Existence part of Iteration theorem
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Iteration theorem: Consider any Peano system $(P, S, 0)$. Let $W$ be an arbitrary set. Let $c$ be a fixed element of $W$ and $g$ be a singular operation on $W$. Then there is a unique function $F: P → W$ such that $F(0) = c$ and $F(S(x)) = g(F(x))$ for all $x$ in $P$.
Why is there a need to prove the existence of F?
Doesn't the phrase "such that $F(0) = c$ and $F(S(x)) = g(F(x))$ for all $x$ in $P$" already define $F$?
natural-numbers
asked Nov 24 at 20:37
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Iteration theorem: Consider any Peano system $(P, S, 0)$. Let $W$ be an arbitrary set. Let $c$ be a fixed element of $W$ and $g$ be a singular operation on $W$. Then there is a unique function $F: P → W$ such that $F(0) = c$ and $F(S(x)) = g(F(x))$ for all $x$ in $P$.
Why is there a need to prove the existence of F?
Doesn't the phrase "such that $F(0) = c$ and $F(S(x)) = g(F(x))$ for all $x$ in $P$" already define $F$?
natural-numbers
asked Nov 24 at 20:37
Little Rookie
625523
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Iteration theorem: Consider any Peano system $(P, S, 0)$. Let $W$ be an arbitrary set. Let $c$ be a fixed element of $W$ and $g$ be a singular operation on $W$. Then there is a unique function $F: P → W$ such that $F(0) = c$ and $F(S(x)) = g(F(x))$ for all $x$ in $P$.
Why is there a need to prove the existence of F?
Doesn't the phrase "such that $F(0) = c$ and $F(S(x)) = g(F(x))$ for all $x$ in $P$" already define $F$?
natural-numbers
asked Nov 24 at 20:37
Little Rookie
625523
Iteration theorem: Consider any Peano system $(P, S, 0)$. Let $W$ be an arbitrary set. Let $c$ be a fixed element of $W$ and $g$ be a singular operation on $W$. Then there is a unique function $F: P → W$ such that $F(0) = c$ and $F(S(x)) = g(F(x))$ for all $x$ in $P$.
Why is there a need to prove the existence of F?
Doesn't the phrase "such that $F(0) = c$ and $F(S(x)) = g(F(x))$ for all $x$ in $P$" already define $F$?
natural-numbers
natural-numbers
asked Nov 24 at 20:37
Little Rookie
625523
asked Nov 24 at 20:37
Little Rookie
625523
asked Nov 24 at 20:37
Little Rookie
625523
asked Nov 24 at 20:37
Little Rookie
625523
asked Nov 24 at 20:37
Little Rookie
625523
625523
add a comment |
add a comment |
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