Got really much me thinking. Why are we regarding non-standard natural numbers as “junk”? I guess the identification of standard natural number as the simplest construciton that is a natural number system.
The thing is a know a perfectly legimate construction for a number that is non-standard and not deliberately a “wrench in the machinery”. The surreal number {1,2,3,4...|}=ω I have sometimes seen characterised as a integer and it’s construction is of the same shape as other integers with lower birthdays (althought they use finite sets, ω is the first to use infinite sets).
The hydra problem seems natural as you can’t have ω-(n1) with finite n that reaches 0, and in fact ω-(n1) is still bigger than any finite number. I can also see how the successor of ω is ω+1 which I guess is the property that successor and addition play nice together.
When geometry was axiomatised it was discovered that there are euclid and non-euclid geometries. They were not called non-standard geometries despite them getting way less attention. In general euclid and non-euclid geometries share some properties (those that stem from aximo not regarding parallel lines) but have different properties in general (ie different parallizaiton rules lead to genuinely different systems). Coudn’t it just be that we are using a way too general system where the formal meaning of a integer captures more entities that we have in mind when we are really interested only in certain kinds of natural numbers? That is ω might be a integer as the axioms read out but when people say integers they don’t mean entities like ω (like when people say space they usually don’t mean minowskian spaces althought those are spaces too).
I do like the rigour that when a mathematician lays out a set of axioms he can know whether all cases are covered without being able to come up with any “viable” exception to them. That is any kind of arithmetic thing that hinges on the differences of finite and infinite numbers is already ambigious based on axioms of arithmetic because finiteness and infiniteness is ortohogonal to the issue (a kind of separation of concerns where you don’t even know how many concerns there are).
Wouldn’t the holistic nature of the truth be viewed as if you have an ambigious delineation on the universe of discourse then you can’t have all properties nailed down. As in if you have a theory of “tallness” that doesn’t allow you to determine an objects color.
Got really much me thinking. Why are we regarding non-standard natural numbers as “junk”? I guess the identification of standard natural number as the simplest construciton that is a natural number system.
The thing is a know a perfectly legimate construction for a number that is non-standard and not deliberately a “wrench in the machinery”. The surreal number {1,2,3,4...|}=ω I have sometimes seen characterised as a integer and it’s construction is of the same shape as other integers with lower birthdays (althought they use finite sets, ω is the first to use infinite sets).
The hydra problem seems natural as you can’t have ω-(n1) with finite n that reaches 0, and in fact ω-(n1) is still bigger than any finite number. I can also see how the successor of ω is ω+1 which I guess is the property that successor and addition play nice together.
When geometry was axiomatised it was discovered that there are euclid and non-euclid geometries. They were not called non-standard geometries despite them getting way less attention. In general euclid and non-euclid geometries share some properties (those that stem from aximo not regarding parallel lines) but have different properties in general (ie different parallizaiton rules lead to genuinely different systems). Coudn’t it just be that we are using a way too general system where the formal meaning of a integer captures more entities that we have in mind when we are really interested only in certain kinds of natural numbers? That is ω might be a integer as the axioms read out but when people say integers they don’t mean entities like ω (like when people say space they usually don’t mean minowskian spaces althought those are spaces too).
I do like the rigour that when a mathematician lays out a set of axioms he can know whether all cases are covered without being able to come up with any “viable” exception to them. That is any kind of arithmetic thing that hinges on the differences of finite and infinite numbers is already ambigious based on axioms of arithmetic because finiteness and infiniteness is ortohogonal to the issue (a kind of separation of concerns where you don’t even know how many concerns there are).
Wouldn’t the holistic nature of the truth be viewed as if you have an ambigious delineation on the universe of discourse then you can’t have all properties nailed down. As in if you have a theory of “tallness” that doesn’t allow you to determine an objects color.