Tagged Questions

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Why are irrational numbers uncountable and rationals contable?

Question 1: Why are irrational numbers uncountable and rationals contable? I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers ...
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Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non-trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as ...
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On the Continuum Hypothesis

Let me start out by saying that I am not a mathematician. I read an article over at Scientific American that discussed the Continuum Hypothesis. I developed the following thought experiment that would ...
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Is there an absolute notion of the infinite?

Skolem's paradox has been explained by the proposition that the notion of countability is not absolute in first-order logic. Intuitively, that makes sense to me, as a smaller model of ZFC might not be ...
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Book/article/tutorial as an introduction to Cardinality

I study CS, but on the first semester I have a lot of mathematics. Of course, there is an introduction to set theory and logic. Recently, we had lectures about cardinality, different kinds of ...
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Cardinality of a set that consists of all existing cardinalities

I have taken a look at the following topics: number of infinite sets with different cardinalities Cardinality of all cardinalities Are there uncountably infinite orders of infinity? Types of ...
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Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. ...