Within surreal number lingo the “construction ordering” is often referred to as “birthday” and −5 being younger than −10 becomes a thing one can prove. 1⁄2 has birthday just after 1 but 1⁄3 has birthday ω.
From the perspective that integers are sets of integers sets of integers do not come “after”. That is if you first create the integers and then you try to form {0,1,2} that is not a new construction as 3 has already been formed. In that perspective “last integer” and “last set” are not that different. There is the distinction of a limit ordinal. If “natural numbers” is a set then making the singleton {natural numbers} can also be thought as a successor ordinal ie ω+1 and how the set “natural numbers” can be thought as the ordinal ω. So there is a process where one can think of giving infinite time to nest sets. One can even give another round to get up to ω*2.
In the other direction one can assume as little magically given base material as possible. Having nothing but how to make new numbers one can get 0 as the limit ordinal of ex nihilo.
Yeah, the surreal numbers are somewhat similar to my intuition, but not the same. Kolmogorov complexity is probably closer. But I don’t have anything precise in mind. Just a feeling that “first numbers, then sets of numbers” makes sense on some level, but there is no way to make the same sense about (all) sets of sets.
I just started reading the reviewed book. Thanks for inspiration!
Within surreal number lingo the “construction ordering” is often referred to as “birthday” and −5 being younger than −10 becomes a thing one can prove. 1⁄2 has birthday just after 1 but 1⁄3 has birthday ω.
From the perspective that integers are sets of integers sets of integers do not come “after”. That is if you first create the integers and then you try to form {0,1,2} that is not a new construction as 3 has already been formed. In that perspective “last integer” and “last set” are not that different. There is the distinction of a limit ordinal. If “natural numbers” is a set then making the singleton {natural numbers} can also be thought as a successor ordinal ie ω+1 and how the set “natural numbers” can be thought as the ordinal ω. So there is a process where one can think of giving infinite time to nest sets. One can even give another round to get up to ω*2.
In the other direction one can assume as little magically given base material as possible. Having nothing but how to make new numbers one can get 0 as the limit ordinal of ex nihilo.
Yeah, the surreal numbers are somewhat similar to my intuition, but not the same. Kolmogorov complexity is probably closer. But I don’t have anything precise in mind. Just a feeling that “first numbers, then sets of numbers” makes sense on some level, but there is no way to make the same sense about (all) sets of sets.
I just started reading the reviewed book. Thanks for inspiration!
Just because you personally don’t understand doesn’t mean that somebody else could not.
But you might also be getting to the distinction that “all sets” is a proper class whereas numbers can be put into a set and are thus a small class.