Natural Numbers:

0 : s.z.z

1 : s.z.s(z)

2 : s.z.s(s(z))

3 : s.z.s(s(s(z)))

etc.

Successor ("add one") function:

S : a.b.c.b(a b c)

Multiply function:

M: a.b.c.a(b c)

Answer the following questions:

#1. Write and show the full evaluation of the following -applications:

1. S 0
2. S 1

#2. Prove (or argue as fully as you can) that S works for all natural numbers.

#3. Write and show the full evaluation of the following -application:

(M 2 3)

#4. Prove (or argue as fully as you can) that M works for all pairs of natural numbers.

#5. BONUS philosophical question: do you think there is a -application that describes the entire universe in every aspect? Why or why not?