MAKE-ORD.html -- ACL2 Version 6.3

### MAKE-ORD

a constructor for ordinals.

    Major Section:  ACL2-BUILT-INS

`Make-ord` is the ordinal constructor. Its use is recommended instead of using `cons` to make ordinals. For a discussion of ordinals, see ordinals.

For any ordinal, `alpha < epsilon-0`, there exist natural numbers `p` and `n`, positive integers `x1, x2, ..., xn` and ordinals `a1 > a2 > ... > an > 0` such that `alpha > a1` and `alpha = w^(a1)x1 + w^(a2)x2 + ... + w^(an)xn + p`. We call `a1` the ``first exponent'', `x1` the ``first coefficient'', and the remainder `(w^(a2)x2 + ... + w^(an)xn + p)` the ``rest'' of alpha.

`(Make-ord fe fco rst)` corresponds to the ordinal `(w^fe)fco + rst`. Thus the first infinite ordinal, `w` (`omega`), is constructed by

    (make-ord 1 1 0)

and, for example, the ordinal `(w^2)5 + w2 + 7` is constructed by:

    (make-ord 2 5 (make-ord 1 2 7)) .

The reason `make-ord` is used rather than `cons` is that it allows us to reason more abstractly about the ordinals, without having to worry about the underlying representation.

To see the ACL2 definition of this function, see pf.