INFINITE EXPRESSIONS

"The infinite is always in potency, never in act" (Aristotle. Metaphysics).

"Mathematics is the science of the infinite" (Hermann Weyl).

"Infinity is but a peculiar turn given to generality" (Charles Sanders Peirce).

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Examples of infinite expressions

1. ([1...]*2]−1) // rep. (1 3 5 5 7 ...)
   
   This expression is equivalent to
   ( (1 3 ...) )
   
   
2. (1÷[1 ...]) // rep. (1 1 1÷2 1÷3 1÷4 ...)
   
   
3. +⊣(1÷[1...]) // rep. (1 + 1÷2 + 1÷3 + ...) (harmonic series)
   
   
4. +⊣(1÷[1...]^2]) // rep. (1 + 1÷1÷2^2) + 1÷3^2) + 1÷4^2) + ...)

Parameterized infinite expressions

1. (p(n) = {n n+1 n+2 n+3 ...})
   is represented by:
   
   ⟨( p(n) = {n...} )⟩
   
   
2. (q(n1 n2) = {n1+n2 (n1+1)*(n2+1) (n1+2)*(n2+2)...})
   is represented by:
   
   ⟨( q(n1 n2) = {⌊n1…⌋+⌊n2…⌋} )⟩

Infinite higher order expressions

They are infinite expressions of infinite expressions. Examples:

1. Infinite sequence of infinite sequences:
   
   ((1 2 3 ...) (1 2 3 ...) (1 2 3 ...) ...)
   is represented as:
   
   ( ( ( 1... )☆) )
   
   
2. Infinite set of infinite sets:
   
   {{1 2 3 ...} {2 3 4 ...} {3 4 5 ...} ...}
   is represented as:
   
   { {[1…]…} }