General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Theorem 31 There is a unary natural number of which /// is the successor
Theorem 32 There are at least 7 unary natural numbers.
Theorem 33 Every unary natural number is either zero or the successor of a unary natural
number. (we will come back to this later)
c○: Michael Kohlhase 33
This seems awfully clumsy, lets introduce some notation
Idea: we allow ourselves to give names to unary natural numbers
(we use n, m, l, k, n1, n2, . . . as names for concrete unary natural numbers.)
Remember the two rules we had for dealing with unary natural numbers
Idea: represent a number by the trace of the rules we applied to construct it.
(e.g. //// is represented as s(s(s(s(o)))))
Definition 34 We introduce some abbreviations
we “abbreviate” o and ‘ ’ by the symbol ’0’ (called “zero”)
we abbreviate s(o) and / by the symbol ’1’ (called “one”)
we abbreviate s(s(o)) and // by the symbol ’2’ (called “two”)
. . .
we abbreviate s(s(s(s(s(s(s(s(s(s(s(s(o)))))))))))) and //////////// by the symbol
’12’ (called “twelve”)
. . .
Definition 35 We denote the set of all unary natural numbers with N1.
(either representation)
c○: Michael Kohlhase 34
Induction for unary natural numbers
Theorem 36 Every unary natural number is either zero or the successor of a unary natural
number.
Proof: We make use of the induction axiom P5:
P.1 We use the property P of “being zero or a successor” and prove the statement by
convincing ourselves of the prerequisites of
P.2 ‘ ’ is zero, so ‘ ’ is “zero or a successor”.
P.3 Let n be a arbitrary unary natural number that “is zero or a successor”
P.4 Then its successor “is a successor”, so the successor of n is “zero or a successor”
P.5 Since we have taken n arbitrary (nothing in our argument depends on the choice)
we have shown that for any n, its successor has property P .
P.6 Property P holds for all unary natural numbers by P5, so we have proven the assertion
c○: Michael Kohlhase 35
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