This time I've task with minimal sets.
"We say that a structure is minimal if is infinite but the only subsets of which are first-order definable with parameters are either finite or cofinite (i.e. complements of finite sets). More generally a set which is first-order definable with parameters is said to be minimal if is infinite, and for every set which is first-order definable in with parameters, either or is finite. "
And now the exercise:
== Let be the graph whose vertices are all the sets of exactly two natural numbers, with joined to iff . Show that is not minimal, but it has infinitely many minimal subsets. ==
I'll be grateful if this problem someone can show me step by step. When I understand this exercise I'll do the rest from my list. Because of this I'm asking for details. Thanks.