$\aleph_0 \oplus \aleph_0 = \aleph_0 \otimes \aleph_0 = \aleph_0$, $\aleph_1 \oplus \aleph_0 = \aleph_1$ and $5 \oplus \aleph_0 = \aleph_0$, computed from absorption and, in the countable cases, independently from the published bijection $\omega \times \omega \approx \omega$

Work in ZF; no choice principle is used anywhere below. With $\oplus$ and $\otimes$ as in Cardinal sum $\kappa \oplus \lambda$, product $\kappa \otimes \lambda$ and exponentiation $\kappa^{\lambda}$, and why they are written apart from the ordinal operations and the alephs as in The successor cardinal $\kappa^{+}$, the alephs $\aleph_\alpha$, the beths $\beth_\alpha$, successor and limit cardinals, and the identifications $\aleph_0 = \omega$ and $\aleph_1 = \omega_1$:

$\aleph_0 \oplus \aleph_0 = \aleph_0, \qquad \aleph_0 \otimes \aleph_0 = \aleph_0, \qquad \aleph_1 \oplus \aleph_0 = \aleph_1, \qquad 5 \oplus \aleph_0 = \aleph_0 .$

Each is an instance of absorption (Absorption: for cardinals $\kappa, \lambda$ with $\kappa$ infinite and $\lambda \le \kappa$, $\kappa \oplus \lambda = \kappa$, and $\kappa \otimes \lambda = \kappa$ when $\lambda \ne 0$). The countable ones are also obtained a second way, from a bijection that was available before cardinal arithmetic existed: $\omega \times \omega \approx \omega$ ($\mathbb{N} \times \mathbb{N} \approx \mathbb{N}$), together with two inclusions and no further input.

What absorption replaces. Step 2.2 is what a reader would have had to do before Absorption: for cardinals $\kappa, \lambda$ with $\kappa$ infinite and $\lambda \le \kappa$, $\kappa \oplus \lambda = \kappa$, and $\kappa \otimes \lambda = \kappa$ when $\lambda \ne 0$ existed: produce a bijection or a pair of injections for each computation separately. Step 2.1 does all four in one line, and the content of the corollary is exactly that the bookkeeping is unnecessary.

Why $\aleph_1 \oplus \aleph_0 = \aleph_1$ has no second computation here. The countable cases are witnessed by explicit maps because $\omega$ is concrete. At $\aleph_1$ no comparable explicit bijection is written down here; the computation goes through Hessenberg: $\kappa \otimes \kappa = \kappa$ for every infinite cardinal $\kappa$, proved in ZF from the canonical well-order of $\kappa \times \kappa$, which Absorption: for cardinals $\kappa, \lambda$ with $\kappa$ infinite and $\lambda \le \kappa$, $\kappa \oplus \lambda = \kappa$, and $\kappa \otimes \lambda = \kappa$ when $\lambda \ne 0$ packages. That is not a gap in this example but the reason the general theorem is worth proving.

The finite summand does not vanish for a trivial reason. $5 \sqcup \omega$ does have more elements than $\omega$ in the naive sense: it carries a tagged copy of $\omega$ and five further points. The equality $5 \oplus \aleph_0 = \aleph_0$ says only that the two sets are equinumerous, and what makes that true is that an infinite well-ordered set absorbs finitely many extra points, the same shift that makes an infinite cardinal a limit ordinal in Every natural number and $\omega$ are cardinals, every infinite cardinal is a limit ordinal, and on the natural numbers the cardinal operations are the published finite counting operations, with $\lvert A \rvert$ in the finite sense equal to $\lvert A \rvert$ in the cardinal sense.

Order does not matter here, and that is not automatic. $\oplus$ is commutative, so $5 \oplus \aleph_0$ and $\aleph_0 \oplus 5$ are the same cardinal. The ordinal $+$ on the very same objects is not commutative, which is precisely why Cardinal sum $\kappa \oplus \lambda$, product $\kappa \otimes \lambda$ and exponentiation $\kappa^{\lambda}$, and why they are written apart from the ordinal operations gives the cardinal operation its own symbol rather than reusing $+$.