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Re: [xml-dev] Infinity

*From*:**Dimitre Novatchev <dnovatchev@gmail.com>***To*: Peter Hunsberger <peter.hunsberger@gmail.com>*Date*: Sat, 3 Mar 2018 14:30:31 -0800

>> of integers. In contrast, the number of real numbers is a 'larger >> infinity' than the number of integers. If you wish to further explore >> this rabbit hole, see <https://en.wikipedia.org/wiki/Aleph_number> and >> work outwards... >> >> > > Actually no, and thankfully the Wikipedia page gets this right. Peter, sorry, but you are mistaken on this. This must be studied even in high school math classes nowadays. The set R of real numbers is denser than the set N of natural numbers (and equivalently the set of all integers). This is due to the fact that there are real numbers that are not rational numbers (transcendental numbers such as Pi or e). And if we need Wikipedia for this, here is what it says (https://en.wikipedia.org/wiki/Transcendental_number): "In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. " To repeat: The sets of real and complex numbers are both uncountable. And almost all real and complex numbers are transcendental. Cheers, Dimitre On Sat, Mar 3, 2018 at 2:05 PM, Peter Hunsberger <peter.hunsberger@gmail.com> wrote: > > On Sat, Mar 3, 2018 at 7:33 AM Norman Gray <norman@astro.gla.ac.uk> >> >> >> It will be, but since there are as many elements in that set as there >> are positive integers (they can be put into a one-to-one >> correspondence), it is no bigger or smaller an infinity than the number >> of integers. In contrast, the number of real numbers is a 'larger >> infinity' than the number of integers. If you wish to further explore >> this rabbit hole, see <https://en.wikipedia.org/wiki/Aleph_number> and >> work outwards... >> >> > > Actually no, and thankfully the Wikipedia page gets this right. Integers and > reals are both of cardinality Aleph naught. The easiest way to conceptualize > this equivalence is to think of them both as being mappable to a set of > points on a line. The set of solutions to all polynomials is traditionally > considered not to be of Aleph naught, (one can think of this as being some > what multidimensional). However, I have recently seen some arguments to the > contrary, though I have not spent the time to dig into them. > > Peter Hunsberger >> >> >> _______________________________________________________________________ >> > -- > Peter Hunsberger -- Cheers, Dimitre Novatchev --------------------------------------- Truly great madness cannot be achieved without significant intelligence. --------------------------------------- To invent, you need a good imagination and a pile of junk ------------------------------------- Never fight an inanimate object ------------------------------------- To avoid situations in which you might make mistakes may be the biggest mistake of all ------------------------------------ Quality means doing it right when no one is looking. ------------------------------------- You've achieved success in your field when you don't know whether what you're doing is work or play ------------------------------------- To achieve the impossible dream, try going to sleep. ------------------------------------- Facts do not cease to exist because they are ignored. ------------------------------------- Typing monkeys will write all Shakespeare's works in 200yrs.Will they write all patents, too? :) ------------------------------------- Sanity is madness put to good use. ------------------------------------- I finally figured out the only reason to be alive is to enjoy it.

**Follow-Ups**:**Re: [xml-dev] Infinity***From:*"Liam R. E. Quin" <liam@w3.org>

**Re: [xml-dev] Infinity***From:*Peter Hunsberger <peter.hunsberger@gmail.com>

**References**:**Infinity***From:*Michael Kay <mike@saxonica.com>

**Re: [xml-dev] Infinity***From:*Elliotte Rusty Harold <elharo@ibiblio.org>

**Re: [xml-dev] Infinity***From:*"Norman Gray" <norman@astro.gla.ac.uk>

**Re: [xml-dev] Infinity***From:*Peter Hunsberger <peter.hunsberger@gmail.com>

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