Function of Reals
Joseph R. Shoenfield · 2018
We now extend our notion of a function to allow reals as arguments. (We could allow all total functions as arguments; but this would complicate matters without really adding anything, since a function can be replaced by its contraction.) We use lower case Greek letters, usually α, β , and γ , for reals. When the value of m is not important, we write α → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203741139/8e9d6847-fedd-45bc-9ee6-91ac5be8d222/content/in67_1.tif"/> for α 1 ,… α m . We use ℝ for the class of reals and ℝ m,k for the class of all ( m + k )-tuples ( α 1 ,…, α m , x 1 ,…, x k ). An ( m,k )–ary function is a mapping of a subset of ℝ m,k into ω . (Thus a (0, k )-ary function is just a k –ary function.) From now on, a function is always an ( m,k )-ary function for some m and k . Such a function is total if its domain is all of ℝ m,k . An ( m,k ) – ary relation is a subset of ℝ m,k . We define the representing function of such a relation as before.