🚀 go-pugleaf

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?
>
> Axiom?

I'm not sure that I understand what, if anything,
you are asking.

Do you recall writing
"The axioms behind this view are silly"?
This is the one axiom behind that view,
the least upper bound axiom. (There are other
axioms that have the same effect, but this
serves as well as any.)
Do you find this axiom silly?

http://en.wikipedia.org/wiki/Least_upper_bound_axiom
< That is, if a nonempty set of real numbers has an
< upper bound, then it has a least upper bound.

Real numbers form a complete ordered field.
Rational numbers form an ordered field.
There is one more axiom for complete ordered fields
than for ordered fields, and one form of it is
the Least Upper Bound Axiom.

I suppose I could ask whether you find the other forms
silly as well, but I picked this one because it seems
the clearest to me, the least technical, if you will.