It has a countably infinite set of ideal classes.

There exists a surjective map from A onto a countably infinite set.

For a countably infinite set, the set of possible order types is even uncountable.

Any countably infinite set can be indexed by .

A random variable is discrete if there are finitely many (or a countably infinite set of) possible values.

The case of a countably infinite set of components is covered formally by allowing .

In particular, the power set of a countably infinite set is uncountably infinite.

A countable set is either a finite set or a countably infinite set.

A countably infinite set of possible strings end in infinite repetitions, which means the corresponding real number is rational.

What about infinite subsets of countably infinite sets?

It is the cardinality (size) of the set of numbers of possible arrangements for all countably infinite sets.

The groups A and S are the only non-identity proper normal subgroups of the symmetric group on a countably infinite set.

Every Hamel basis of this space is much bigger than this merely countably infinite set of functions.

Well-formed formulas are built freely using these connectives from a countably infinite set of propositional variables p.

If A is a countably infinite set, the corresponding Cantor cube is a Cantor space.

Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example).

Examples of countably infinite sets are the natural numbers, the even numbers, the prime numbers, and also all the rational numbers, i.e., the fractions.

The set of axioms may be empty, a nonempty finite set, a countably infinite set, or be given by axiom schemata.

A countably infinite set endowed with the cofinite topology is locally connected (indeed, hyperconnected) but not locally path connected.

In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a countably infinite set of axioms.

Let P (propositions) and A (actions) be two finite sets of symbols, and let V be a countably infinite set of variables.

However, some countably infinite sets of nonempty sets can be proven to have a choice function in ZF without any form of the axiom of choice.

The observed data points may be discrete (taking values in a finite or countably infinite set) or continuous (taking values in an uncountably infinite set).

The second sense is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer.

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms.