FoL = First Order Logic

2. We want to find an example of a FoL expression that has only infinite models. We pick a predicate symbol L(x, y), which will eventually receive the meaning "r y (but in logical expressions we only use L(x,y). We want to define in FoL that L(z,y) is a total order without mazimum. (Again: In 2.1 through 2.5 you must not use ", since we work with the syntax here.) 2.1 Define reflerivity of L by a FoL- expression. 2.2 Define anti-symmetry of L by a FoL expression. 2.3 Define transitivity of L by a FoL expression 2.4 Define totality of L by a FoL expression. 2.5 Give a FoL-expression that says that the total order relation L has no maximal object. 2.6 For any natural integer n 0, let (Sn, be any total order structure, where Sn is a set of cardinality n. Show by induction on n that (Sn, has a maximum element; i.e.. (Sn, S) does not satisfy the expression found in 2.5 2.7 Let E be the conjunction of the expressions in 2.1 through 2.5. Use 2.6 to show that E has no finite model 2. We want to find an example of a FoL expression that has only infinite models. We pick a predicate symbol L(x, y), which will eventually receive the meaning "r y (but in logical expressions we only use L(x,y). We want to define in FoL that L(z,y) is a total order without mazimum. (Again: In 2.1 through 2.5 you must not use ", since we work with the syntax here.) 2.1 Define reflerivity of L by a FoL- expression. 2.2 Define anti-symmetry of L by a FoL expression. 2.3 Define transitivity of L by a FoL expression 2.4 Define totality of L by a FoL expression. 2.5 Give a FoL-expression that says that the total order relation L has no maximal object. 2.6 For any natural integer n 0, let (Sn, be any total order structure, where Sn is a set of cardinality n. Show by induction on n that (Sn, has a maximum element; i.e.. (Sn, S) does not satisfy the expression found in 2.5 2.7 Let E be the conjunction of the expressions in 2.1 through 2.5. Use 2.6 to show that E has no finite model