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I was reading papers 1 and 2 on numerosities. These present a way of comparing the size of sets as equivalences of the size of the set intersected with finite subsets. I am trying to extend the work of these papers from subsets of the real numbers to subsets of the surreal numbers. However, I am having trouble understanding how the labeling system from 1 became the system in 2. I understand it is due to needing uncountable sets, but I do not have any intuition behind it.

The first paper defines numerosity as follows: $l_A:A \to \mathbb{N}$ where $l_A$ is finite to 1. Then define sets $A_n=\{ a| l_A(a) \leq n \}$, and then num(A) is an equivalence of the sequence $|A_n|$ on some ultrafilter of $\mathbb{N}$.

The second paper defines numerosity as follows: Take some base set $A$ and make a universe $V(A)$ where $V_0(A)=A$, $V_{n+1}(A)=V_n(A) \cup P(V_n(A))$, and $V(A)=\bigcup V_n(A)$. Then let set $L$ be the set of all finite subsets of $V(A)$.

A label set $B$ is a label set if For all $s, t \in B$, their union and intersection is in $B$. For all $s \in B$, $s \cap L = \emptyset$(This means that s contains only infinite sets and atoms). $\bigcup\limits_{s \in S} V(s) = V(A)$ The largest possible $B$ which contains all finite subsets of infinite subsets and atoms is called $B_{\max}$

The label function for a label set is $l(a) = \bigcap\limits_{\mu \in I_a} \mu$ where $I_a = \{\mu \in B \mid a \in V(\mu)\}$

And then num($E$) = $\lim\limits_{\lambda \uparrow V(A)} |E \cap \lambda|$, where $\lim\limits_{\lambda \uparrow V(A)} \phi (\lambda)$ is an equivalence of functions over an ultrafilter of $L$, which is implied to be $B$ but I'm not sure.

My main question is, how does this new definition actually allow uncountable sets, and how is it related to the first definition?

The paper outlines a method of closure that is a label set, namely for a directed set $D$, $\overline{D} = G(\{s \in B_{\max} \mid \exists t \in D, s \sqsupseteq t\})$, but I'm not entirely sure what $\sqsupseteq$ represents (its Definition 2.13), or how to choose $D$. My secondary question is, how do you produce a labelling set, and therefore function, that has a specific property?

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  • $\begingroup$ The paper 1 has an issue on page 63, starting with words "We remark that the above construction shows that the assignment of numerosities to labelled sets is not uniquely determined." It says then that depending on definition they can define numerosity of positive even numbers $num(even_+)$ as being equal to the numerosity of positive odd numbers $num(odd_+)$ or greater than that by 1. But these definitions contradict their own axioms, because from axioms it follows that $num(odd_+)=1/4num(\mathbb Z)$ and $2num(even_+)+1=1/2num(Z)$. Solving the equations gives $num(odd_+)=1/2+num(even_+)$ $\endgroup$
    – Anixx
    Commented Jun 26 at 23:35
  • $\begingroup$ This may be relevant: math.stackexchange.com/questions/4934215/… $\endgroup$
    – Anixx
    Commented Jun 27 at 0:15
  • $\begingroup$ Anixx I know its not unique if you use a general ultrafilter, but in the second paper they show how to refine the filter to have specific values, specifically choosing a filter of ${0,...,m!^{m!}}$ for integers that allow all arithmetic sequences to be $\alpha/n$ and so all power sequences are $\alpha^{\frac{1}{n}}$. So it is possible to do. Also it seems reasonable for the numerosity for the rationals to not be countable, since all countable ordinals embed into the rationals. $\endgroup$
    – opfromthestart
    Commented Jun 27 at 1:28
  • $\begingroup$ And what axioms do they state that they contradict? $\endgroup$
    – opfromthestart
    Commented Jun 27 at 1:33
  • $\begingroup$ They label sets of negative numbers isomorphically to the positive numbers (see page 59). Particularly , $num(\mathbb Z)=num(\mathbb N_0)\oplus\mathbb{N}\text{\ }\{0\})$. From this follow the equations above. $\endgroup$
    – Anixx
    Commented Jun 27 at 10:52

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