One Step Past Omega

At a glance

• Part 1: 1 +𝜔 =𝜔. Label the first bead 0 and each following bead with the next whole number.
• Part 2: 𝜔 +1 ≠𝜔. The first row has a last position, while 𝜔 does not.
• Part 3: Ordinal addition is not commutative because 1 +𝜔 ≠𝜔 +1.
• Part 4: 3 +𝜔 =𝜔, but 𝜔 +3 ≠𝜔.

Key idea: order is part of the number

Ordinal addition joins ordered rows. The positions are not merely counted; their order must be preserved. A finite block placed before 𝜔 can be absorbed by shifting the labels of the infinite tail. A finite block placed after 𝜔 creates final positions that 𝜔 did not have.

Worked solution, part by part

1. Relabel 1 +𝜔

The top strand contains one new first bead, followed by a complete copy of 𝜔. Label the positions in their order:

More explicitly, pair the extra first bead with 0 in 𝜔. Pair the bead that used to be position 0 in the copied 𝜔-strand with 1, the bead that used to be position 1 with 2, and continue by pairing old position 𝑛 with new position 𝑛 +1.

This pairing preserves the order and uses every position on both sides. Therefore the strand has the same order type as 𝜔:

2. Inspect 𝜔 +1

The bottom strand first contains every position of 𝜔, then one gold bead is attached after all of them. That gold bead is the last position of the new strand.

The ordinal 𝜔 has no last position: after every finite position 𝑛, there is a next position 𝑛 +1. An order-preserving pairing cannot match a row with a last position to a row with no last position, because being last is determined by the order.

Therefore

The new ordinal is named 𝜔 +1: it is one step after every finite ordinal.

3. Test commutativity

Commutativity would require every pair of addends to give the same result when swapped. Parts 1 and 2 give one counterexample:

So

One counterexample is enough to show that ordinal addition is not commutative.

4. Move a block of three

For 3 +𝜔, place three positions first and then a copy of 𝜔. Relabel the three initial positions 0, 1, and 2. Relabel old position 𝑛 in the copied 𝜔-tail as 𝑛 +3. The entire row is again ordered like

so

For 𝜔 +3, the three added positions come after every finite position. The resulting row has a last position, namely the third added position. Since 𝜔 has no last position,

Check

The conclusions use the addition rule in the worksheet's stated left-to-right order. The relabelings for 1 +𝜔 and 3 +𝜔 preserve first, next, and every later position. The exclusions for 𝜔 +1 and 𝜔 +3 use a feature that every order-preserving pairing must preserve: whether a last position exists.

Another facet

These rows all contain countably many positions, so counting how many positions they have cannot distinguish them. Ordinal arithmetic records more information: it remembers their order. That is why moving a finite block from before 𝜔 to after 𝜔 changes the ordinal sum.