5
$\begingroup$

My question concerns a proof given on page 118 in the text Introduction to Set Theory 3rd ed. by Hrbacek and Jech.

The authors on page 117 prove a version of the transfinite recursion theorem (Theorem 4.11) that says given unary operations $G_1, G_2$, and $G_3$ there is an operation $F$ such that \begin{align*} &F(0)=G_1(0),\\ &F(\alpha+1)=G_2(F(\alpha))\quad\text{for all ordinals $\alpha$, and}\\ &F(\alpha)=G_3(F_\restriction \alpha)\quad\text{for all limit ordinals $\alpha$}.\\ \end{align*}

They then leave it up to the reader to devise a parametric version of Theorem 4.11. I have determined this to be as follows: Given binary operations $G_1, G_2$, and $G_3$ there is an operation $F$ such that for all z

\begin{align*} &F(z,0)=G_1(z,0),\\ &F(z,\alpha+1)=G_2(z,F_z(\alpha))\quad\text{for all ordinals $\alpha$, and}\\ &F(z,\alpha)=G_3(z,F_z\restriction \alpha)\quad\text{for all limit ordinals $\alpha$}.\\ \end{align*}

Next they make use of the parametric version of Theorem 4.11 in the proof of Theorem 3.6 on page 118. Theorem 3.6 states:

\begin{align*} &\text{Let G be an operation.}\\ &\text{For any set $a$ there is a unique infinite sequence $\langle a_n|n\in N \rangle$ such that} \\ &(a) a_0=a\\ &(b) a_{n+1}=G(a_n,n)\quad\text{for all $n\in N$}\\ \end{align*}

The proof for Theorem 3.6 given in the text is as follows: \begin{align*} &\text{Let G be an operation. We want to find, for every set $a$, a sequence}\\ &\text{$\langle a_n|n\in N \rangle$ such that $a_0=a$ and $a_{n+1}=G(a_n,n)$ for all $n\in N.$}\\ &\text{By the parametric version of the Transfinite Recursion Theorem 4.11,}\\ &\text{there is an operation $F$ such that $F(0)=a$ and $F(n+1)=G(F(n),n)$ for all $n\in N.$}\\ &\text{Now we apply the Axiom of Replacement: There exists a sequence $\langle a_n|n\in N \rangle$}\\ &\text{that is equal to $F\restriction \omega$ amd the Theorem follows.}\\ \end{align*}

Now, I understand everything in the proof of Theorem 3.6 except how the parametric version of Theorem 4.11 is used to derive the operation $F$ in the proof. Can can someone please help me fill in blanks?

  • 0
    I thought this was going to be an easy question for someone.2012-10-21
  • 0
    I'd suggest that you haven't received any help because the nature of your question requires that you find someone who's read Jech recently. In Googling for parameterized transfinite recursion, two of the first three results are your question and Jech's book. I'd suggest explaining the notion of parameterization for those who are interested enough to answer but not interested enough to pull out our Jech.2012-10-21

1 Answers 1