Re: A New (And Slightly Stupid) Set Theory

In sci.math, ken quirici
<kquirici@xxxxxxxxx>
wrote
on 28 Apr 2005 09:34:16 -0700
<1114706056.082608.267960@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>:
> The Ghost In The Machine wrote:
>> This is going to look a little silly, but Herc has given me an idea.
>> Probably a very bad idea, to be sure, but an idea nonetheless.
>>
>> Let the empty set be a level -1 set. Any finite set is a level 0
> set.
>> The whole numbers is a level 1 set, as it can be construected from
>> all possible level 1 sets.
>>
>> The real numbers is a level 2 set.
>>
>> Herc has proven that, for any mapping of a level 1 set to a level 2
> set,
>> one can represent the level 2 sets as level 1 set binaries:
>>
>> {0000000... AND 0} <-> 0
>> {1000000... AND 1} <-> 1
>> {0100000... AND 2} <-> 2
>> {1100000... AND 2} <-> 3
>> ...
>>
>> if I'm using this notation correctly.
>>
>> Since this is clearly a 1-1 correspondence, card(R) = card(N).
>>
>> I don't have time to explore this further but just thought I'd throw
>> this out there for discussion, lambastion, dissection, revulsion, ...
>> :-)
>>
>> --
>> #191, ewill3@xxxxxxxxxxxxx
>> It's still legal to go .sigless.
>
> This is going to look even sillier, but can't you diagonalize the
> set of binaries in your construction? We could use the same approach
> with decimal representations of the reals between 1 and 1 -
>
> 00000000000000000
> 10000000000000000
> 20000000000000000
> ...
> 01000000000000000
> 02000000000000000
>
>
> etc.
>
> and even though it appears we're going to hit every real (decimal
> representation) eventually, we don't.
>
> I think it has something (even sillier still) to do with the
> rationals being 'single-threaded' and the reals being 'multi-threaded'.
> If anyone can make sense of that, a tip 'o the hat.

Well, one of the issues is that the rationals are not closed [*],
in the sense that, if one has a monotonic sequence

a_1 < a_2 < a_3 < ...

and another monotonic sequence

b_1 > b_2 > b_3 > ...

such that a_i < b_i for every i (or perhaps a_i < b_j for
all pairs (i,j)), and all of the a's and b's are in Q,
and if there's at most one number between all the a's and
all the b's, then that the number c squished in the middle
is not necessarily in Q. For example,

a_i = floor(sqrt(2 * n^2)) / n
b_i = ceil(sqrt(2 * n^2)) / n

However, the reals *are* closed.

>
> Thanks.
>
> Ken
>

[*] a better term might be "complete"; I'd have to look up the
term here. This is one reason we have reals at all... :-)

Cauchy and Dedekind did it better, admittedly. :-)

--
#191, ewill3@xxxxxxxxxxxxx
It's still legal to go .sigless.
.

Relevant Pages

  • Re: A New (And Slightly Stupid) Set Theory
    ... > This is going to look a little silly, but Herc has given me an idea. ... > one can represent the level 2 sets as level 1 set binaries: ... This is going to look even sillier, ... rationals being 'single-threaded' and the reals being 'multi-threaded'. ...
    (sci.math)
  • Re: A New (And Slightly Stupid) Set Theory
    ... >> This is going to look a little silly, but Herc has given me an idea. ... You asked us to consider a mapping from N to R, ... But from a computability standpoint R is denumerable, ... And of course the IEE754 approximation only has about 2^64 reals, ...
    (sci.math)
  • Re: computation and cardinality
    ... > ken quirici wrote: ... As a cardinal, it's the cardinality ... > but I sure don't agree with it in the case of sets of reals. ... Set Theory, ...
    (sci.logic)
  • Re: just 5 quick answers then I can summarise and GO
    ... ken quirici wrote: ... > There are reals that aren't computable, ... that seems stronger than computability. ... polynomial qwith rational coefficients such that q= 0. ...
    (sci.math)
  • Re: just 5 quick answers then I can summarise and GO
    ... ken quirici wrote: ... > There are reals that aren't computable, ... that seems stronger than computability. ... polynomial qwith rational coefficients such that q= 0. ...
    (sci.logic)