Not about tetration but what do you think about this paper?

[Ansus]

There are very few good math forums that would allow free discussion about preprints and that have competent people.

so, here I invite to discuss this new paper of mine:

https://arxiv.org/abs/2411.00296

[MphLee]

Very interesting. I certainly lack expertise in integration and surreal numbers, but I am fascinated by the potential philosophical implications. I regret that I cannot provide you with useful comments regarding your constructions; unfortunately, I do not have sufficient time to study them, but they are stimulating material for future reflections, at least for me.

Some brief superficial comments: a typo at page 2 "satisfying the Hard field requirement"; In the conclusion I see some examples of computations. These are quite incredible values but also intriguing. It is not immediate to me how all of this still satisfies Euclid principle.

A personal comment: I find this concept of numerosity very intriguing. Due to my ignorance, I was unaware that it was studied so thoroughly! Essentially, it is an alternative way to Cardinality to specify the concept of the size of a multiplicity.
The point seems to be this. Upon initial philosophical analysis, following the beautiful historical introduction by Katerina Trlifajov´a in the article "Sizes of Countable Sets" that you cite, one could say that the size of a multiplicity must adhere to at least two principles: Euclid's principle (PW) and Hume's principle (HP). The first states that the size of a multiplicity is greater than the size of any of its parts (PW); the second states that multiplicities have equal sizes if and only if there exists a perfect correspondence between the individuals that compose them (HP).
Now, as is well known, the two principles cannot hold simultaneously if we wish to extend the notion of size to any type of multiplicity. Sets that possess parts in a one-to-one correspondence with a proper subset of themselves are defined as infinite. Therefore, if we are determined to have a universal notion of "size," i.e., without restrictions on the type of multiplicity, we must either uphold Euclid's principle or Hume's principle. Historically, the more successful approach has been to regard Hume's principle as fundamental for defining a notion of size. This is the path that has led to the concept of Cardinality. It seems that the alternative path, which holds Euclid's principle as fundamental, leads instead to the notion of Numerosity. This, at first glance, appears to be closely related to the concept of measure, e.g., the measure of a sigma algebra, and thus to integration theory.
Thank you for sharing your paper here. I do not understand much in detail, but it seems very interesting.

I would like to add a side note: there are other ways to extend the notion of "size." For example, Euler's characteristic has similar properties, \(\chi (X\cup Y)=\chi X + \chi Y - \chi(X\cap Y) \), on more general objects than simple collections of individuals, i.e., sets, and on algebraic-geometric structures, it connects to Grothendieck's algebraic K-theory where, if I understand correctly, a sort of "arithmetic" of "generalized dimensions" is obtained. Similarly interesting is the concept of "magnitude" by Leinster, which unifies the cardinality of finite sets, classical dimension, area, volume, and even Hausdorff dimension in the case of fractal sets.

[Ansus]

Thank you for the reply!

Indeed, Euler's characteristic is the finite part of numerosity in my definition. For instance, the closed interval \([0,1]\) has numerosity \(\omega_1+1\). The more-than-one-dimensional case for numerosity is not in the scope of this paper though.

This suggestion of integration of of surreals very much differs from the previous suggestions. They strived to prefer linearity against infinite miltiplier, and so far there is no successful accepted proposal for this.

In my theory, we cannot move an infinite factor from under integral like a real constant. This post explains why I think it is more natural: https://mathoverflow.net/questions/47590...nsidered-e

I am also proud that there are two integral-based expressions of the constant representing numerosity of continuum. This makes \(No(\omega_2)\) surreal numbers (countable surreals and continuum-sized surreals) to be closed under integration: \(\omega_1=\frac1\pi \int_0^1 \omega dx=\frac 1\pi \int_0^\infty \ln \omega dx\).

This makes the set of "geometric surreals", the algebraic closure of surreals that are enough to denote numerosity in Euclidean space.

I do not strive to study surreals as a whole, but for now only these smaller sets (countable and continuum-sized), of which countable are the most important as they immediately follow the finite numbers and already studied as Hardy fields and transseries.

Also, this theory provides an alternative for the theory of distributions and hyperfunctions, by defining the surreal-valued delta function. As such, we can ascribe surreal-valued derivatives to many discontinuous functions.

[Ansus]

In the older version of the paper I considered a multi-dimensional case (see Appendix 3 here: https://arxiv.org/pdf/2411.00296v2) and mentioned equivalence of the Euler's characteristic.

[Ansus]

What also is interesting, is if we want a closure undder operation of definite integration of real-valued, but unbounded functions, in such a way that it preserves order (that is, if \(f(x)>g(x)\), then \(\int_a^b f(x)dx > \int_a^b g(x)dx\)) we get the set of countable surreals. And if we put countable surreals under the integral, we get continuum-sized surreals. Thus, the both sets come naturally from the attempts to find a closure of reals under integration. And it seems, one cannot get even greater surreals this way (putting continuum-sized surreals under the integral will not produce anything even greater than continuum-sized).