The set of all finite and countable ordinals is also an ordinal, called $\omega_1$, and is the first uncountable ordinal.
Thus, the smallest infinite cardinal is $\omega =\aleph_0$, the next one is $\omega_1=\aleph_1$, which is the first uncountable cardinal, then comes $\omega_2=\aleph_2$, etc.
The first uncountable cardinal is ℵ1, which is the first ordinal after ℵ0 not in 1 1 correspondence with any ordinal preceding it.
Further, the first uncountable ordinal ω1 is the cardinality of the universe, so the Continuum Hypothesis holds.
But the perfect set property for co-analytic sets implies that the first uncountable cardinal, $\aleph_1$, is a large cardinal in the constructible universe $L$ (see Section 7), namely a so-called inaccessible cardinal (see Section 10), which implies that one cannot prove in ZFC that every co-analytic set has the perfect set property.
We show that, for every scattered compact space K such that K((ω1) = Ø, where ω1 design the first uncountable ordinal, there exists on b(K) an equivalent norm such that the dual norm is LUR.
SKQ is the first kriging-based model able to take into account the three types of variability steming from quantitative factors, qualitative factors, and uncountable sources (random errors).
In these terms, the continuum hypothesis can be stated as follows: The cardinality of the continuum is the smallest uncountable cardinal number.
Then there's the nearly uncountable smaller cons he orchestrated against the people he was close to.
He was the inspiration for uncountable calendars, cards, T-shirts and mugs — not to mention best-selling books of cartoons.
