All Questions
Tagged with ordinals solution-verification
88
questions
2
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1
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104
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Define the sum of transfinite ordinal sequences according to finite ordinal sum.
I know it is possibile to define finite sum of ordinals: if $(\alpha_i)_{i\in n}$ is a sequence of lenght $n$, with $n$ in $\omega$, then the symbolism $\sum_{i\in n}\alpha_i$ has a (formal) meaning; ...
- 4,906
-2
votes
1
answer
57
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Prove that the order type of $\alpha\cdot\beta$ is the antilexicographic order in $\alpha\times\beta$. [closed]
This question is related to this one, but not a duplicate, since I am struggling with injectivity and monotonicity, rather than proving that $\{\alpha\cdot\eta + \xi:\eta<\beta\textrm{ and }\xi<\...
- 4,906
2
votes
1
answer
67
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Is possibile to define an exponentiation with respect an ordinal operation?
It is well know the following resul holds.
Theorem
For any $(M,\bot,e)$ monoid there exists a unique esternal operation $\wedge_\ast$ from $X\times\omega$ into $X$ such that for any $x$ in $M$ the ...
- 4,906
0
votes
1
answer
67
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An ordinal $\nu$ is a natural iff there is no injection $f$ of $\nu$ into $X$ in $\mathscr P(\nu)\setminus\{\nu\}$.
Let's we prove the following theorem.
Theorem
An ordinal $\nu$ is a natural if and only if there is no injection of $\nu$ into $X$ in $\mathscr P(\nu)\setminus\{\nu\}$.
Proof.
Let's we assume there ...
- 4,906
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
3
votes
1
answer
60
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Cofinalities of ordinals $\omega_1 \omega, \omega_2 \omega_1, \omega_2 \omega_1 \omega$?
I am trying to understand cofinalities so I would really appreciate it if you could confirm whether the following line of reasoning is correct. (All multiplication is ordinal multiplication)
$$
cf(\...
- 481
1
vote
2
answers
87
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Solution verification: proof that the supremum of a set of cardinals is a cardinal
I'm trying to show that if $X$ is a set of cardinals, then $\sup X$ is a cardinal.
This is part (ii) of Lemma 3.4 in the third edition of Jech on page 29.
The proof in Jech is pretty terse and I don't ...
- 11.9k
1
vote
1
answer
126
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a proof idea: Every well-ordered set has an order-preserving bijection to exactly one ordinal.
I have seen a proof of the statement, and its usually by transfinite induction. And I'm trying to find out why my proof doesn't work, it seems too simple:
Let $X$ be a well-ordered set. Define $X^{<...
- 438
2
votes
1
answer
143
views
Dubious proof in Halmos's book: Two similar ordinal numbers are always equal
I'm studying ordinal numbers using Naive set theory of Halmos.
I think there was a small mistake in his proof of the statement "if two ordinal numbers are similar, then they are equal"
The ...
- 1,350
0
votes
1
answer
35
views
Is $\operatorname{cf}(2^{<\kappa})=\operatorname{cf}(\kappa)$?
As the title says, I am wondering whether $\operatorname{cf}(2^{<\kappa})=\operatorname{cf}(\kappa)$ where $\kappa$ is an infinite cardinal. I believe I have a proof: Define $f:\kappa\to 2^{<\...
- 448
0
votes
0
answers
94
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if $\alpha \beta$ is successor $\Rightarrow \alpha$ is successor y $\beta$ is successor
Please could you look at the demonstration I propose below for this implication and tell me if I am wrong.
What other way or idea is there to demonstrate the implication?
Proof:
The implication is ...
- 11
1
vote
1
answer
51
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"Hyper-pigeonhole principle": size of tail-avoiding sequence families
I'm working on a problem in complex analysis and I wanted to check my logic on a specific point.
Let $A$ be an infinite set with cardinality $\aleph^A := |A|$ and initial ordinal $\omega^A$.
By abuse ...
- 5,830
1
vote
0
answers
49
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Prove that for all $\alpha \in \text{On}\setminus \{1\}$ there exists an arbitrarily large $\beta \in \text{On}$ such that :
$\alpha^{\beta}= \alpha.\beta$.
In order to do this, it is suggested to prove that for all $\alpha, \beta >2$, $\alpha^{\beta}= \alpha.\beta$ iff $\alpha^{\beta}= \beta$.
Here's my proof of this ...
- 3,330
0
votes
0
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143
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Supremum of a set of ordinals
The definition of supremum (from wikipedia) is as follows:
If $(P,\le)$ is a partially ordered set and $S$ is a subset of $P$ then $x\in P$ is an upper bound for $S$ if $y\le x$ for all $y\in S$. ...
user10575
1
vote
1
answer
101
views
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