Answer Happy

Guided Project Let x be a set. Define the successor of x to be the set xt = xU{x}. 1. Determine the successors and the successors of the successors of the sets 0,{0}, and {a,b,c}. 29.3 Set Constructions 333 We now introduce the following notation. Let 0 = 0,1 = 0+, 2 = 1+, and so on. 2. Write down 0,1,2,3, and 4 as sets in two different ways; first using the def- initions made above, and then using only the symbol 0, set brackets, and appropriate set notation. It may seem intuitively obvious that if we “do this forever," then we will have de- fined the natural numbers. However, as simple and as attractive as this approach may be, it is not what we call mathematically rigorous. What we need is a statement that explains that we can do this forever. We will take this statement as an axiom, and thus it will not be proved.