- #1
member 587159
Hello everyone. I have read a proof but I have a question concerning the notation. To give some context, I will write down this proof as written in the book.
Theorem: There is a unique binary operation ##+: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}## that satisfies the following two properties for all ##n,m \in \mathbb{N}##
1) n + 1 = s(n)
2) n + s(m) = s(n + m)
(s is the successor function as described in the Peano Postulates)
Proof: Uniqueness: I'm going to skip this here as it is bot important for my question.
Existence:
For ##p \in \mathbb{N}##, we can apply the recursiob theorem to the set ##\mathbb{N}##, the element ##s(p) \in \mathbb{N}## and the function ##s: \mathbb{N} \rightarrow \mathbb{N}## to deduce that there is a unique function ##f_p: \mathbb{N} \rightarrow \mathbb{N}## such that ##f_p(1) = s(p)## and ##f_p \circ s = s \circ f_p##. Let ##+: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}## be defined by ##c + d = f_c(d)## for all ##(c,d) \in \mathbb{N} \times \mathbb{N}##. Let ##n,m \in \mathbb{N}##. Then ##n + 1 = f_n(1) = s(n)##, which is part 1) and ##n + s(m) = f_n(s(m)) = s(f_n(m)) = s(n + m)##, which is part 2).
Now, here comes this silly question. Why does the author use the notation ##f_c(d)##? It seems that he's 'hiding' that ##f_c## depends on 2 variables ##c,d## instead of 1. Although I do understand the proof, I feel uncomfortable with this notation.
Thanks in advance
Theorem: There is a unique binary operation ##+: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}## that satisfies the following two properties for all ##n,m \in \mathbb{N}##
1) n + 1 = s(n)
2) n + s(m) = s(n + m)
(s is the successor function as described in the Peano Postulates)
Proof: Uniqueness: I'm going to skip this here as it is bot important for my question.
Existence:
For ##p \in \mathbb{N}##, we can apply the recursiob theorem to the set ##\mathbb{N}##, the element ##s(p) \in \mathbb{N}## and the function ##s: \mathbb{N} \rightarrow \mathbb{N}## to deduce that there is a unique function ##f_p: \mathbb{N} \rightarrow \mathbb{N}## such that ##f_p(1) = s(p)## and ##f_p \circ s = s \circ f_p##. Let ##+: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}## be defined by ##c + d = f_c(d)## for all ##(c,d) \in \mathbb{N} \times \mathbb{N}##. Let ##n,m \in \mathbb{N}##. Then ##n + 1 = f_n(1) = s(n)##, which is part 1) and ##n + s(m) = f_n(s(m)) = s(f_n(m)) = s(n + m)##, which is part 2).
Now, here comes this silly question. Why does the author use the notation ##f_c(d)##? It seems that he's 'hiding' that ##f_c## depends on 2 variables ##c,d## instead of 1. Although I do understand the proof, I feel uncomfortable with this notation.
Thanks in advance
Last edited by a moderator: