Are The Specifiable Real Numbers Countable

I'll assume this definition of specifiable:

A real number is specifiable if it can be written in a finite sequence of symbols from a finite alphabet.

If that's the definition you're thinking of, then the answer to your question is yes, since the set of finite sequences of symbols from a finite alphabet is countable.

That is not necessarily true for an infinite sequence. See GeorgCantor's diagonalization argument (CantorsProof).

Note that the specifiable real numbers are not a subset of the rational numbers. For example, the definition

x = square root (2)

uniquely specifies an irrational number using a finite sequence of symbols from a finite alphabet.

See AreTheSpecifiableRealsWellDefined, though - it's not clear the above is a coherent definition. In fact, it depends on whether the "language" in which the numbers are specified is well-defined - i.e. whether each sequence of symbols maps unambiguously to a real number.

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There are numerous interesting (and countable) subsets of the reals (which of course are not countable).

Among them:

• The integers
• The rationals
• The "constructable" numbers -- expressible as a finite sequence of addition, subtraction, multiplication, division, and Pythagorean sum (sqrt (a*2 + b*2)) -- essentially, those numbers whose magnitude can be computed with compass and straightedge (see SquaringTheCircle?)
• The "simple algebraic" numbers -- expressible as a finite sequence of addition, subtraction, multiplication, division, and integer root extraction (square root, cube root, etc.). All roots of polynomials of degree 4 or lower are in this category; higher-order polynomials may have roots which are not so expressible.
• The algebraic numbers.

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CategoryMath