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.
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.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)