I am having trouble thinking of integers in terms of ZF set theory, does the following make sense?
From elementary algebra the picture of the natural numbers ... 0,1,2 ... is that of a number line with zero at the origin and then marked off in unit intervals, 1, 2, 3 ... etc
Would it correct to view the natural numbers in ZF set theory as a set of concentric circles, the inner most circle labeled 0, the next circle labeled 1, the next circle labelled 2 and so on.
The inner most circle, "set 0" , representing the integer 0 contains nothing, no elements, and in particular it does not contain the symbol "0". The next circle, set 1 contains 1 element, the inner most circle, or, by name, 0. Set 1 does not contain itself, that is set 1 does not contain "1" else it would contain two elements 0 and 1. The next circle, the third circle, circle 2 by name, contains, in union, two elements, set 1 and, the set 0 which is nested inside of set 1. Again set 2 does not contain itself, that is set 2 does not contain 2, only 0 and 1. This continues outward with each circle representing the next greater integer, with any set n containing in union, n nested elements 0, 1, 2 .... n-1. set n does not contain itself, does not contain n. In set language, n is distinct from {n} and the name of {n} is the name given to the successor of n.
Is that a basic picture that I can then use to consider what is meant by a singleton and an ordered pair?
I find that it is possible to do a lot of math by following rules and even to get correct answers but yet not know what exactly one is taking about and where the rules run into trouble. That is my motivation in asking this perhaps unsophisticated question. Thanks, Dale
From elementary algebra the picture of the natural numbers ... 0,1,2 ... is that of a number line with zero at the origin and then marked off in unit intervals, 1, 2, 3 ... etc
Would it correct to view the natural numbers in ZF set theory as a set of concentric circles, the inner most circle labeled 0, the next circle labeled 1, the next circle labelled 2 and so on.
The inner most circle, "set 0" , representing the integer 0 contains nothing, no elements, and in particular it does not contain the symbol "0". The next circle, set 1 contains 1 element, the inner most circle, or, by name, 0. Set 1 does not contain itself, that is set 1 does not contain "1" else it would contain two elements 0 and 1. The next circle, the third circle, circle 2 by name, contains, in union, two elements, set 1 and, the set 0 which is nested inside of set 1. Again set 2 does not contain itself, that is set 2 does not contain 2, only 0 and 1. This continues outward with each circle representing the next greater integer, with any set n containing in union, n nested elements 0, 1, 2 .... n-1. set n does not contain itself, does not contain n. In set language, n is distinct from {n} and the name of {n} is the name given to the successor of n.
Is that a basic picture that I can then use to consider what is meant by a singleton and an ordered pair?
I find that it is possible to do a lot of math by following rules and even to get correct answers but yet not know what exactly one is taking about and where the rules run into trouble. That is my motivation in asking this perhaps unsophisticated question. Thanks, Dale