🚀 go-pugleaf

On 3/24/2011 8:05 PM, Albrecht wrote:
> On 24 Mrz., 15:08, James Burns<burns...@osu.edu>  wrote:
>> Albrecht wrote:
>>> On 24 Mrz., 10:31, William Hughes
>>> <wpihug...@gmail.com>  wrote:
>>>> Absolute piffle.  There is no reason that defining the
>>>> existence of the limit of finite ratios means that you
>>>> have to define either infinite quantities, or the ratio
>>>> of infinite quantities.
>>
>>> So you admit that an infinite decimal number has crucial
>>> other properties than a finite decimal number? You admit
>>> that a number like 3.1415... is crucial different from a
>>> number like 0.75? But you insist that the "behaviour" of
>>> a number like 3.1415... should be like the "behaviour" of
>>> a number like 0.75?
>>
>>> The axioms behind this view are silly.
>>
>> If I am not mistaken, there is only one axiom that
>> distinguishes infinite decimal numbers (real numbers)
>> from finite decimal numbers (rational numbers, or close
>> enough for our purposes).
>>
>> <  For every non-empty set of real numbers, if it has
>> <  an upper bound, then it has a least upper bound.
>>
>> Upper bound and least upper bound are defined in the
>> expected ways:
>>
>> Upper bound of set A == a real number greater than or
>> equal to every element of A
>>
>> Least upper bound of set A == an upper bound of A
>> less than or equal to every upper bound of A
>>
>> I just wanted to be sure:
>> do you find this axiom silly?
>
> No. I have the axiom of infinity in mind. And the
> combination of the power set axiom with the axiom of
> infinity. Without these axioms the interpretation of
> the diagonal argument of Cantor is completely
> different from the common interpretation.

Yes, very different. No set of real numbers, for example.
(Well, no proof that the real numbers exist, unless
you add another axiom saying all sets are finite.)
Also, no set of rational numbers, no set of integers,
no set of naturals. Very different.

Are you saying, then, that pretty much all of
everyday mathematics is silly? Or do you see parts
that do not look silly?

---

Please consider an argument for the not-silliness
of the completeness of the reals (see Least Upper Bound
Axiom) which I suspect is a better suspect for the
origin of your chuckles.

When any calculation much more difficult than
"2 + 2 = 4" is done, very often it will proceed in
stages, "successive approximations" it is often called.
Even something as simple as dividing 7 into 3
(longhand) produces a digit, then another, then another
in an unending stream.

This series of partial results, these successive
approximations, form a Cauchy series: all the
partial results agree with each other to any
desired degree after a certain point in the
series. None of this so far needs real numbers
in order to be true.

A Cauchy series is a Cauchy series whether or not
there is anything that the series converges to.
After all, all we are given is that the partial
results get close to /each other/. But, it would be
convenient if there were a single /thing,/ call it
what you will, that all of the partial results
draw close to, the limit of the series.

And, then, there is the question of alternative
methods of calculation. What does it mean to have
two ways of calculating "the same number" if none
of the partial results agree at any stage? The
"same number" seems to mean that the partial results
from one series agrees with the partial results
from the other series to any desired degree if
one follows the two series out far enough.

One way of building the set of real numbers is
to collect all the Cauchy series of rational
numbers that "calculate the same number" and
identify the collection as a real number and
the limit of those Cauchy series. (Rational
numbers already exist and pose no logical
problems.)

 From this identification of a real number as
a collection of agreeable Cauchy series follows
two very desirable properties:
(1) Every series of successive approximations
converges to an answer.
(2) If two series of successive approximations
get arbitrarily close, then they converge
to the same answer.
If these properties seem like common sense, it is
because we have all grown up using real numbers.
They are not inevitable.

If you have an order field ('+', '-', '*', '/',
and '<' with their usual meanings) that satisfies
properties 1 and 2, then you have the real numbers,
with all of its silly consequences.

It may be possible to devise an alternate system,
the not-silly-real numbers, if you like. In order
to do that, though, you will have to give up
something. Are you sure that you want to?