first order language in logic

Apr 2010
7
1
Let L = {f } be a first-order language containing a unary function
symbol f , and no other non-logical symbols.
1.Write down a sentence χ of L which is satisfiable in some structure
with an infinite domain but is false in every structure with a finite domain.
What can you say about the size of the domains of the models of the sentence
2.Write down a sentence ρ such that whenever A |= ρ and A is finite,
then A contains an even number of elements and, further, every finite set
with an even number of elements is the domain of some model of ρ. What
can you say about the size of the domains of the models of the sentence ¬ρ?

Could anyone please give me some hints how to deal with this problem? I'm not sure where to start. Any help is appreciated!
 

emakarov

MHF Hall of Honor
Oct 2009
5,577
2,017
Let L = {f } be a first-order language containing a unary function
symbol f , and no other non-logical symbols.
1.Write down a sentence χ of L which is satisfiable in some structure
with an infinite domain but is false in every structure with a finite domain.
What can you say about the size of the domains of the models of the sentence
You probably have equality symbol in the language as well. (This differs from one definition to the next.)

Describe f as a successor function in arithmetic. For 0 use the element that that does not have a predecessor (i.e., which is not a successor of anything). Then say that f is injective.