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Understanding Canters Diagonal Argumentative Essays

This article is about a concept in set and number theory. Not to be confused with matrix diagonalization. See diagonalization (disambiguation) for several other uses of the term in mathematics.

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.[1][2][3] Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874.[4][5] However, it demonstrates a powerful and general technique that has since been used in a wide range of proofs,[6] including the first of Gödel's incompleteness theorems[2] and Turing's answer to the Entscheidungsproblem. Diagonalization arguments are often also the source of contradictions like Russell's paradox[7][8] and Richard's paradox.[9]

Uncountable set[edit]

In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one). He begins with a constructive proof of the following theorem:

If s1, s2, … , sn, … is any enumeration of elements from T, then there is always an element s of T which corresponds to no sn in the enumeration.

The proof starts with an enumeration of elements from T, for example:

s1 =(0,0,0,0,0,0,0,...)
s2 =(1,1,1,1,1,1,1,...)
s3 =(0,1,0,1,0,1,0,...)
s4 =(1,0,1,0,1,0,1,...)
s5 =(1,1,0,1,0,1,1,...)
s6 =(0,0,1,1,0,1,1,...)
s7 =(1,0,0,0,1,0,0,...)
...

Next, a sequence s is constructed by choosing the 1st digit as complementary to the 1st digit of s1 (swapping 0s for 1s and vice versa), the 2nd digit as complementary to the 2nd digit of s2, the 3rd digit as complementary to the 3rd digit of s3, and generally for every n, the nth digit as complementary to the nth digit of sn. For the example above, this yields:

s1=(0,0,0,0,0,0,0,...)
s2=(1,1,1,1,1,1,1,...)
s3=(0,1,0,1,0,1,0,...)
s4=(1,0,1,0,1,0,1,...)
s5=(1,1,0,1,0,1,1,...)
s6=(0,0,1,1,0,1,1,...)
s7=(1,0,0,0,1,0,0,...)
...
s=(1,0,1,1,1,0,1,...)

By construction, s differs from each sn, since their nth digits differ (highlighted in the example). Hence, s cannot occur in the enumeration.

Based on this theorem, Cantor then uses a proof by contradiction to show that:

The set T is uncountable.

The proof starts by assuming that T is countable. Then all its elements can be written as an enumeration s1, s2, … , sn, … . Applying the previous theorem to this enumeration produces a sequence s not belonging to the enumeration. However, this contradicts s being an element of T and therefore belonging to the enumeration. This contradiction implies that the original assumption is false. Therefore, T is uncountable.

Interpretation[edit]

The interpretation of Cantor's result will depend upon one's view of mathematics. To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. It does not rule out the possibility that the latter are subcountable.[citation needed] In the context of classical mathematics, this is impossible, and the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers.[citation needed]

Real numbers[edit]

The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from the above result. To prove this, an injection will be constructed from the set T of infinite binary strings to the set R of real numbers. Since T is uncountable, the image of this function, which is a subset of R, is uncountable. Therefore, R is uncountable. Also, by using a method of construction devised by Cantor, a bijection will be constructed between T and R. Therefore, T and R have the same cardinality, which is called the "cardinality of the continuum" and is usually denoted by or .

An injection from T to R is given by mapping strings in T to decimals, such as mapping t = 0111… to the decimal 0.0111…. This function, defined by f(t) = 0.t, is an injection because it maps different strings to different numbers.

Instead of mapping 0111… to the decimal 0.0111…, it can be mapped to the baseb number: 0.0111…b. This leads to the family of functions: fb(t) = 0.tb. The functions fb(t) are injections, except for f2(t). This function will be modified to produce a bijection between T and R.

Construction of a bijection between T and R

This construction uses a method devised by Cantor that was published in 1878. He used it to construct a bijection between the closed interval [0, 1] and the irrationals in the open interval (0, 1). He first removed a countably infinite subset from each of these sets so that there is a bijection between the remaining uncountable sets. Since there is a bijection between the countably infinite subsets that have been removed, combining the two bijections produces a bijection between the original sets.[10]

Cantor's method can be used to modify the function f2(t) = 0.t2 to produce a bijection from T to (0, 1). Because some numbers have two binary expansions, f2(t) is not even injective. For example, f2(1000…) = 0.1000…2 = 1/2 and f2(0111…) = 0.0111…2 = 1/4 + 1/8 + 1/16 + … = 1/2, so both 1000… and 0111… map to the same number, 1/2.

To modify f2(t), observe that it is a bijection except for a countably infinite subset of (0, 1) and a countably infinite subset of T. It is not a bijection for the numbers in (0, 1) that have two binary expansions. These numbers have the form m/2n where m is an odd integer and n is a natural number. Put these numbers in the sequence: r = (1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, …). Also, f2(t) is not a bijection to (0, 1) for the strings in T appearing after the binary point in the binary expansions of 0, 1, and the numbers in sequence r. Put these eventually-constant strings in the sequence: s = (000…, 111…, 1000…, 0111…, 01000…, 00111…, 11000…, 10111…, ...). Define the bijection g(t) from T to (0, 1): If t is the nth string in sequence s, let g(t) be the nth number in sequence r; otherwise, g(t) = 0.t2.

To construct a bijection from T to R, start with the tangent function tan(x), which is a bijection from (−π/2, π/2) to R (see the figure shown on the right). Next observe that the linear functionh(x) = πx – π/2 is a bijection from (0, 1) to (−π/2, π/2) (see the figure shown on the left). The composite function tan(h(x)) = tan(πx – π/2) is a bijection from (0, 1) to R. Composing this function with g(t) produces the function tan(h(g(t))) = tan(πg(t) – π/2), which is a bijection from T to R.

The function h: (0,1) → (−π/2,π/2)

The function tan: (−π/2,π/2) → R

General sets[edit]

A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every setS, the power set of S—that is, the set of all subsets of S (here written as P(S))—has a larger cardinality than S itself. This proof proceeds as follows:

Let f be any function from S to P(S). It suffices to prove f cannot be surjective. That means that some member T of P(S), i.e. some subset of S, is not in the image of f. As a candidate consider the set:

T = { sS: sf(s) }.

For every s in S, either s is in T or not. If s is in T, then by definition of T, s is not in f(s), so T is not equal to f(s). On the other hand, if s is not in T, then by definition of T, s is in f(s), so again T is not equal to f(s); cf. picture. For a more complete account of this proof, see Cantor's theorem.

Consequences[edit]

This result implies that the notion of the set of all sets is an inconsistent notion. If S were the set of all sets then P(S) would at the same time be bigger than S and a subset of S.

Russell's Paradox has shown us that naive set theory, based on an unrestricted comprehension scheme, is contradictory. Note that there is a similarity between the construction of T and the set in Russell's paradox. Therefore, depending on how we modify the axiom scheme of comprehension in order to avoid Russell's paradox, arguments such as the non-existence of a set of all sets may or may not remain valid.

The diagonal argument shows that the set of real numbers is "bigger" than the set of natural numbers (and therefore, the integers and rationals as well). Therefore, we can ask if there is a set whose cardinality is "between" that of the integers and that of the reals. This question leads to the famous continuum hypothesis. Similarly, the question of whether there exists a set whose cardinality is between |S| and |P(S)| for some infinite S leads to the generalized continuum hypothesis.

Analogues of the diagonal argument are widely used in mathematics to prove the existence or nonexistence of certain objects. For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument. Also, diagonalization was originally used to show the existence of arbitrarily hard complexity classes and played a key role in early attempts to prove P does not equal NP.

Version for Quine's New Foundations[edit]

The above proof fails for W. V. Quine's "New Foundations" set theory (NF). In NF, the naive axiom scheme of comprehension is modified to avoid the paradoxes by introducing a kind of "local" type theory. In this axiom scheme,

{ sS: sf(s) }

is not a set — i.e., does not satisfy the axiom scheme. On the other hand, we might try to create a modified diagonal argument by noticing that

{ sS: sf({s}) }

is a set in NF. In which case, if P1(S) is the set of one-element subsets of S and f is a proposed bijection from P1(S) to P(S), one is able to use proof by contradiction to prove that |P1(S)| < |P(S)|.

The proof follows by the fact that if f were indeed a map ontoP(S), then we could find r in S, such that f({r}) coincides with the modified diagonal set, above. We would conclude that if r is not in f({r}), then r is in f({r}) and vice versa.

It is not possible to put P1(S) in a one-to-one relation with S, as the two have different types, and so any function so defined would violate the typing rules for the comprehension scheme.

See also[edit]

References[edit]

  1. ^Georg Cantor (1891). "Ueber eine elementare Frage der Mannigfaltigkeitslehre". Jahresbericht der Deutschen Mathematiker-Vereinigung 1890–1891. 1: 75–78 (84–87 in pdf file).  English translation: Ewald, William B. (ed.) (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2. Oxford University Press. pp. 920–922. ISBN 0-19-850536-1. 
  2. ^ abKeith Simmons (30 July 1993). Universality and the Liar: An Essay on Truth and the Diagonal Argument. Cambridge University Press. pp. 20–. ISBN 978-0-521-43069-2. 
  3. ^Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill. p. 30. ISBN 0070856133. 
  4. ^Gray, Robert (1994), "Georg Cantor and Transcendental Numbers"(PDF), American Mathematical Monthly, 101 (9): 819–832, doi:10.2307/2975129, JSTOR 2975129 
  5. ^Bloch, Ethan D. (2011). The Real Numbers and Real Analysis. New York: Springer. p. 429. ISBN 978-0-387-72176-7. 
  6. ^Sheppard, Barnaby (2014). The Logic of Infinity (illustrated ed.). Cambridge University Press. p. 73. ISBN 978-1-107-05831-6. Extract of page 73
  7. ^"Russell's paradox". Stanford encyclopedia of philosophy. 
  8. ^Bertrand Russell (1931). Principles of mathematics. Norton. pp. 363–366. 
  9. ^Keith Simmons (30 July 1993). Universality and the Liar: An Essay on Truth and the Diagonal Argument. Cambridge University Press. p. 27. ISBN 978-0-521-43069-2. 
  10. ^See page 254 of Georg Cantor (1878), "Ein Beitrag zur Mannigfaltigkeitslehre", Journal für die Reine und Angewandte Mathematik, 84: 242–258 . This proof is discussed in Joseph Dauben (1979), Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, ISBN 0-674-34871-0 , pp. 61–62, 65. On page 65, Dauben proves a result that is stronger than Cantor's. He lets "φν denote any sequence of rationals in [0, 1]." Cantor lets φν denote a sequence enumerating the rationals in [0, 1], which is the kind of sequence needed for his construction of a bijection between [0, 1] and the irrationals in (0, 1).

External links[edit]

An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.
An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijectionf(x)=2x from the natural to the even numbers demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.
Illustration of the generalized diagonal argument: The set T = {n∈ℕ: nf(n)} at the bottom cannot occur anywhere in the range of f:ℕ→P(ℕ). The example mapping f happens to correspond to the example enumeration s in the above picture.

Argumentative Essays

Summary:

The Modes of Discourse—Exposition, Description, Narration, Argumentation (EDNA)—are common paper assignments you may encounter in your writing classes. Although these genres have been criticized by some composition scholars, the Purdue OWL recognizes the wide spread use of these approaches and students’ need to understand and produce them.

Contributors: Jack Baker, Allen Brizee, Elizabeth Angeli
Last Edited: 2013-03-10 11:46:44

What is an argumentative essay?

The argumentative essay is a genre of writing that requires the student to investigate a topic; collect, generate, and evaluate evidence; and establish a position on the topic in a concise manner.

Please note: Some confusion may occur between the argumentative essay and the expository essay. These two genres are similar, but the argumentative essay differs from the expository essay in the amount of pre-writing (invention) and research involved. The argumentative essay is commonly assigned as a capstone or final project in first year writing or advanced composition courses and involves lengthy, detailed research. Expository essays involve less research and are shorter in length. Expository essays are often used for in-class writing exercises or tests, such as the GED or GRE.

Argumentative essay assignments generally call for extensive research of literature or previously published material. Argumentative assignments may also require empirical research where the student collects data through interviews, surveys, observations, or experiments. Detailed research allows the student to learn about the topic and to understand different points of view regarding the topic so that she/he may choose a position and support it with the evidence collected during research. Regardless of the amount or type of research involved, argumentative essays must establish a clear thesis and follow sound reasoning.

The structure of the argumentative essay is held together by the following.

  • A clear, concise, and defined thesis statement that occurs in the first paragraph of the essay.

In the first paragraph of an argument essay, students should set the context by reviewing the topic in a general way. Next the author should explain why the topic is important (exigence) or why readers should care about the issue. Lastly, students should present the thesis statement. It is essential that this thesis statement be appropriately narrowed to follow the guidelines set forth in the assignment. If the student does not master this portion of the essay, it will be quite difficult to compose an effective or persuasive essay.

  • Clear and logical transitions between the introduction, body, and conclusion.

Transitions are the mortar that holds the foundation of the essay together. Without logical progression of thought, the reader is unable to follow the essay’s argument, and the structure will collapse. Transitions should wrap up the idea from the previous section and introduce the idea that is to follow in the next section.

  • Body paragraphs that include evidential support.

Each paragraph should be limited to the discussion of one general idea. This will allow for clarity and direction throughout the essay. In addition, such conciseness creates an ease of readability for one’s audience. It is important to note that each paragraph in the body of the essay must have some logical connection to the thesis statement in the opening paragraph. Some paragraphs will directly support the thesis statement with evidence collected during research. It is also important to explain how and why the evidence supports the thesis (warrant).

However, argumentative essays should also consider and explain differing points of view regarding the topic. Depending on the length of the assignment, students should dedicate one or two paragraphs of an argumentative essay to discussing conflicting opinions on the topic. Rather than explaining how these differing opinions are wrong outright, students should note how opinions that do not align with their thesis might not be well informed or how they might be out of date.

  • Evidential support (whether factual, logical, statistical, or anecdotal).

The argumentative essay requires well-researched, accurate, detailed, and current information to support the thesis statement and consider other points of view. Some factual, logical, statistical, or anecdotal evidence should support the thesis. However, students must consider multiple points of view when collecting evidence. As noted in the paragraph above, a successful and well-rounded argumentative essay will also discuss opinions not aligning with the thesis. It is unethical to exclude evidence that may not support the thesis. It is not the student’s job to point out how other positions are wrong outright, but rather to explain how other positions may not be well informed or up to date on the topic.

  • A conclusion that does not simply restate the thesis, but readdresses it in light of the evidence provided.

It is at this point of the essay that students may begin to struggle. This is the portion of the essay that will leave the most immediate impression on the mind of the reader. Therefore, it must be effective and logical. Do not introduce any new information into the conclusion; rather, synthesize the information presented in the body of the essay. Restate why the topic is important, review the main points, and review your thesis. You may also want to include a short discussion of more research that should be completed in light of your work.

A complete argument

Perhaps it is helpful to think of an essay in terms of a conversation or debate with a classmate. If I were to discuss the cause of World War II and its current effect on those who lived through the tumultuous time, there would be a beginning, middle, and end to the conversation. In fact, if I were to end the argument in the middle of my second point, questions would arise concerning the current effects on those who lived through the conflict. Therefore, the argumentative essay must be complete, and logically so, leaving no doubt as to its intent or argument.

The five-paragraph essay

A common method for writing an argumentative essay is the five-paragraph approach. This is, however, by no means the only formula for writing such essays. If it sounds straightforward, that is because it is; in fact, the method consists of (a) an introductory paragraph (b) three evidentiary body paragraphs that may include discussion of opposing views and (c) a conclusion.

Longer argumentative essays

Complex issues and detailed research call for complex and detailed essays. Argumentative essays discussing a number of research sources or empirical research will most certainly be longer than five paragraphs. Authors may have to discuss the context surrounding the topic, sources of information and their credibility, as well as a number of different opinions on the issue before concluding the essay. Many of these factors will be determined by the assignment.

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