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.
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.