# Thread: first order language in logic

1. ## first order language in logic

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!

2. 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.