Left Compatible with Ordinal Addition (Multiplication) for alephs

Question

I have a question about strictly increasing properties of addition and multiplication on Ord. Do they work for alephs?

My attempt:

• Subset is Left Compatible with Ordinal Addition:

$\forall \alpha, \beta, \gamma \in Ord: \beta \prec \gamma \Rightarrow \alpha + \beta \prec \alpha + \gamma$

• Subset is Left Compatible with Ordinal Multiplication:

$\forall \alpha, \beta, \gamma \in Ord: [(\beta \prec \gamma \wedge 0 \prec \alpha) \Rightarrow \alpha \cdot \beta \prec \alpha \cdot \gamma]$ ;

• For alephs, the sum and product operations are trivial:

$\forall \alpha, \beta \in Ord: \aleph_{\alpha} + \aleph_{\beta} = \aleph_{\alpha} \cdot \aleph_{\beta} = \max \{\aleph_{\alpha}, \aleph_{\beta} \}$

My questions:

1. $0 \prec \aleph_{\alpha} \Rightarrow \aleph_{\alpha} + 0 \prec \aleph_{\alpha} + \aleph_{\alpha} = \aleph_{\alpha} \Rightarrow \aleph_{\alpha} \prec \aleph_{\alpha}$ ?
2. $1 \prec \aleph_{\alpha} \Rightarrow \aleph_{\alpha} \cdot 1 \prec \aleph_{\alpha} \cdot \aleph_{\alpha} = \aleph_{\alpha} \Rightarrow \aleph_{\alpha} \prec \aleph_{\alpha}$ ?

Are there something wrong? I hope your helps.

Answer 1

Cardinal addition and multiplication are not cancellative: $\kappa<\lambda$ does not imply $\mu+\kappa<\mu+\lambda$ or $\mu\cdot\kappa<\mu\cdot\lambda$ (note that cardinal addition and multiplication is commutative so we don't have to specify that we're adding on the left). Your examples demonstrate this.

Note that we see this already in the realm of the natural numbers, once we replace $+$ and $\cdot$ each with $\max$ (which after all is how they behave at infinite cardinals).