Questions tagged [cardinals]
This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.
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Injective monotonic mapping from rationals $\mathbb Q^2$ to $\mathbb R$
Exercise: $f: \mathbb Q^2\to\mathbb R$. Where $\mathbb Q$ is the set of rational
numbers.
$f$ is strictly increasing in both
arguments.
Can $f$ be one-to-one?
This question is related to many ...
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Can a countable union of subgroups of uncountable index in G be equal to G? [closed]
Let G be a group and $\{H_i\}_{i<\omega}$ be a countable family of subgroups of $G$, each of them of uncountable index. Can $G=\bigcup_{i<\omega} H_i$?
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Is the cardinality of $\varnothing$ undefined?
It is intuitive that the cardinality of the empty set is $0$.
However we are asked to demonstrate this using given definitions/axioms in Tao Analysis I 4th ed ex 3.6.2.
My question arises as I think ...
- 3,325
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How does measure theory deal with higher cardinalities?
The second part of the definition of a sigma-algebra is that countable unions of measurable sets are measurable. The second property of a measure is that the measure of countable unions of measurable ...
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On the Singular Cardinal Hypothesis
I'm trying to find the proof of this result.
If for each $\lambda\geq2^\omega$, $\lambda^\omega\le\lambda^+$, then the SCH holds.
I'm not sure where to look. So if you have any info about this, please ...
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Why is $\{0,1\}^{\Bbb N}$ uncountable? [duplicate]
In the book Measure and Integral : An introduction to real analysis, in chapter 8 Lp spaces, theorem 8.18, the authors give a counterexample to show that $l^\infty$ is not separable:
Question: why is ...
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Cardinal sum of powers
I'm trying to solve this exercise. Can anybody please help me:
If $\kappa$ ist an infinite cardinal number with $cf(\kappa) = \kappa$ and for all $\mu < \kappa$ the inequality $2^{\mu} \leq \kappa$ ...
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For all cardinals $\kappa, \lambda$ with $\lambda \geq cf(\kappa)$ the inequality $\kappa^{\lambda} > \kappa$ holds [duplicate]
I genuinely have no idea why the proposition in the title holds or how to show it. I am kind of new to cardinals and ordinals and very confused. If someone could explain, I would really appreciate ...
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Cardinal power towers
I am not an expert on large cardinals. I could not find any reference (and terminology) for the following question:
We start with
$$\lambda:=\aleph_0 \text{ [tet] } \omega = \aleph_0 ^ {\aleph_0 ^ {\...
- 11
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Would well-founded Scott cardinals work in ZCA + Ranks?
Does original Zermelo's set theory + Regularity + Ranks, prove that every set is of equal size to some element of a Scott cardinal? The original Zermelo does include an axiom of Choice, and it admits ...
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The cardinality of specific set $A\subset \mathbb{N}^{\mathbb{N}}$
Let $A$ be a set of total functions from the naturals to the naturals
such that for every $f\in A$ there is a finite set $B_f\subset \mathbb{N}$ , such that for every $x\notin B_f$ , $f(x+1)=f(x)+1$.
...
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Induction does not preserve ordering between cardinality of sets?
Consider building a binary tree and consider it as a collection of points and edges. Here is one with five levels, numbered level $1$ at the top with $1$ node to level $5$ at the bottom with $16$ ...
- 6,360
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A family of $\kappa^{<\omega}$ such that for every member in $\kappa$ is contained by all but finitely many elements of the family
Suppose that $\kappa$ is an uncountable cardinal. Let $\kappa^{<\omega}$ denote the family of all finite subsets of $\kappa$. Does there exist a family $S\subset\kappa^{<\omega}$ such that for ...
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How should I should prove $\mathbb{R}\sim\{0,1\}^{ \mathbb{N}}$ [duplicate]
I've seen some argument about the binary representation, but I think it is not accurate because under some extreme cases, the rounding or bit constraint would results distinct reals also have the same ...
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Dependence of the equation $a+1=a$ for infinite cardinals $a$ on the axiom of choice
let $A$ be a set such that for all $n \in $ N $ A ≉ N_n$
where $N_n = \{ 0 ,1 ,2 ...... n-1\} $
and $a$ be the Cardinality of $A$ meaning ($|A| = a$)
is it possible to prove that $a+1=a$ without ...
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