How many irrationals are there which are unique upto addition with rationals?

Question

I'm studying real analysis and I know from previous courses that there are countably infinite rationals but uncountably infinite irrationals. However, I haven't done a formal proof about uncountability of irrationals.

I've been thinking about which irrationals are "unique" (in terms of being able to be expressed in terms of other irrationals / rationals), and for this informal thought, I've come up with the following situation which I'll describe formally:

Suppose we define equivalence classes on irrationals, such that

$$[r] = \{ p + r : \forall p \in \mathbb{Q} \}$$

Or stated in other words,

$$x \in [y] \lor y \in [x] \rightarrow x - y \in \mathbb{Q}$$

How many such distinct equivalent classes would exist? Countably infinite? uncountably infinite?

Answer 1

Since the equivalence relation you are considering on $\mathbb{R}$ -- a very classical one, as users have pointed out above -- is none other than the congruence modulo the additive subgroup $\mathbb{Q}$, the cardinality of the quotient set $\mathbb{R}/\mathbb{Q}$ is by definition the index $(\mathbb{R}\colon\mathbb{Q})$ and -- as elementary group theory teaches us -- must therefore satisfy the cardinal relation: $$|\mathbb{R}|=(\mathbb{R}\colon\mathbb{Q})|\mathbb{Q}| \tag{*}.$$

On the right hand side we behold a product of two cardinals, at least one of which is infinite, namely $|\mathbb{Q}|=\aleph_0$. Elementary cardinal theory then tells us that the product of the two will be equal to the larger of the two (by reasons of multiplicative absorption of infinite cardinals). Were this larger cardinal to be equal to $\aleph_0$, we would reach the conclusion that $|\mathbb{R}|=\aleph_0$. On the other hand, any decent course of analysis should have taught us in its opening classes focused on the elementary construction of $\mathbb{R}$ that the cardinal relation $|\mathbb{R}|=2^{\aleph_0}$ holds.

Since by virtue of one of Cantor's famous theorems $2^{\mathbf{x}}>\mathbf{x}$ is valid for any cardinal $\mathbf{x}$, the above made assumption leads to a contradiction. Therefore, the larger of the two cardinals in question on the right hand side of relation $(^{*})$ above must be the index $\left(\mathbb{R} \colon \mathbb{Q}\right)$ and we can finally conclude that $|\mathbb{R}|=\left(\mathbb{R} \colon \mathbb{Q}\right)$.

So not only is this cardinality of equivalence classes uncountably infinite, we actually know which uncountable infinity it is: the power of the continuum.

Answer 2

Every class is of the form $[x]_{\sim}=x+\mathbb{Q}$ where $x$ is irrational, so $[x]$ is countable. Since $\sim$ is an equivalence relation, $\mathbb{R}/_{\sim}$ is a decomposition into disjoint sets, therefore there must exist uncountably many equivalence classes. Your construction is not new and it is sometimes used to construct counterexamples in topology/set theory. There is for example the Dowker's Example in topological dimension theory.

Answer 3

"However, I haven't done a formal proof about uncountability of irrationals."

(Rudin PMA) Theorem 2.14. Let $A$ be the set of all sequences whose elements are the digits $0$ and $1$. This set $A$ is uncountable.

The elements of $A$ are sequences like $1, 0, 0, 1, 0, 1, 1, 1, ... $

You can look the proof up by finding the book for free with a google search.

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Define $f:A \to \mathbb{R}$ by for example $f( (1,0,1,0,0,1,1,1,0,0,1,...)) \to 0.10100111001...$, so $f$ takes any sequence of $A$, concatenates the sequence and places it after a "$0.$". Then $f$ is well-defined and injective, so $f(A)$ is therefore uncountable.

Therefore $\mathbb{R} \supset A$ is uncountable.

Also, the union of two countable sets is countable, (e.g. by Rudin's PMA Theorem 2.12).

We also know that the rational numbers $\mathbb{Q}$ are countable.

Therefore, if the irrational numbers $\mathbb{R} \setminus \mathbb{Q}$ were countable, then $\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$ would be countable, contradicting the fact that $\mathbb{R}$ is uncountable.

Therefore $\mathbb{R} \setminus \mathbb{Q}$ is uncountable.