i have a theoretical question: if a set of phrases $\sum$ has the following property foreach $a,b \in \sum$ $a \Rightarrow b$ or $b \Rightarrow a$. can we build an infinite set $\sum$ so that it follows the above properties, but that for each $a \in \sum$ exists $b \in \sum$ so that $a \not \Rightarrow b$?
wondering about that, can such a thing exist?
thank you very much
For all r in R, let p(r) be r <= x
and $\Sigma$ = { p(r) : r in R }.
If r <= s then p(s) implies p(r),
If s <= r then p(r) implies p(s),
Thus (p(s) implies p(r)) or (p(r) implies p(s)).
However, p(r+1) does not follow from p(r).
If you want a denumerable collection, use Z instead of R.