Tagged Questions
4
votes
2answers
59 views
Showing there is only one isomorphism between well ordered sets using transfinite induction
I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
2
votes
1answer
87 views
Question on the use of a parametric version of Transfinite Recursion Theorem in Introduction to Set Theory 3rd ed. by Hrbacek and Jech
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 ...
0
votes
1answer
82 views
Equivalent statement of transfinite/ordinal recursion
I am trying to prove that the "standard" statement of transfinite/ordinal recursion:
"Suppose $G$ is a definite operation on partial functions on ordinals. Then there is a unique definite operation ...
1
vote
2answers
129 views
What does $\upharpoonright$ in $G(F\upharpoonright\alpha)$ mean?
More formally, we can state the Transfinite Recursion Theorem as follows. Given a class function $G\colon V\to V$, there exists a unique transfinite sequence $F\colon\mathrm{Ord}\to V$ (where ...
2
votes
1answer
121 views
Well ordering and maximal principle
I have to prove that given any poset $(P,\preceq)$ there exists a chain $S$ such that it is maximal (meaning that if $S\subseteq S'$ then $S=S'$). The book contains a proof using the axiom of ...
2
votes
2answers
277 views
Injection from the set of countable ordinals $\Omega$ into $\mathbb{R}$
I'm reading through this and I'd like to define an injective function from the set of countable ordinals $\Omega$ into $\mathbb{R}$ using transfinite induction (or maybe transfinite recursion?).
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