Let the signature $s$ be $s=\{E\}$ (a 'graph', kind of). Is it possible to create a $s$-structure that represents a infinitely long circle (I.E. a circle with an infinite number of nodes $(v_0,v_1,...,v_{\infty},v_0)$)?
My thoughts so far:
(1) Using a countable infinite universe:
Something like this $\mathcal C=(\mathbb N, E=\{ (u,v), (v,u)\;|\; u + 1 = v \})$. Although this is a line, not a circle. And I personally don't know how I can connect the two ends in infinity.
(2) Using an uncountable infinite universe: $\mathcal C = (\mathbb R, E=\{ (-1, 1), (1, -1) \}\cup\{ (u,v),(v,u)\;|\; u,v\in[-1,1], u+\varepsilon=v \})$. Although there are of course no neighbouring numbers inside a uncountable infinite range. So this does not work either.
Is it possbile to define a $s$-structure representing an infinite cirle? If not why?
(I have no Idea what tags I should use. I'm sorry if it's not the right one.)