Homework 3: Set Theory, Fall 1998 Due October 19

John T. Baldwin

October 24, 1998

1. The Hessenberg sum of ordinals is defined as follows: If alpha and α′ are written in Cantor normal form as

α = ∑_{i<k} ω^{β_{i}} · m_{i}

and

α′ = ∑_{i<k} ω^{β′_{i}} · m′_{i}

(I am allowing some m_{i} to be zero to save typing.) then

α ⊕ α′ = ∑_{i<k} ω^{β′_{i}} · (m_{i} + m′_{i}).

Now show:

1. ⊕ agrees with + if α and β are finite.

2. ⊕ is commutative.

3. How do you determine which of two ordinals written in Cantor normal form is larger?

4. If β < β′ then α ⊕ β < α ⊕ β′.

5. Show the following further form of ‘indecomposable’. If α = ω^{δ} some δ then for any β, γ < α, β ⊕ γ < α.

2. Show that the Hercules slays the hydra no matter what strategy he follows. (Hint: Cantor normal form is key.)

problems 14,15, 18, 19 (cardinals). Feel free to e-mail me at jbaldwin@uic.edu or phone to make an appointment

to discuss any difficulties that arise. Office: 417 SEO. Office phone: 312-413-2149