@DanielFischer i get what you saying but, you can't make question for an exam based on UB.. so they just gave simplifying assumptions so we can just work out how the stack would look like .
here we dont have much snow, once in 2-3 years , and for a few days
i also prefer the summer because i like the sea.
we defined $\omega$ to be the collection of all natural numbers* where a natural number* is an ordinal $\alpha$ s.t $\alpha = 0$ or $\alpha = \beta +1$ and each $\gamma \lt \alpha$ is also a natural number*
So we'd like to show $\bigcup \omega \subset \omega$, which amounts to showing $\alpha \in \omega \implies \alpha \subset \omega$, i.e. the transitivity of $\omega$, and then $\omega \subset \bigcup \omega$.
Modulo the small problem that the ordinals form a proper class, so we should use a different term. But I'm only looking at the set $\{ \alpha : \alpha \leqslant \omega\}$ of ordinals, and there saying "linear order" is totally legit.
im showing now by induction on $\beta$ that $(\alpha +\beta ,<) $ is iso. to $(\alpha \times \{0\} \cup \{1\} \times \beta , \lt_{lex} )$
and $(\alpha \beta , <)$ iso. to $(\beta \times \alpha , \lt_{lex})$
so $\beta =0$ is fine,assuming it holds for $\beta$ i show the addition is true for $\beta +1$. im not sure how to show that $\alpha (\beta +1) = \alpha \beta + \beta$ is iso. to $(\beta+1)\times \alpha$
in the addition case given $\sigma$ the iso of $\beta$ i defined $\sigma \cup \{<\alpha + \beta , <1,\beta > >\}$
You need some more. $\sigma_{\gamma_1}$ is an order-isomorphism between an initial segment of $\alpha + \gamma_2$ and an initial segment of $\{0\}\times \alpha \cup \{1\} \times \gamma_2$. If you know enough about order-isomorphisms between well-oredered sets, that does it. Otherwise, you have to prove it.
Apps aren't my thing. Nor are phones having an OS. I like my phones dumb. (Actually, I dislike them too, but I acknowledge that sometimes a telephone is useful, so it's something I'm prepared to put up with. To some extent.)
