Cardinality of a linear continuum

Question

A linear continuum is a totally ordered set with more than one element, which is both dense and satisfies the least upper bound property. Is it true that every linear continuum has the same cardinality as the reals?

Answer 1

No, there are larger linear continua. Let $\kappa$ be any infinite cardinal; then $X=\kappa\times[0,1)$ with the lexicographic order is a linear continuum. It’s easily seen to be densely ordered. If $A\subseteq X$ is non-empty and bounded above by some $\langle\alpha,x\rangle$, let

$$\beta=\min\left\{\gamma<\kappa:\big(\{\gamma\}\times[0,1)\big)\cap A=\varnothing\right\}\;.$$

If $\beta$ is a limit ordinal, then $\sup A=\langle\beta,0\rangle$. If $\beta=\gamma+1$ for some $\gamma<\kappa$, let $$S=\{y\in[0,1):\langle\gamma,y\rangle\in A\}\;.$$ If $\sup S=1$, then again $\sup A=\langle\beta,0\rangle$. Otherwise, $\sup A=\langle\gamma,\sup S\rangle$.