Re: (sketch of a) Proof that the set of Real Numbers doesn't exist

From: Piotr Sawuk (piotr5_at_unet.univie.ac.at)
Date: 01/21/05 

Date: 21 Jan 2005 02:50:01 GMT

In article <REM-2005jan09-004@yahoo.com>,
        rem642b@Yahoo.Com (tinyurl.com/uh3t) writes:
>> From: piotr5@unet.univie.ac.at (Piotr Sawuk)
>> I already figured out that there would exist a bijection between
>> an infinite set and its powerset
>
> That can never happen.

It could, but not in ZFC!
>
>> if the notion of "function" would not be limited to countable steps
>> with finite amount of operations, or something like that.
>
> That's completely irrelevant.

no, whole ZFC has been built on those assumptions.
>
>> Seeing R as the set of numbers between rational numbers does help a
>> lot in understanding maths,
>
> That's almost but not quite correct. Between any two distinct rational
> numbers, there are not only an infinite number of real numbers, but an
> infinite number of rational numbers among those reals. So in some sense
> both rationals and reals are numbers between rational numbers.

well, rational numbers are a set of numbers with their own rules,
and if you obey certain limitations you will never leave that set.
if you loosen some of those limitations, then you get a bigger
countable set into which rational could be densely mapped. you
will never succeed to remove all limitations in countably many
steps, and no matter what you do you are left with holes in the
set such that the countable set is dense within the set of such
holes, thereby according to continuum-hypothesis you require the
powerset of some countable set in order to assign values to those
holes. for some rational number r you could define a real number
as everything between that number and the set of all rational
numbers larger than r, and you could assign the value r to that
hole. in my understanding this is how dedekind defined real
numbers. therefore one could say that real numbers are the numbers
between rational numbers, inside of the gaps formed by them.
but this also does mean that in some way there are equally
many real numbers as rational ones, even though you could not
create a bijection between them. It's like with multidimensional
spaces: maybe you could create a bijection between each of them,
but by limiting yourself to preserving a certain order or certain
axioms you narrow down the set of spaces which can be morphed
into eachother through such operations. it's all a matter of
isomorphisms allowed, and according to the limitations on bijective
functions only sets with the same cardinality are isomorphic.
it still doesn't say if there exists a bigger set of functions
obeying the bijectivity-axioms (inside of a different kind of
logic) such that R is isomorphic to Q or not.
>
> What you need is this: Between any two disjoint nonempty open sets of
> rational numbers, there may or may not be a rational number, but there
> is always at least one real number. Examples:
>
> Let A be the set of rationals whose square is strictly between 1 and 4,
> and let B be the set of rationals whose square is strictly between 4
> and 9. Between A and B there's exactly one rational number, namely 2.
>
> Let A be the set of rationals whose square is strictly between 1 and 2,
> and let B be the set of rationals whose square is strictly between 2
> and 9. Between A and B there's no rational number at all, but there is
> exactly one real number, namely the positive square root of 2.
>
> Let A be the set of rationals whose square is strictly between 1 and 2,
> and let B be the set of rationals whose square is strictly between 4
> and 9. Between A and B there are an infinite number of both rationals
> and irrational real numbers.

yes, that's what I mean in this posting with "real numbers are the gaps
between the rational numbers", except that you still would need to prove
that rational numbers are dense in real numbers, while with clopen sets
assigned to rational numbers and pairs of open sets assigned to irrational
numbers you get this quite naturally.
>
>> extending natural numbers into infinity such that some numbers
>> simply have an infinite length
>
> I don't know what you mean by that.

usually in logic you can only write formulas of finite size. if you
would allow formulas of infinite size, then the set of "natural" numbers
would be extended to countable sequences of "something". this extension
is quite different from mere "for all" or "there exists", as it allows
for uncountably many expressions and so some proofs wouldn't work anymore.
>
>> with the additional attribute that all such infinite numbers would be
>> taboo in usage and interpretation
>
> Whether it's taboo or not depends on what you meant above.
> Until you state precisely *how* you would extend natural numbers,
> nothing you say about such has any meaning.

I extend them by using a set of axioms within which ZFC is true,
and where removing some axiom of ZFC does keep up most of the
results of ZFC neccessary for doing maths. removing said axiom
would then be called "lifting the taboo".
>
>> Real numbers do circumvent this taboo
>
> Whether they do or not depends on what taboo you're talking about,
> which again you haven't defined. Maybe if you say how you intended to
> extend integers, and how that in any way relates to reals, we might be
> able to guess what taboo you're talking about, or maybe even then you'd
> have to explicitly state what the taboo is.

you could define real numbers as the result you get by applying the
division-algorithm on infinite sequences of digits (0-9) together
with the property that the result should be between 0 and 1, together
with an additonal sequence denoting that the result should be
multiplied by 10 to the power of this "number". and of course
recursively any such Real number could get divided in the same
way except that the difference between these additional exponents
would be used instead of assuming the result to be between 0 and 1.
>
>> I do have a problem in understanding why an open set does not have a
>> minimum
>
> Before you can even understand what you just said, you need to know the
> definition of a metric space, the definition of an "open set" within a
> metric space, and the definition of the "Cartesian" or "real" metric on
> the rational numbers. Do you know all of that already, so we can
> discuss the "why" you asked, or do you need to learn that before we can
> start?

I do know that open sets are *defined* as sets without minimum or
maximum, and I suspect you could also proof that this property does
hold even after extending such sets to real numbers, the kind of
real metric I know does have the property that for a special bilinear
function < x , y > the squareroot of < x , x > does define the
metric for the pairs (0,x), and the rest follows because of the
bi-linearity of that function. however, I also learned in logic
that if some set of axioms is unable to say anything about the
properties of some set, then one could create a new set of axioms
such that the old axioms are still true but something can now
be said about some of those sets. since open sets are *defined*
as sets without minimum or maximum one wouldn't be able to tell
if all open sets defined in the usual set of axioms would keep
up the property of being "open" in this extension. that's why
I am asking: what does cause the open sets to actually exist?

in the second version of my proof I showed a nice alternative
to just dropping them altogether: make it so that not all
intervals can be used for creating open sets by simply
removing a finite amount of points. I don't know much about
logic, but I have the feeling that the idea of extending
a set of axioms should get ported to maths in some way.

my point here is that according to goedel's incompleteness
theorem maths is incomplete, and just like logicians did
create their turing-degrees and L-spaces (of which I only
have a bad understanding), so also mathematicians would
need to define some algorithm for extending maths such
that certain areas previously inaccessible could get entered. 

-- 
Better send the eMails to netscape.net, as to
evade useless burthening of my provider's /dev/null...
P

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