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Cantor's diagonal argument | A Wisdom Archive on Cantor's diagonal argument | | Cantor's diagonal argument A selection of articles related to Cantor's diagonal argument | |
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Cantor's diagonal argument
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| ARTICLES RELATED TO Cantor's diagonal argument | | | |  Cantor's diagonal argument: Encyclopedia II - Mathematics - Notation, language, and rigorMost of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence o ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration, pure and applied mathematics, and aesthetics, Mathematics - Notation, language, and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Notation, language, and rigor |
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| | | | | | | Cantor's diagonal argument: Encyclopedia II - Ontological argument - Anselm's argument The ontological argument was first proposed by Anselm in Chapter 2 of the Proslogion. While Anselm did not propose an ontological system, he was very much concerned with the nature of being. He argued that there are necessary beings – things that cannot not exist – and contingent beings – things that may or may not exist, but whose existence is not necessary.
Anselm presents the ontological argument as part of a prayer directed to God. He starts with a definition of God, or a necessary assumption about the nature of God, or perhaps both.
"Now we believe that [the Lord] is som ...
See also:Ontological argument, Ontological argument - Anselm's argument, Ontological argument - Philosophical assumptions underlying the argument, Ontological argument - A modern description of the argument, Ontological argument - Criticisms and Objections, Ontological argument - Gaunilo's island, Ontological argument - Necessary nonexistence, Ontological argument - Existence as a property, Ontological argument - Miscellaneous, Ontological argument - Revisionists, Ontological argument - Descartes' ontological arguments, Ontological argument - Plantinga's modal form and contemporary discussion, Ontological argument - Bibliography Read more here: » Ontological argument: Encyclopedia II - Ontological argument - Anselm's argument |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Controversy over Cantor's theory - Footnote The quote "Later generations will regard set theory as a disease from which one has recovered" is from Kline[1982], and is apparently his translation of a quote from Poincaré's speech "The future of mathematics" given in 1908. There has been considerable dispute about what Poincaré actually intended to imply. Another translation reads "I think, [...] that it is important never to introduce any conception which may not be completely defined by a finite number of words. Whatever may be the remedy adopted, we can promise ourselves the joy of ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Footnote |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Ontological argument - Philosophical assumptions underlying the argument In order to understand the place this argument has in the history of philosophy, it is important to understand the essence of the argument in the context of the Influence of Hellenic philosophy on Christianity.
First, it is important to realize that Anselm's argument stemmed from the philosophical school of Realism. Realism was the dominant philosophical school of Anselm's day. According to Realism, and in contrast to Nominalism, things such as "greenness" and "bigness" were known as universals, which had a real existence outside the ...
See also:Ontological argument, Ontological argument - Anselm's argument, Ontological argument - Philosophical assumptions underlying the argument, Ontological argument - A modern description of the argument, Ontological argument - Criticisms and Objections, Ontological argument - Gaunilo's island, Ontological argument - Necessary nonexistence, Ontological argument - Existence as a property, Ontological argument - Miscellaneous, Ontological argument - Revisionists, Ontological argument - Descartes' ontological arguments, Ontological argument - Plantinga's modal form and contemporary discussion, Ontological argument - Bibliography Read more here: » Ontological argument: Encyclopedia II - Ontological argument - Philosophical assumptions underlying the argument |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Controversy over Cantor's theory - Reception of the argument From the start, Cantor's Theory was controversial among mathematicians and (later) philosophers.
I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there (Kronecker)
Later generations will regard [Cantor's] set theory as a disease from which one has recovered (Poincare 1908, see endnote)
Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Reception of the argument |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Controversy over Cantor's theory - Cantor's argument Cantor's 1891 argument is that there exists an infinite set (which he identifies with the set of real numbers), which has a larger number of elements, or as he puts it, has a greater 'power' (Mächtigkeit), than the infinite set of finite whole numbers 1, 2, 3, ...
There are a number of steps implicit in his argument, as follows
That the elements of no set can be put into one-to-one correspondence with all of its subsets. This is known as Cantor's theorem. It depends on very few of the assumptions of set theory, and (as J ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Cantor's argument |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Controversy over Cantor's theory - Naïve objections Objections to Cantor's proof (together with objections to Gödel's theorem) are a standard feature of mathematical Usenet discussions. These are generally flawed in some way.
Many of these objections depend on objections to step two of the argument. These typically use applications of the pigeonhole principle, or other assumptions that require "counting" all the natural numbers. Thus they rely on the assumption that we can "count" all such numbers by a process that at some point comes to an end. This is what Cantorians deny. They say ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Naïve objections |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Controversy over Cantor's theory - Objections to Cantor's theorem As shown above, most objections to Cantor's theorem (i.e. the theorem that no set can be correlated one-one with the set of all of its subsets) result from misunderstanding it (for it relies on mostly logical assumptions and steps).
Wittgenstein, however, disparages it as trivial, a result that might have been well known before the invention of set theory, "and familiar even to school-children". The child wonders, given a list of decimals, how to write a number different from any on the list. "The method says: Not at all: change the f ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Objections to Cantor's theorem |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Definable number - Other notions of definability The notion of definability treated in this article has been chosen primarily for definiteness, not on the grounds that it's more useful or interesting than other notions. Here we treat a few others:
Definable number - Definability in other languages or structures.
The language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the natural numbers. Since no variables of this language range over the reals, we can ...
See also:Definable number, Definable number - General facts, Definable number - Notion does not exhaust unambiguously described numbers, Definable number - Other notions of definability, Definable number - Definability in other languages or structures, Definable number - Definability with ordinal parameters Read more here: » Definable number: Encyclopedia II - Definable number - Other notions of definability |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Controversy over Cantor's theory - Objection to the axiom of infinity One of the most common (and also the most respectable) objections to Cantor's theory of infinite number involves the axiom of infinity. It is generally recognised view by all logicians that this axiom is not a logical truth. Indeed, as Mark Sainsbury (1979, p.305) has argued "there is room for doubt about whether it is a contingent truth, since it is an open question whether the universe is finite or infinite". Bertrand Russell for many years tried to establish a foundation for mathematics that did not rely on this axiom. Mayberry (2000, p.1 ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Objection to the axiom of infinity |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Mathematics - Notation language and rigor Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Notation language and rigor |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Mathematics - Mathematical tools Old:
Abacus
Napier's bones, slide rule
Ruler and compass
Mental calculation
New:
Calculators and computers
Programming languages
Computer algebra systems (listing)
Internet shorthand notation
statistical analysis software
SPSS
SAS programming language
R programming language
...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Mathematical tools |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Mathematics - Common misconceptions Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.
Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudosci ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Common misconceptions |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Primitive recursive function - Examples
Primitive recursive function - Addition.
Intuitively we would like to define addition recursively as:
add(0,x)=x
add(n+1,x)=add(n,x)+1
In order to fit this into a strict primitive recursive definition, we define:
add(0,x)=P11(x)
add(S(n),x)=S(P13(add(n,x),n,x))
(Note: here P13 is a func ...
See also:Primitive recursive function, Primitive recursive function - Definition, Primitive recursive function - Examples, Primitive recursive function - Addition, Primitive recursive function - Subtraction, Primitive recursive function - Limitations, Primitive recursive function - Bibliography Read more here: » Primitive recursive function: Encyclopedia II - Primitive recursive function - Examples |
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| | | | | | Cantor's diagonal argument: Encyclopedia II - Mathematics - Major themes in mathematics An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists.
Mathematics - Quantity.
Quantity starts with counting and measurement.
Natural numbers
...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Major themes in mathematics |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Mathematics - Overview of fields of mathematics As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the emp ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Overview of fields of mathematics |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Mathematics - Inspiration pure and applied mathematics and aesthetics Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that ins ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Inspiration pure and applied mathematics and aesthetics |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Controversy over Cantor's theory - Introduction Georg Cantor's argument that there are sets that have a cardinality (or "power" or "number") that is greater than the (already infinite) cardinality of the whole numbers 1,2,3,... has probably attracted more hostility than any other theoretical argument, before or since. Logician Wilfrid Hodges has commented on the energy devoted to refuting this "harmless little argument". What had it done to anyone to make them angry with it?
This article summarises the argument and ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Introduction |
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| | | | | Cantor's diagonal argument: Encyclopedia II - Mathematics - Is mathematics a science? Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".
If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is < ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Is mathematics a science? |
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