All Questions
Tagged with ordinals transfinite-induction
39
questions
0
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1
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51
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Is $\operatorname{rank}(A)\subseteq\operatorname{TC}(A)$?
This question came to me when I was thinking about rank and transitive closures. Let $A$ be a set of rank $\alpha$ and let $\operatorname{TC}(A)$ denote the transitive closure of $A$. Is it true then ...
- 448
4
votes
1
answer
182
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Colourful class function
Background. We're in $\mathsf{ZFC}$, and I can use the principle of $\epsilon$-induction, but not (directly) the $\epsilon$-recursion.
Problem. Let $F : V \to V$, where $V$ is the class of all the ...
- 509
1
vote
2
answers
104
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Transfinite recursion to construct a function on ordinals
I am asked to use transfinite recursion to show that there is a function $F:ON \to V$ (here $ON$ denote the class of ordinals and $V$ the class of sets) that satisfies:
$F(0) = 0$
$F(\lambda) = \...
- 839
2
votes
1
answer
42
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If $X_1$,$X_2$ are wosets isomorphic to ordinals $\alpha_1,\alpha_2$ then $X_1\times X_2$ is isomorphic to $\alpha_2\cdot \alpha_1$
I want to prove the following:
Let $X_1$,$X_2$ be wosets, isomorphic to ordinals $\alpha_1,\alpha_2$ respectively. Then $X_1\times X_2$, with the lexicographic order, is isomorphic to $\alpha_2\cdot\...
- 4,827
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1
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89
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How to prove this alternative version of transfinite recursion
There are several formulations of transfinite recursion. I am interested in the following one.
Let $(V, \in)$ be a model of ZF. Let $g_1$ be a set and $G_2,G_3 : V \to V$ be two definable class ...
- 576
2
votes
1
answer
107
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I don't understand this proof: Trichotomy for ordinals (by double transfinite induction)
Hello everyone!
The proof is from this book: https://st.openlogicproject.org/
I think I understand the black bracket (checked). What worries me is the blue bracket (question mark).
As far as I ...
- 537
2
votes
1
answer
75
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if $\omega^{\alpha} =A \cup B$ then $A$ or $B$ has order type $\omega^{\alpha}$
Show that if $\omega^{\alpha} =A \cup B$ then $A$ or $B$ has order type $\omega^{\alpha}$ where $\alpha \geq 1$ is a ordinal number.
Hint: Use induction on $\alpha$
I don't know how to start with the ...
- 133
0
votes
1
answer
62
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Obscure question in Smullyan and Fitting: "strengthening of definition by finite recurrence"
Context: self-study from Smullyan and Fitting's "Set Theory and the Continuum Problem" (revised 2010 edition), chapter 3, section 8, Definition by Finite Recursion.
They give Theorem 8.1
...
- 5,057
0
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1
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143
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Transfinite Construction, an intuitive interpretation.
Theorem (Transfinite Construction).
Let $W$ be a well-ordered set, and $E$ an arbitrary class. Assume:
For each $x\in W$, there is a given rule $R_x$ that associates with each $\varphi\colon W(x)\to E$...
- 2,504
1
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1
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161
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Questions about the induction on cardinals
From Hereditary Cardinality and Rank :
For an infinite cardinal $\kappa$, $$\forall x,\ \textrm{hcard
}x<\kappa\rightarrow\textrm{rank }x<\kappa$$
We can show this by induction on $\kappa$. ...
- 12k
1
vote
1
answer
101
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Explain why transfinite induction does not assume that a property must be true for zero.
THIS QUESTION IS NOT A DUPLICATE OF THIS ONE!!!
So I would like to discuss the following proof of transfinite induction which is taken by the text Introduction to Set Theory wroten by Karell Hrbacek ...
- 4,906
1
vote
1
answer
120
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Exercises on transfinite induction
Define by transfinite recursion $V(0)=\emptyset, V(\alpha+1)=\mathcal P(V(\alpha)), V(\alpha)=\cup_{\beta < \alpha}V(\beta)$ for $\beta$ a limit ordinal.
I'm trying to show the following properties:...
- 12k
0
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0
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64
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Question on the hierarchy of named Ordinals
I was wondering how many named transfinite ordinals there are, and what the notations and names are, arranged as a hierarchy which ignores the different levels on a single level of Ordinals (ω+1, 2ω, ...
- 218
1
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1
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70
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How to show that the Zermelo hierarchy is really a hierarchy?
In Cameron's Sets, Logic and Categories (p. 48-49), he sets out prove the following fact about the Zermelo hierarchy $V$, namely, that $V_\alpha\subseteq V_{s(\alpha)}$ for all ordinals $\alpha$. His ...
- 1,779
1
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2
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331
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Recursion theorem for ordinals proof
I'm trying to understand the proof of the recursion principle of ordinals, the theorem is:
The proof of this theorem uses this other theorem:
The proof is pretty long (I'm sorry) so I will try to ...
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