First-order logic without sets?

Thread starter jordi
Start date Mar 7, 2009
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Logic Sets

In summary, In Stefan Bilaniuk's book "A Problem Course in Mathematical Logic", he states that when reading set theory books, they mostly resort to the concept of sets without defining them. However, he found a problem when trying to use 1st order logic as the foundation for studying set theory--all books he has checked up to this point resort to sets without defining them. He also found a comment in a pdf stating that "setting up propositional or first-order logic formally requires that we have some set theory in hand." This suggests that first order logic cannot be used without set theory.

Mar 23, 2009

Hurkyl

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Science Advisor

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jordi said:

False. Proof: let (blank) := set

"X is a set" is meaningful. "X belongs to the set of all sets" is not meaningful, because the set of all sets does not exist.

More detail please. In this thread, we have used the word "set" in at least three different senses:
(1) A set in some model of ZFC
(2) A set in some model of ZFC
(3) A synonym for "logical predicate"

Furthermore, the corresponding logical predicate would be

(blank) is a predicate

which, incidentally, is also meaningless.

(Unless, of course, you meant it as a metalogical predicate, or as a predicate about strings -- but then that wouldn't have anything to do with the "logic = set theory" correspondence)

And, for the record, there are forms of set theory that do have a set of all sets.

<h2>1. What is first-order logic without sets?</h2><p>First-order logic without sets is a formal system of mathematical logic that allows for the representation and manipulation of mathematical statements without the use of sets. It is based on the principles of first-order logic, which allows for the quantification of variables and the use of logical connectives, but does not include the concept of sets as a foundational element.</p><h2>2. How does first-order logic without sets differ from traditional first-order logic?</h2><p>First-order logic without sets differs from traditional first-order logic in that it does not include the use of sets as a foundational element. This means that statements and proofs in first-order logic without sets do not rely on the concept of sets, and instead focus on the quantification of variables and the use of logical connectives.</p><h2>3. What are the advantages of using first-order logic without sets?</h2><p>One advantage of using first-order logic without sets is that it allows for a more streamlined and concise representation of mathematical statements and proofs. It also eliminates the need for dealing with the complexities of set theory, which can be difficult for some mathematicians to grasp.</p><h2>4. Are there any limitations to using first-order logic without sets?</h2><p>One limitation of first-order logic without sets is that it cannot fully capture the complexity of some mathematical concepts that rely heavily on the use of sets, such as topology or category theory. Additionally, it may not be suitable for certain applications that heavily rely on set theory, such as database systems.</p><h2>5. How is first-order logic without sets used in practice?</h2><p>First-order logic without sets is commonly used in mathematical and philosophical research, particularly in areas that do not require heavy use of set theory. It is also used in computer science, particularly in the development of formal verification methods for software and hardware systems.</p>

Related to First-order logic without sets?

1. What is first-order logic without sets?

First-order logic without sets is a formal system of mathematical logic that allows for the representation and manipulation of mathematical statements without the use of sets. It is based on the principles of first-order logic, which allows for the quantification of variables and the use of logical connectives, but does not include the concept of sets as a foundational element.

2. How does first-order logic without sets differ from traditional first-order logic?

First-order logic without sets differs from traditional first-order logic in that it does not include the use of sets as a foundational element. This means that statements and proofs in first-order logic without sets do not rely on the concept of sets, and instead focus on the quantification of variables and the use of logical connectives.

3. What are the advantages of using first-order logic without sets?

One advantage of using first-order logic without sets is that it allows for a more streamlined and concise representation of mathematical statements and proofs. It also eliminates the need for dealing with the complexities of set theory, which can be difficult for some mathematicians to grasp.

4. Are there any limitations to using first-order logic without sets?

One limitation of first-order logic without sets is that it cannot fully capture the complexity of some mathematical concepts that rely heavily on the use of sets, such as topology or category theory. Additionally, it may not be suitable for certain applications that heavily rely on set theory, such as database systems.

5. How is first-order logic without sets used in practice?

First-order logic without sets is commonly used in mathematical and philosophical research, particularly in areas that do not require heavy use of set theory. It is also used in computer science, particularly in the development of formal verification methods for software and hardware systems.