User blog comment:LittlePeng9/ITTM galore/@comment-10429372-20140321162236/@comment-25418284-20140321214530

@Wyth You have misconceived how the encoding works. Understandable, because it's complicated and the explanation I wrote is quite shoddy.

The encoding for zero and successor ordinals is simple enough. For limit ordinals, suppose we have a countably infinite collection of already-encoded ordinals like so:


 * A = 1001011101010100...
 * B = 0101100111010100...
 * C = 0101001011010100...
 * D = 0101101101011110...

(These are random pretend encodings. Don't try to analyze them.) We need to create a new ordinal, the supremum of {A, B, C, D, ...} by mashing all these encodings into a single one. We know that w^2 has the same cardinality as w, this is hardly difficult -- just "interleave" all the encodings like so:

A: 1 0 0 1 0 1 1 1 0 1 0 1 0 1 0 0 ... B: 0   1   0   1   1   0   0   1  ... C:   0       1       0       1    ... D:       0               0        ... E:               1                ... ... (1)1000011000111111011000100011010 ...  (encoding of sup{A, B, C, D, E, ...})

That's LittlePeng's way. Here's another, as given in the ITTM article:

A: 10 0 1   0    1     ... B:  0 1  0   1    1    ... C:     0  1   0    1   ... D:         0   1    0  ... E:              1    1 ... ... (1)10001010100101111101 ...  (encoding of sup{A, B, C, D, E, ...})