All Questions
Tagged with well-orders transfinite-recursion
6
questions
1
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1
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81
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Any subset of a well-ordered set is isomorphic to an initial segment of this well-ordered set.
I wanted to prove the fact for which I have a sketch of proof: Let $(W,\leq )$ be a well-ordered set and $U$ be a subset of $W$. Then considering the restriction of $\leq $ to $U\times U$, we have ...
- 375
4
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0
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105
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Why does proof of Zorn's lemma need to use the fact about ordinals being too large to be a set?
I'm not understanding why its necessary to invoke the knowledge about ordinals in order to prove Zorn's lemma.
The Hypothesis in Zorn's lemma is
Every chain in the set Z has an upper bound in Z
Then ...
- 41
1
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1
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Transfinite Recursion Theorem - Particular case - Enderton
I have the following theorem for any formula $\gamma(x,y)$:
Theorem of Transfinite Recursion: Given a well-ordered set $A$ such that for any $f$ there is a unique $y$ such that $\gamma(f,y)$ holds, ...
- 3,630
1
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Comparability theorem for well ordered sets using transfinite recursion
Similar questions have already been asked here and here. But I am asking for verification of my proof. Halmos leaves out an "easy" transfinite induction argument, which I have struggled to ...
- 4,119
5
votes
2
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650
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Finding a well-ordering of the natural numbers of a given order type
Let $X$ be the set of all well-orderings of the set of natural numbers, and let $O$ be the set of countable ordinals, i.e. the set of ordinals that are order types of the well-orderings in $X$. Then ...
- 10.4k
7
votes
1
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524
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Direct proof of principle of transfinite induction
This is a problem from the book Set theory by You-Feng Lin.
Principle of Transfinite Induction
Let $(A,\le)$ be a well-ordered set. For each $x \in A$, let $p(x)$ be a statement about $x$. If for ...
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