Intereting Posts

I am trying to show $\int^\infty_0\frac{\sin(x)}{x}dx=\frac{\pi}{2}$
Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer
Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
Help with $\int \frac 1{\sqrt{a^2 – x^2}} \mathrm dx$
Trouble in understanding a proof of a theorem related to UFD.
how to show $f$ attains a minimum?
Find all such functions defined on the space
$p^3 + 2$ is prime if $p$ and $p^2 + 2$ are prime?
Fourier transform of $\Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) /\sqrt{ \cosh(p/2)}$
Calculate $x$, if $y = a \cdot \sin{}+d$
Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?
Solving Bessel's ODE problem with Green's Function
How is the hyperplane bundle cut out of $(\mathbb{C}^{n+1})^\ast \times \mathbb{P}^n$?
Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$
pullback of rational normal curve under Segre map

Continuum hypothesis states, there is no set with cardinality between the integers and the reals.

There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + not-CH are consistent.

What if ZFC and not-CH. Thus, we have an axiom which states, *there* *is* a cardinality between $\aleph_0$ and $2^{\aleph_0}$.

- Zero vector of a vector space
- When do surjections split in ZF? Two surjections imply bijection?
- Proofs given in undergrad degree that need Continuum hypothesis?
- ¿Can you help me with axiom of regularity?
- How does (ZFC-Infinity+“There is no infinite set”) compare with PA?
- Where is axiom of regularity actually used?

Can a such set be defined?

- Prove that $\mathbb{|Q| = |Q\times Q|}$
- Ideals in the ring of endomorphisms of a vector space of uncountably infinite dimension.
- How many total order relations on a set $A$?
- Cardinality of a set of closed intervals
- Is there an easy proof for ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$?
- How to show $\kappa^{cf(\kappa)}>\kappa$?
- Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?
- What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?
- A question about cardinal arithmetics without the Axiom of Choice
- Can sets of cardinality $\aleph_1$ have nonzero measure?

In some sense, yes, you can always construct a set of size $\aleph_1$. Specifically $\omega_1$ is a set of size $\aleph_1$. And if the continuum hypothesis fails, it serves as a counterexample.

You might want to ask whether or not you can construct a set of real numbers of this particular size, and the answer to that will depend on your notion of “construct”, but if you mean define “in a reasonable way” the answer is consistently negative.

- Is it possible to create a volumetric object which has a circle, a square and an equilateral triangle as orthogonal profiles?
- My proof that a harmonic series diverges..
- Is there a way to solve for an unknown in a factorial?
- In how many ways can a number be expressed as a sum of consecutive numbers?
- If polynomials are almost surjective over a field, is the field algebraically closed?
- Finding a choice function without the choice axiom
- Quaternion distance
- Gradient steepest direction and normal to surface?
- Why is the Auslander Reiten theory not working in this example?
- Sum of two periodic functions is periodic?
- Prove the ring $a+b\sqrt{2}+c\sqrt{4}$ has inverse and is a field
- Viewing Homotopies as Paths in $\mathcal{C}^0(X,Y)$
- A matrix and its transpose have the same set of eigenvalues
- Why is conjugation by an odd permutation in $S_n$ not an inner automorphism on $A_n$?
- prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$