Infinity, limits and mollwollfumble's hypothesis

[mollwollfumble]

Hi, 8 years ago I asked the "what if" question "what if the infinite limit of n is not equal to the infinite limit of 2n?" This led to to rediscovery of dozens of systems of infinite numbers that are being largely overlooked by mathematicians, including Conway's surreals, Robinson's hyperreals, du Bois-Reymond's infinitary calculus, the Veronese continuum and Hahn series. These five systems of infinite numbers are largely equivalent and in each an "actual infinity" small-omega can be defined. Enough introduction, now for the cartoons. In the first cartoon, with title "Your infinity is too small", I list the most useful systems of infinite numbers - and those with the funniest names.





To be continued (because forum only allows 4 images at a time).

 

[mollwollfumble]

mollwollfumble's hypothesis is that, given the above definitions of limit and infinite, "every sequence can be expressed as the sum of a smooth function and an oscillating function", where the oscillating function has zero median, or zero mean, as required. Set the limit of the oscillating function to zero, and that gives a unique limit to every sequence obtained by simply evaluating the smooth function at infinity omega.

Your task, should you choose to accept it, is to find a counterexample.

Corollaries are:
* Every sequence has a limit
* There is no such thing as a divergent series
* Every definite integral has a solution

Which i think would have a lot of practical applications, don't you?




Simple example, all three limits are already well known and can be found on wikipedia.



Tough example.



Can you find a sequence, divergent series or definite integral that can't be easily evaluated in this way?

 

[mollwollfumble]

A definition of "smooth function" and some practical applications from mollwollfumble's hypothesis.

First some practical applications, definite integrals. These could be used in renormalisation in quantum physics, or to be more specific in regularisation in quantum physics. I suspect that quantum physicists already know of these, but mathematicians don't.



A formal definition of "order of magnitude", developed by Du Bois-Reymond and beloved by computer scientists.



All of the following are examples of "smooth functions". I don't have a formal definition, just an extrapolation from examples. You can help by trying for a formal definition. Don't be bamboozled by the number of examples, just concentrate on one or two. Those smooth functions created by exp, log, power and polynomial form a system of infinite numbers known as the "logarithmico-exponential numbers", which are incomplete as a definition of infinite numbers because they exclude the functions log*, busy beaver, ackerman and tetration. Du Bois Reymond, and later Borel who extended his work, were already familiar with log* and tetration.



That's all i really need to say. So get cracking and find a counterexample to the hypothesis that "every sequence can be expressed as the sum of a smooth function and a pure oscillatory".

 

[Otis]

... So get cracking and find a counterexample to the hypothesis that "every sequence can be expressed as the sum of a smooth function and a pure oscillatory".

Have you already found a counterexample (i.e., your thread is a recreational challenge)?

?

 

[mollwollfumble]

Have you already found a counterexample (i.e., your thread is a recreational challenge)?

?

No, and yes. I intend it as fun exercise for forum readers rather than as a scholarly exercise. How pathological can a sequence, series, definite integral or function get?

I haven't found a counterexample myself yet, even f(n) = |tan(n)| works, although that sequence is quite nasty.

One possible way of devising a counterexample occurred to me since posting. An asymptotic series is divergent but equals a definite integral. If the method evaluated the divergent series to a different value than the definite integral then that would serve as a counterexample of the method applied to definite integrals, not necessarily a counterexample to the method applied to divergent series.

 

[mollwollfumble]

Another way to find a counterexample would be to find a sequence that does not, in the infinite limit, have a median.

One possibility there would be the sequence of digits of pi. If the digits are truly random then the median in the limit is 4.5. But if it can be shown that the sequence of digits is not random then the median may be 4, or 5. In this case perhaps the mean of the median is required.

 

[mollwollfumble]

No, and yes. I intend it as fun exercise for forum readers rather than as a scholarly exercise. How pathological can a sequence, series, definite integral or function get?

I haven't found a counterexample myself yet, even f(n) = |tan(n)| works, although that sequence is quite nasty.

One possible way of devising a counterexample occurred to me since posting. An asymptotic series is divergent but equals a definite integral. If the method evaluated the divergent series to a different value than the definite integral then that would serve as a counterexample of the method applied to definite integrals, not necessarily a counterexample to the method applied to divergent series.

Because asymptotic series more often than not involve the Bernoulli numbers, it becomes necessary to look at the sequence of Bernoulli numbers. The Bernoulli numbers themselves have an asymptote that is a pure fluctuation times a smooth function. So it has a mean in the infinite limit of zero, using the new definition of limit from the first post in this thread, the finite fimit extended to infinite numbers (ie. not shift-invariant) from cartoon 676.

In this case the pure fluctuation is Re(exp(i*n*pi/2)), which goes 0,-1,0,1,0,-1,0,1,... which is multiplied by the smooth function sqrt(pi*n)*(n/2*pi*e)^n.
It's trivial to show that the median of this is exactly zero. So the sequence of Bernoulli numbers satisfies mollwollfumble's hypothesis that every sequence can be split into the sum of a smooth function and a pure fluctuation, using either the mean or the median - in this case both.

 

[mollwollfumble]

The other indication that asymptotic series satisfy the hypothesis that "on the hyperreals no sequence diverges" is the way they are generated. Many asymptotic sequences can be generated using the identity 1/(1-x) ~ sum(n=0 to infinity) x^n. Using omega as a non-shift-invariant infinity this becomes (1-x^(omega+1))/(1-x) = sum(n=0 to omega) x^n. Then the asymptotic series is generated in exactly the same way as before, and yields exactly the same result when the infinite pure fluctuation is dropped.

The whole thing is very similar to how renormalisation (or to be specific, regularisation) is handled in quantum physics.

Until further notice, the lack of forum members finding any pathological sequence weird enough to disprove mollwollfumble's hypothesis will be taken as confirmation of that hypothesis.

 

[JeffM]

Until further notice, the lack of forum members finding any pathological sequence weird enough to disprove mollwollfumble's hypothesis will be taken as confirmation of that hypothesis.

I love it.

"Hey I have a really cool proof."

"Great. What is it?"

"I hypothesized something at a homework help site, asked for a disproving example, and was ignored. When do I win the Fields prize?"

 

[mollwollfumble]

Thanks a million. I had been deliberately provocative.

Do you know a better forum for discussing wildly speculative extensions to conventional maths? I mean, Hilbert did, didn't he, in 1900, back in the days before the publication of speculative hypotheses was banned. This work was accepted on ArXiv (which means nothing), and I have given a talk on it to a Uni Math department.

 

[Otis]

... Until further notice, the lack of [any counterexamples] will be taken as confirmation of [mollwollfumble's] hypothesis.

Understood. (It's a free world, brother.)

I'm interested, but it's not near the top of my list. So, a lack of discussion over the past 20 days could mean that none of the regular contributors at the forum have time right now, for this recreational pursuit. Yet, a lot of other people read threads in the forum. Perhaps, there will be better replies for you sometime in the future. Maybe one of them will be from me!

?

 

[pka]

The other indication that asymptotic series satisfy the hypothesis that "on the hyperreals no sequence diverges" is the way they are generated. Many asymptotic sequences can be generated using the identity 1/(1-x) ~ sum(n=0 to infinity) x^n. Using omega as a non-shift-invariant infinity this becomes (1-x^(omega+1))/(1-x) = sum(n=0 to omega) x^n. Then the asymptotic series is generated in exactly the same way as before, and yields exactly the same result when the infinite pure fluctuation is dropped.
The whole thing is very similar to how renormalisation (or to be specific, regularisation) is handled in quantum physics.
Until further notice, the lack of forum members finding any pathological sequence weird enough to disprove mollwollfumble's hypothesis will be taken as confirmation of that hypothesis./QUOTE]
The following is a free down-loadable calculus textbook based upon the hyper-real number system.
In Elementary Calculus: An Infinitesimal Approach

In Elementary Calculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this. The chapters and whole book is a free down-load at http://www.math.wisc.edu/~keisler/.

 

[mollwollfumble]

Love that quote. Godel would have loved it, too.

I have spoken with three researchers in systems of infinite numbers about this. And written a 150 page monograph. Two of the researchers were too wrapped in their own work to kelp much, but the third was a marvel. He (correctly as it turned out) told me that I should be looking at working with Robinson's hyperreals instead of Conway's surreals, the two systems may be identical but Robinson's work is far more advanced. He also had grave concerns about me using a statistical mean of zero to separate smooth from fluctuating parts. He was right with that, too. Median, for instance, worked far better than mean with the nasty sequence |tan(n)|

Hardy wrote great monographs on both du Bois-Reymond's infinite calculus and about divergent series. It annoys me greatly that he didn't think to combine the two.

What I'm missing now is finding someone who likes to play around with pathological sequences. Such sequences can be deterministic, like |tan(n)| or generated using random numbers, chaos theory, fractals, Cauchy-Hamel functions, or other methods from the great sequence database. Where can I find such a person?

 

[Otis]

... the third [researcher with whom I conferred] was a marvel ... What I'm missing now is finding someone who likes to play around with [these topics] ... Where can I find such a person?

If that is not a rhetorical statement, then did you consider asking the marvel?

\(\;\)

 

[mollwollfumble]

The marvel's interest was in keeping track of the latest developments in the theories of infinite numbers. He has written a good book about the paradoxes that need to be overcome when working with infinite numbers. He did offer me the opportunity to work for a further 2 years with him to produce a PhD. I balked at that, having already spent two solid years on it I had had enough. But he had no interest at all in divergent sequences. Perhaps the person I'm looking for is one who has contributed a large number of sequences to the "On-Line Encyclopedia of Integer Sequences". But anyone with a burning passion for recreational mathematics would be a good start.

 

[mollwollfumble]

No, and yes. I intend it as fun exercise for forum readers rather than as a scholarly exercise. How pathological can a sequence, series, definite integral or function get?

I haven't found a counterexample myself yet, even f(n) = |tan(n)| works, although that sequence is quite nasty.

One possible way of devising a counterexample occurred to me since posting. An asymptotic series is divergent but equals a definite integral. If the method evaluated the divergent series to a different value than the definite integral then that would serve as a counterexample of the method applied to definite integrals, not necessarily a counterexample to the method applied to divergent series.

I just found an entire book on the topic of asymptotic series on the hyperreals. I think I can rule out that as a counterexample of the hypothesis.