I am trying to understand the principle of recursion:
Let ℕ be the natural numbers.
Let T be a set.
Let a ∈ T.
Let g: T → T be a mapping.
$$\forall x \in N: f \left({x}\right) = \begin{cases} a & : x = 0 \\ g > \left({f \left({n}\right)}\right) & : x = n + 1 \end{cases}$$ Wiki
Could someone elaborate on what the ':' symbol implies? Judging from the reference at Wolfram I believe my concept of what exactly a mapping is may be faulty, especially in the latter half of the equality.
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$\begingroup$There are three different uses of the colon in that passage. In $g:T\to T$, it’s simply part of the standard notation for a function from one set to another: $g$ is a function from the set $T$ into the set $T$. In terms of ordinary English punctuation, the colon after $\forall x\in\Bbb N$ could just as well be a comma: ‘for each $x$ in $\Bbb N$, $f(x)$ is equal to ...’. The two colons in the definition of $f(x)$ should be read ‘if’.
$\endgroup$ 2 $\begingroup$For each $x$ in $N$, $f(x)=a$ if $x=0$, and $f(x)=g\big(f(n)\big)$ if $x=n+1$.
In mappings (such as $f: X \to Y$) it just means, in my eyes, "that maps to", while in the function defined, it just means "such that." Nothing too special.
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