One Step Past Omega
Page 1 · Build the idea
Rows have order types
Ordinals describe positions in order. To form , place the entire row for first, then the entire row for . The first infinite ordinal is : . It has a first position and a next position after every position, but no last position. Two rows represent the same ordinal when their positions can be paired in order.
Rehearse with finite rows
and both have three positions.
Pair first with first, second with second, and third with third.
For finite rows, reversing the two lengths still gives five positions.
Now remove the last position
The missing last position in is why the order of the addends may matter.
Page 2 · Follow the infinite order
Write on the diagram. Use scratch paper for explanations.
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On the strand, write 0 under the first bead, then 1, 2, 3, and 4 under the next four. Describe how the labeling continues. Does this pair every position with in order? Find .
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On the strand, the gold bead comes after every position in the copy of . Does this strand have a last position? Does ? Use that difference to decide whether .
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An operation is commutative if swapping the addends never changes the result. Is ordinal addition commutative? Write an equation or inequality from Parts 1 and 2 as evidence.
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Without drawing another infinite strand, decide whether and whether . Explain both decisions.