infinite set to be non-denumerable?
How did Cantor define the first transfinite number?
What is a cardinal number?
What is an ordinal number?
REVIEW
If two sets M and N can be put into one-to-one correspondence with each
other, we say they have the same cardinality. Cantor discovered this idea
applies equally well to finite and infinite sets.
Cantor decides that any set in one-to-one correspondence with N is
denumerable or countably infinite. He invents a new number to represent
the size of all sets equivalent to N. This is the first transfinite number, ℵ
0
(pronounced “Aleph-naught”)
Cantor defines c (for continuum), a new cardinal number for the real
numbers in the interval (0,1) which are non-denumerable.
The real numbers, transcendental numbers, and irrational numbers all
have cardinality c, whereas the natural numbers, integers, rationals and
algebraic numbers have cardinality ℵ
0
REVIEW
Cantor had identified the first transfinite number, ℵ and then the second
0
transfinite number c.
It seemed to be the case that ℵ < c (inequality was unproven, yet) 0 What if there were more transfinite numbers than just two? By what procedure would Cantor order the size of these transfinite numbers? How could it be determined if one was greater, less than, or equal to another? To order the transfinite numbers Cantor needed two definitions and the Cantor-Schröder-Berstein theorem. WITH THIS THEOREM, IT CAN BE SHOWN THERE ARE NOT JUST TWO “LEVELS” OF INFINITY, BUT INFINITE INFINITIES… HOW CAN WE ORDER CARDINAL NUMBERS? WE NEED TWO DEFINITIONS AND THE CANTOR-SCHRÖDER-BERNSTEIN THEOREM HOW CAN WE ORDER CARDINAL NUMBERS? For transfinite sets A and B represent the cardinality of A by A, and the cardinality of B by B. Definition 1: If A and B are sets, then A≤B if there exists a one-to-one correspondence from all of members of A to a subset of the members of B. Example: We can put the natural numbers N and a subset of the rationals Q, into the one-to-one correspondence: So we conclude, by definition 1, that N≤Q. We know (from last lecture) that this is satisfied, because in fact N=Q=ℵ 0 HOW CAN WE ORDER CARDINAL NUMBERS? Definition 2: A Purchase answer to see full attachment

  
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