Integration described by first-order logic?

  • Context: Graduate 
  • Thread starter Thread starter schlynn
  • Start date Start date
  • Tags Tags
    Integration Logic
Click For Summary
SUMMARY

The discussion centers on the feasibility of describing a Riemann integral using first-order logic (FOL). Participants assert that any adequate description necessitates qualifiers such as "∀A ⊆ X," which cannot be expressed in FOL. The inability to define integration without existential quantifiers or infinite disjunctions further supports the conclusion that first-order logic is insufficient for this purpose. The conversation highlights the limitations of FOL in formal logic, particularly in the context of mathematical integration.

PREREQUISITES
  • Understanding of first-order logic (FOL)
  • Familiarity with Riemann integration concepts
  • Basic knowledge of model theory
  • Experience with quantifiers in formal logic
NEXT STEPS
  • Research the limitations of first-order logic in mathematical descriptions
  • Explore second-order logic and its applications in integration
  • Study the formal definitions of Riemann integrals
  • Investigate alternative logical frameworks for mathematical expressions
USEFUL FOR

Mathematicians, logicians, and students of formal logic seeking to understand the relationship between first-order logic and mathematical concepts like integration.

schlynn
Messages
88
Reaction score
0
Is it possible, to describe a Riemann integral with just first-order logic? And if so could someone point me to somewhere that has such a description of it.
 
Physics news on Phys.org
As far as I know any description of integration requires some qualifier of the form "##\forall A \subset X##" which is not expressible in first order logic. But I'd be happy to be proven wrong!
 
I don't know much formal logic beyond my intro model theory class, but...

In my attempt to define it, I couldn't do it without being able to say [tex]``\exists n\in\mathbb N:\exists x_1,...\exists x_n:\enspace...\text{''}[/tex] or [tex]``\bigvee_{n=1}^\infty \exists x_1,...\exists x_n:\enspace...\text{''}[/tex] which isn't kosher in any version of FOL I know.
 

Similar threads

Undergrad About the undecidability of first-order-logic
  • · Replies 26 ·
Replies
26
Views
4K
Graduate Second-order arithmetic
  • · Replies 34 ·
2
Replies
34
Views
4K
Graduate Second order logic and completeness
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Why the emphasis on first-order logic?
  • · Replies 9 ·
Replies
9
Views
2K
Graduate What are your insights on The Hardest Logic Puzzle Ever?
  • · Replies 2 ·
Replies
2
Views
3K
High School Traditional logic and its usefulness in the past
  • · Replies 21 ·
Replies
21
Views
3K
MHB Infinite Natural Numbers: First-Order Logic Formula Explained
  • · Replies 2 ·
Replies
2
Views
2K
MHB Attempting to add logic mathematics to a college paper - need your input
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Undecidability and multivalent logic
  • · Replies 3 ·
Replies
3
Views
2K
High School What is the usefulness of formal logic theory?
  • · Replies 1 ·
Replies
1
Views
3K