All Questions
21
questions
4
votes
0
answers
103
views
Prove that any continuous function $f:S_\Omega \rightarrow \mathbb{R}$ is eventually constant.
Let $\Omega$ be the first non-numerable ordinal number ( $\aleph_1$ is the first cardinal number greater than $\aleph_0$ when treated as an ordinal number is denoted by $\Omega$ ) and let $[0,\Omega)$ ...
- 147
1
vote
1
answer
69
views
The cardinality of the preimage of an ordinal
Define a cardinal as an ordinal $\kappa$ such that for all ordinals $\alpha < \kappa$, $\alpha$ injects into $\kappa$ but is not bijective to $\kappa$.
Let $\kappa$ be an infinite cardinal. I'm ...
- 12k
0
votes
0
answers
16
views
What is the difference between $G(f_{|0})$ and $G(f_{|1})$
I just wanted a quick reality check from the SE community to make sure that I am correctly concluding the distinction between $G(f_{|0})$ and $G(f_{|1})$...where, in English, $f_{|a}$ is saying "...
- 5,064
3
votes
0
answers
71
views
Is it wrong to define functions on ordinals? [duplicate]
After familiarizing myself with the ordinal numbers and their arithmetic, I’ve been (perhaps sloppily) using function notation to refer to certain combinations of arithmetic operations on ordinals. ...
0
votes
1
answer
111
views
Give a bijection from $\Bbb N\to\omega^{\omega}$ based upon counting consecutive similar digits of binary strings
Question
Give a bijection from $\omega^{\omega}\to\Bbb N$ based upon counting consecutive similar digits of binary strings: There is a near-bijection given in attempt 1 below; the aim is to fix it to ...
- 9,299
0
votes
1
answer
225
views
Give a bijection from $\omega^{\omega}\to\Bbb N$ [duplicate]
Question
Give a bijection from $\Bbb N\to\omega^{\omega}$
Where $\omega^{\omega}$ is the set of ordinals expressible in Cantor normal form as:
$$\sum_{i=k}^0\omega^i\cdot a_i:a_i\in\Bbb N_{\geq0}$$
...
- 9,299
2
votes
1
answer
167
views
How is cardinality exactly defined as a function and why is it different from the ordinals
Once we construct the definition of the ordinals:
$$0=\{\} \, \, 1=\{0\} \,\, 2=\{0,1\} \,\,3= \{0, 1,2\} \,\, ...$$
And we want to describe the cardinality of the set $S$:
$$S=\{3,2,4\}$$
Intuitively ...
- 310
0
votes
0
answers
69
views
On what set is the group action of the sequence truncation function, the identity function?
Let $\omega^\omega$ be the set of finite sequences of positive natural numbers and let $\lvert x\rvert$ be the ordinal of $x\in\omega^\omega$ given by using the terms as coefficents in Cantor normal ...
- 9,299
-2
votes
1
answer
45
views
Unknown notation/omega
What does the 3rd term in $\omega\times\omega\times\omega^{\operatorname*{\omega}\limits_{\smile}}$ with semicircle below the last $\omega$ in definitions 5.3.6
here mean?
- 3,978
0
votes
1
answer
45
views
Let $f$ be a mapping, $\beta$ be an ordinal, $X=\{\alpha\mid f(\alpha)\le \beta\}$, and $\gamma=\sup X$. Is $\gamma\in X$?
Let $f:\operatorname{Ord}\to\operatorname{Ord}$ be a mapping consisting of only addition, multiplication, and exponentiation operations, and $\beta$ be an ordinal. Let $X=\{\alpha\mid f(\alpha)\le \...
- 17.6k
-1
votes
1
answer
90
views
When does an injective function $\omega^{<\omega}\to\Bbb N$ have positive density in the integers? [closed]
Under what minimal conditions must the range of a definable, injective function $\omega^{<\omega}\to\Bbb N$ have positive density in the integers?
I'm very inexperienced in such matters but it ...
- 9,299
4
votes
1
answer
1k
views
Show that there exists a fixed point for this (set theoretic) class function
I see that this question might be trivial but I can't seem to figure it out myself:
Suppose that $F:ON\to ON$ is a class function: that is, for every ordinal $\alpha$ there is unique ordinal $F(\...
- 608
0
votes
1
answer
49
views
Where is the mistake in this set-theoretic argument?
Let $\omega$ be the first infinite ordinal and for all $n\in \omega $ define $n=\{0,1,2,...,n-1\}$. In particular, $2=\{0,1\}$.
Let $f:\omega\rightarrow \omega$, $f(x)=x^2 $.
Now $4=f(2)=f(\{0,1\})=...
- 104
5
votes
2
answers
94
views
Is $f: \mathcal{P}(\omega) \to \omega \cup \lbrace \omega \rbrace, \ x \mapsto \bigcup \lbrace n \in \omega: n \subset x \rbrace $ surjective?
Intro: I am currently preparing for an exam and I have found this particular question in a previous exam of the same class. It is a 'simple' question where you only have to write the answer without ...
- 3,499
1
vote
0
answers
73
views
Ordinal arithmetic and functions
I have two function $G$ and $F$ defined on ordinals and I know that
$$G(\alpha +\omega )\subseteq F(\gamma +\alpha+\omega)$$ when $G(\alpha)\subseteq F(\gamma)$ and $\alpha$ is a limit ordinal. I ...
- 4,160