Hi Dimitre,
Two different things, go read the Wikipedia page Norman pointed at, as I said it gets it right. The fact that reals are transcendent does not affect their Cantor classification. And yes, I did learn this while is was in grade school, but not in class.... :)
>> 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
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Truly great madness cannot be achieved without significant intelligence.
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