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Searching for an asymptotic to exp[0.5]
If we compute fake f(y^(2^m)) and then substitute y = x^(1/2^m) and let m go to +oo ,

Call the resulting coef g(t) such that g(t) belongs to x^t ,

Then , we get a very good approximation ( for all fake methods ! ) towards

f(x) =~ integral[0,oo] g(t) x^t dt.

The ratio of LHS to RHS as x Goes to oo is conjectured therefore to be - for most f - equal to sqrt(2 pi).

-----

From fake function theory it is logical to conjecture

F(x) = lim_n C sum_{i = n - sqrt(n)}^{n} t_i x^i.

Where t_i are the Taylor coef.

----

I found the Continuüm hypothesis is strongly related to fake function theory and tetration.

Basically I found a way to describe the cardinality of sets ( all sets ?? At least most , another intresting idea ! ) as Taylor series of real-entire functions.

The cardinality of a set is then

Lim x -> w f(x).

Where f is a real-entire Taylor series.

( and w is the card of the integers )

There is no need for AC.

We proceed by noting card( ~ exp(w) ) = card( 2^w ).
And also card( A + w ) = card(A) for card(A) >= w ( Card A nonfinite ).

Then we notice card ( growth C ) = card ( growth D ) IFF growth C = growth D.


( sheldon's growth of f := lim slog( f^[n] ) / n )

Then we use fake-half(x) = t(x).

And note

1) card(w) < f(w) < card(exp(w))

2) card(w) = card( fractions )
Card(exp(w)) = card( reals )

3) from fake function theory we know f(f(w)) = exp(w) and f(w) exists !!

4) the continuüm hypothese is basically 1).

5) the " tail " of f(w) is sufficiënt so arguments that the first Degree V Taylor polynomial bijects to w and thus does not increase card is NO PROBLEM.

6) as for the error terms card(w + w) = card(w).
Card(2^w) = card( w^j k^w + w^l ) for k >= 2 , j,l >= 0.



Qed.

Regards

Tommy1729

Ps : the new conjectures are

1) card of any set = Some Taylor of w.

2) it seems we have an uncountable number of card between card(N) and card® because any growth between 0,1 exists as a distinct card.
Therefore the idea of real Beth Numbers seems to destroy the idea of countable alephs or at least aleph 2.
The consequences are not yet understood ...


Will set theory , calculus and tetration finally be Friends after the " long war " ?


Some names might be pleasant.

Tetrational set theory.
Fake set theory ??

Asymptotic set theory ??

Beth set theory ???


With special thanks to Sheldon as cofounder of the essential fake function theory.


Regards

Tommy1729
The master
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Messages In This Thread
RE: Searching for an asymptotic to exp[0.5] - by tommy1729 - 09/04/2016, 07:21 PM

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