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