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Preintuitionism | A Wisdom Archive on Preintuitionism | | Preintuitionism A selection of articles related to Preintuitionism | |
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Controversy over Cantor's theory, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Footnote, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objection to the axiom of infinity, 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 - Objections to the power set axiom, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Reception of the argument, Preintuitionism
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ARTICLES RELATED TO Preintuitionism | |
| | |  Preintuitionism: Encyclopedia II - Controversy over Cantor's theory - Objections to Hume's principleAs argued above, many naïve objections depend on implicitly denying Hume's principle, and are therefore question-begging. Wittgenstein explicitly denies the principle, arguing that our concept of number depends essentially on counting. "Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one"
The expressions "divisible into two parts" and "divisible without limit" have completely different forms. This is, of course, the same case as the one in which someone operat ...
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 Hume's principle |
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| | | | Preintuitionism: 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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| | | | Preintuitionism: 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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| | | | Preintuitionism: Encyclopedia II - Controversy over Cantor's theory - Preface The pure mathematicians and applied mathematicians who object to Cantor's theory of sets claim that Cantor introduced into mathematics an element of fantasy that should be expunged. The basic "anti-Cantorian" argument was stated most elegantly and concisely by Hermann Weyl when he wrote:
...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall ...
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 - Preface |
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| | | | Preintuitionism: 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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| | | | Preintuitionism: 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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| | | | Preintuitionism: 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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| | | | Preintuitionism: 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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| | | | Preintuitionism: 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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| | | | Preintuitionism: Encyclopedia II - Preintuitionism - Arguments over the excluded middle It was for this assertion, among others, that Poincaré was considered to be similar to the intuitionists. For Brouwer though, the Pre-Intuitionists failed to go as far as necessary in divesting mathematics from metaphysics, for they still used principium tertii exclusi or the "Law of excluded middle". (Note: It actually reads "principle of the excluded third", but it is not commonly known by that name.)
The principle of the excluded middle does lead to some strange situations. Such as the question in regard to the future, "Wil ...
See also:Preintuitionism, Preintuitionism - The introduction of natural numbers, Preintuitionism - The principle of complete induction, Preintuitionism - Arguments over the excluded middle, Preintuitionism - Other Pre-Intuitionists Read more here: » Preintuitionism: Encyclopedia II - Preintuitionism - Arguments over the excluded middle |
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