User blog:PsiCubed2/A Surjection from the Natural Numbers to the SVO

This is something I've done long ago and forgotten about, until Edwin Shade's post reminded me of it.

I present here a way to notate any ordinal below the SVO by a natural number. It should be noted that in my system every natural number represents some ordinal, though the same ordinal can be represented by different numbers.

The system goes like this:

(1) Ord(1) = 0

(2) Ord(2) = 1

(3) Ord(xy) = Ord(x)+Ord(y)

(4) If q as then nth prime (with n>1) then:

If n=2a3b5c7d.... then:

Ord(q) = φ(...,Ord(c+1),Ord(b+1),Ord(a+1))

And that's it.

For example, to calculate the ordinal corresponding to the number 100:

(1) Ord(100) = Ord(5x5x2) = Ord(5)+Ord(5)+Ord(2)

(2) Ord(2) = 1

(3) To calculate Ord(5):

(i) 5 is the 3rd prime

(ii) 3 = 2031

(iii) So b=1 and a=0

(iv) So Ord(5) = φ( Ord(1+1), Ord(0+1) ) = φ( Ord(2), Ord(1) ) = φ(1,0) = ε₀

(4) So the number 100 corresponds to the ordinal:

Ord(100) = Ord(5x5x2) = Ord(5)+Ord(5)+Ord(2)= ε₀+ε₀+1 = ε₀×2+1.

Neat, huh?

(credit for the idea of using primes in this way goes to a wiki member called Emlightened, who used a similar construct to create a bijection from the Natural numbers to ε₀)