First-Order Logic: Finite & Infinite Domains

In summary, the conversation is discussing a first-order language L containing a unary function symbol f and no other non-logical symbols. The first sentence χ is asked to be written, which should be satisfiable in a structure with an infinite domain but false in every structure with a finite domain. The second sentence ρ is also asked to be written, such that if A satisfies ρ and A is finite, then A contains an even number of elements and any finite set with an even number of elements is the domain of some model of ρ. The question also asks for the size of the domains of models of ¬ρ. The person asking for help is unsure of how to approach this problem and is seeking hints.
  • #1
kazuyak
4
0
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!
 
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  • #2
Does your definition of a first order language include the symbol "=", and is this to be interpreted as the identity in the models (i.e. are they "normal" models)?
 

Related to First-Order Logic: Finite & Infinite Domains

What is First-Order Logic?

First-Order Logic (FOL) is a formal logic system used in mathematics, computer science, and philosophy to represent and reason about statements involving quantifiers, predicates, and variables. It allows for precise and unambiguous statements about the world.

What are Finite and Infinite Domains in First-Order Logic?

Finite and infinite domains refer to the set of objects or individuals that are being quantified over in a logical statement. A finite domain has a limited number of elements, while an infinite domain has an unbounded or potentially infinite number of elements.

What is the difference between First-Order Logic and Second-Order Logic?

The main difference between First-Order Logic and Second-Order Logic is that the latter allows quantification over sets or collections of objects, while the former only allows quantification over individual objects. This makes Second-Order Logic more expressive and able to capture more complex statements.

How is First-Order Logic used in computer science?

In computer science, First-Order Logic is used as a foundation for formal methods, which are techniques for specifying and verifying the correctness of computer systems. It is also used in automated reasoning and artificial intelligence, where logical inference is used to solve problems and make decisions.

What are the limitations of First-Order Logic?

While First-Order Logic is a powerful and widely used formal logic system, it has its limitations. It cannot handle self-referential statements or statements about its own truthfulness, and it does not have the ability to reason about uncertainty or probability. Additionally, it may struggle with expressing some concepts in natural language that are intuitive to humans but difficult to formalize in logical terms.

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