🚀 go-pugleaf

On 3/27/2011 7:46 AM, Albrecht wrote:
> On 27 Mrz., 05:45, Barb Knox<s...@sig.below> wrote:
>> In article
>> <iml34b$k0...@charm.magnus.acs.ohio-state.edu>,
>> Jim Burns<burns...@osu.edu>  wrote:
>>> 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?
>>
>> Whatever he does next, he will be sure that
>> it is correct.
>>
>> Your exposition is helpful and to the point,
>> but it will not get through.

[ Piggyback to Barb Knox:
   Thank you for your kind words.
   Whether or not it gets through,
   I think I understand the reals
   a bit better every time I explain them.
]

> You must be a sybil. You love this job?

How could she not, what with the high pay and perks?

How does declaring the naturals, integers,
rationals and reals to form proper classes
instead of sets get rid of any silliness?
Isn't the proper class of naturals still
larger than the proper class of reals?