It is said Compactness fails in S.O , but holds in F.O.
So consider describing a well-order in N.
"transitive", reflective, symmetric canbe described in F.O.
and adding others to make a well order
if we add \forall x, \exists z x<z
I think we can make an infinite structure with an order "<", implying no largest element.
What I would like to say is that F.O can also describe some kind of infinity.
we can than apply Compact. Theorem,
to this set :{
\phi1 = exists x, y, x!=y
\phi2 = exists x,y,z ,x !=y, y!=z, z!=x,
...
a formular describe the well order
a formular says no largest element
}
This will lead to contradiction by the similar tech. we use in SO too.
So what is wrong here
thanks