> Theorem 1 (Zarach). It is consistent with ZFC- that ω1 exists but is singular and hence that a countable union of countable sets can be uncountable.
Proof. Consider the model W as constructed in the L ́evy collapse of אω above. Note that every cardinal אVn is collapsed in V [Gn ] and hence in W , but אVω remains a cardinal in every V [Gn ] and hence also in W . Thus, ω1W = אVω , which has cofinality ωinV andhenceinW,aswitnessedbythesequence אVn |n<ω ∈V ⊆W. So W satisfies that ω1 is singular. In particular, W satisfies that ω1 is a countable union of countable sets, as ω1W = {אVn | n ∈ ω}.

Afternoon nap dream: Reading a wikibook with a dark grey background. One of the pages talks about generalising to infinity and there's a paragraph saying something along the lines of there is no way to get more elements unless (forgot), followed by a set builder notation
$$\{f |f(k) \mapsto f(forgot)|(forgot) \mapsto (forgot) | (forgot)\}$$

The article then ends with "But what if there exists something that can compute every computing matter, that consciousness is (forgot) real"

I then click the link to the next page, which talks about something titled "Conscious Fir" and the 1st paragrap…

Ok, so that means, choose a coordinate frame, you have some mass with initial position and velocities $(\mathbf{x}_i,\mathbf{v})$ and your hole has some position vector $\mathbf{x}_f$. There is a maximum to the force applied which means the magnitude of the acceleration vector has a maximum, and the angle of hitting determines the relative proportions of y and x components of the acceleration. So in other words, you want to solve for $a$ the following vector equation:

$$\mathbf{x}_f - \mathbf{x}_i=\int_0^1 \mathbf{v} + \mathbf{a}t dt$$

Basically, the blue vector is the displacement of the ball's current position to the hole's centre, the red vector is the initial velocity, the green vectors are the possible directions of impulses, and the dark red vectors are the final velocities after the kick. The green vectors are bounded by a dotted circle which means all the possible directions with maximum impulse is given by that circle. This particular scenario is produced so that the ball will never get into the hole in a single kick