A new (?) construction of R
Denis Feldmann
(http://en.wikipedia.org/wiki/Construction_of_real_numbers) give a
really incredible (and elementary) construction in terms of equivalence
classes of quasi-morphims of Z, ie f such that f(m+n)-f(m)-f(n) is a
bounded set ; see
http://en.wikipedia.org/wiki/Construction_of_real_numbers#Construction_from_the_group_of_integers
for details
José Carlos Santos
Cute! I had never been exposed to this construction.
Best regards,
Jose Carlos Santos
Herman Jurjus
If memory doesn't betray me, many years ago someone posted a
construction very similar to this, on this very newsgroup (sci.math), to
construct [0, infty) directly from N. But google is not my friend, this
time.
--
Cheers,
Herman Jurjus
David C. Ullrich
Heh - that _is_ very interesting.
Took me a second to see what the isomoprhism from the usual reals
to these gizmos is. If r is a usual real define f_r : Z -> Z by saying
f_r(n) = the floor of r*n (or the closest integer or whatever).
Then f_r is an almost-homomorphism, and the map taking r to
the equivalence class of f_r is what we want.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Angus Rodgers
<denis.feldm...@neuf.fr> wrote:
What are the equivalence classes of almost endomorphisms of the
additive group Z[i] of Gaussian integers?
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
victor_me...@yahoo.co.uk
> On Fri, 05 Sep 2008 13:26:11 +0200, Denis Feldmann
>
> >The wikipedia entry on construction of reals
> >(http://en.wikipedia.org/wiki/Construction_of_real_numbers) give a
> >really incredible (and elementary) construction in terms of equivalence
> >classes of quasi-morphims of Z, ie f such that f(m+n)-f(m)-f(n) is a
> >bounded set ; see
> >for details
>
> What are the equivalence classes of almost endomorphisms of the
> additive group Z[i] of Gaussian integers?
That is, Z^2. Probably M_2(R)?
Victor Meldrew
"I don't believe it!"
Angus Rodgers
Could be! I should probably have mentioned commuting with i. 8-P
Angus Rodgers
In 1946 Kolmogorov gave a related definition of a positive real number
as a sequence xi of positive integers such that:
(1) k.xi(n) <= xi(kn) < k.(xi(n) + 1)
(2) for each n there exists k such that k.xi(n) < xi(kn)
There's a brief account in the paper I mentioned in the recent thread
"Rudin and Dedekind cuts" (started 20 Aug):
M. H. Stone, "The real number system reviewed", L'enseignment
mathematique, tome XV (1969), pp. 261-267.
But this account is rather brief, and the treatment in terms of almost
homomorphisms is neater. Anyway, the reference in Stone's paper is:
"Kolmogorov, A. N., Uspekhi Matematicheskhik Nauk, 1 (1) (1946),
217-219. This paper contains no proofs, but proofs are provided by
N. I. Kavun, Uspekhi, 2 (5) (1947), 199-229."
This is at least of some interest as a previous attempt to formalise
Eudoxus's definition in terms of the (positive) integers directly,
rather than proceeding via the rationals.
David C. Ullrich
<denis.feldm...@neuf.fr> wrote:
First reaction: Keen.
Second reaction: Come to think of it, this is really not that
different from more "obvious" things - it's simply defining
a real as a sequence of integers n_j such that n_j/j
converges to the real in question.
Third reaction: Otoh, there _is_ something here that really
is different from anything I've thought of or seen, namely
the idea that we can define multiplication using _composition_.
Angus Rodgers
<dull...@sprynet.com> wrote:
>Third reaction: Otoh, there _is_ something here that really
>is different from anything I've thought of or seen, namely
>the idea that we can define multiplication using _composition_.
That's a very natural idea when you consider the application of real
numbers in measurement. It's in the Behrend paper (second reference
in the Arthan paper, which is the first arXiv paper linked to in the
Wikipedia article); and in Bourbaki's "General Topology"; and in the
first volume (section 4.6) of Behnke et al., "Fundamentals of Mathe-
matics" (English translation, 1974, of second edition of German text
whose first edition was published in 1958).
(These treatments all date from the 1950s. I don't know if the 1946
Russian paper by Kolmogorov mentions multiplication as composition.)
For suitable definitions of number and magnitude (such definitions
aren't unique, of course), the scalar product a.x is defined for any
number a and magnitude x. Since the set of numbers itself satisfies
the definition of a set of magnitudes, the product a.b is defined in
the same way; and we have (a.b)x = a.(b.x) for all numbers a, b, and
all magnitudes x; so the product of numbers a.b is also composition,
when each number a is identified with the function x |-> a.x (which
is indeed one way of defining numbers in terms of magnitudes).
This is related to the definition of the logarithmic and exponential
functions (because the positive real numbers under multiplication
form a set of magnitudes, and so x^a is an instance of a.x); and it's
related to the idea of dimensional analysis (which introduces a wider
definition of multiplication).
I'm always meaning to get around to writing this all out in complete
detail. (There are many reasons why I haven't. One reason is that I
get entangled in deep foundational issues, which I'm unlikely to be
able to resolve until I'm a lot more comfortable with mathematics.
I'm pretty sure it can all be done in a pleasantly neat fashion, so
I'd rather hold off until I can avoid making a dog's dinner of it!)