## FANDOM

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We know that $$\omega$$ and $$\varepsilon_0$$ and $$\Gamma_0$$ are coutable. In this blog, I will provide some actual injections from $$\mathbb{N}_0$$ to these ordinals. These formulas aren't particularly groundbreaking, but they're a fun diversion.

I will use $$n\gamma$$ as a shorthand for $$\gamma \times n$$ throughout this blog.

### Proof that $$|\omega| = \aleph_0$$

The mapping is just $$a \mapsto a$$. Nothing to do here.

### Proof that $$|2\omega| = \aleph_0$$

Every ordinal here is of the form $$a$$ or $$\omega + a$$. The mapping is $$a + 2b \mapsto a\omega + b$$ where $$a$$ is 0 or 1.

### Proof that $$|\omega^2| = \aleph_0$$

An ordinal less than $$\omega^2$$ is of the form $$a\omega + b$$. Thus $$2^b3^a$$ is a valid notation for all ordinals less up to $$\omega^2$$. (The mapping I've made is only a partial function, but with infinite sets, partial surjections are equivalent to bijections.)

### Proof that $$|\omega^\omega| = \aleph_0$$

All the ordinals we're looking for are polynomials in $$\omega$$. There exists an injection from polynomials with coefficients in $$\mathbb{N}_0$$ to prime factorizations of nonnegative integers:

$2^{a_0}3^{a_1}5^{a_2} \ldots p_{n + 1}^{a_n} \mapsto a_n\omega^n + \cdots + a_1\omega + a_0$

where $$p_n$$ denotes the $$n$$th prime.