Page 111 - 'Blast_Into_Math

Blast into Math! Prime nummers: indestructimle muilding mlocks

and since each natural number is assigned to a unique element,

if n = m.
z n = z m

Now we need to assign unique natural numbers to the elements of

k+1

S j .
j=1

All the elements of S k+1 which are already in

k

N
S j = {z n } n=1
j=1

have already been assigned to a unique natural number. So, we look at

N
S k+1 \{z n } n=1 .

Since S k+1 is finite, either

N
S k+1 \{z n } n=1 = ∅

or there is m ∈ N such that

N m
S k+1 \{z n } n=1 = {s n } n=1 .

In this case we define

z N+n = s n , for n =1,...,m.

If

N
S k+1 \{z n } n=1 = ∅,

then we continue to

N
S j \{z n } n=1

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