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Topic: Real Analysis!!!
Replies: 12   Last Post: Aug 13, 2012 7:28 AM

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 Zaljohar@gmail.com Posts: 2,625 Registered: 6/29/07
Re: Real Analysis!!!
Posted: Aug 12, 2012 1:24 PM

On Aug 12, 8:20 pm, Zuhair <zaljo...@gmail.com> wrote:
> On Aug 12, 2:09 pm, Zuhair <zaljo...@gmail.com> wrote:
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> > On Aug 11, 8:21 pm, Zuhair <zaljo...@gmail.com> wrote:
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> > > /
>
> > > This will be sufficient to do all Real Analysis.
>
> > It would be extensive to show that but here is an outline:-
>
> > The set N of all natural numbers is definable here and is non empty,
> > so this theory proves infinity.

>
> > A natural number is defined as a cardinal of some finite set
>
> > Def.) x is a natural number <-> Exist y. y is finite & x= |y|
>
> > To define finite we first must define the relation "smaller than"
> > denoted by <, between any two "cardinals"

>
> > For cardinals x,y:  x < y iff  Exist m,n. x=|m| & y=|n| & m is a
> > proper subset of n.

>
> > The relation > is the converse relation of <
>
> > Now a set s of cardinals is prefinite iff  < is well ordering on s & >
> > is well ordering on s

>
> > Any set x is said to be finite iff Exist s. s is a set of cardinals &
> > s is prefinite & x equinumerous to s.

>
> > Clearly N is definable. It is easy to define Natural summation,
> > multiplication and exponentiation. where N is closed on them.

>
> > Now the set Z of all integers can be defined as ordered pairs of
> > "naturals" of the form (0,x) ; (x,0) ; (0,0) to represent +x,-x, and
> > Zero
> > where Zero is unsigned. It is easy to define Integer summation,
> > subtraction, multiplication, exponentiation where Z is closed on them.

>
> > Now the set Q of all rationals can be defined as ordered pairs of
> > "integers" that are minimal of all fractionally similar ones, provided
> > that (0,0) is not the first projection. Fractional similarity of two
> > ordered pairs of integers (x,y) , (z,w) is defined as Exist k: (x*k=z
> > & y*k=w) or (z*k=x & w*k=y), the minimal one of those is the one with
> > the least integer value of first projection. Also it is easy to define
> > summation, subtraction, multiplication, division, exponentiation where
> > Q is close on them.

>
> > In a similar manner to how Q is defined we can easily define a set of
> > all exponents that will be closed on summation, subtraction,
> > multiplication, division, exponentiation and rooting.

>
> > Reals are defined as Dedekind cuts as in the head post. However if we
> > seek unique Reals for each point on the continuous number line, then
> > we can restrict those to particular kind of Dedekind sets where
> > rationals are of the form  (m*10^n + r)/10^n  where r is the first 10
> > positive whole rationals, in set A and of course for each n there is
> > such a number and m is the same over all members of A. These are the
> > "decimal" Dedekind cuts, and are unique for each point on the real
> > number line.

>
> > Of course m-multi-dimentional space is the set of all ordered m-tuples
> > of Reals.

>
> > Now relations between reals are definable here, so we have the full
> > Real Analysis.

>
> > So this theory is sufficient to formulate all of ordinary mathematics.
>
> > Zuhair
>
> It is also useful to show that this theory do not have one singleton
> in its universe of discourse.
>
> The empty set exists here an its unique, this simply follows from
> Comprehension and Extensionality.
>
> Now lets denote the empty set by 0.
>
> Now we know that |0|={|0|}
>
> Also we know that ||0|| is not equal to |0| by Hum's principle!
>
> Now both ||0|| and |0| are singletons. QED
>
> Of course this would lead to having infinitely many singletons.
>
> Zuhair

Another merit of this theory is that it has Boolean structure! And the
Category of sets in it are Cartesian closed! something that NF and its
fragments lack.