Set of provably true statements

Definition

A set of provably true statements refers to a collection of assertions within a formal system that can be demonstrated as true using the axioms and rules of inference of that system. This concept is crucial in understanding the limits of what can be proven within a given logical framework and connects closely with ideas like consistency, completeness, and decidability in recursive function theory.

5 Must Know Facts For Your Next Test

1. A set of provably true statements is often used to illustrate the differences between what can be known versus what can be computed or proven in mathematics and logic.
2. This set is not necessarily finite; it can be infinite, depending on the axioms and rules used in the logical system.
3. Not all true statements are provably true; some may be true but cannot be derived from the existing axioms and rules of inference.
4. In recursive function theory, understanding sets of provably true statements helps to establish boundaries around computability and decidability.
5. The relationship between provably true statements and recursive functions helps to clarify the structure and limitations of mathematical systems.

Review Questions

• How does the concept of a set of provably true statements relate to the idea of completeness in a logical system?
  • The concept of a set of provably true statements directly ties into the idea of completeness, which states that every statement that is true in a logical system can also be proven within that system. If a logical system is complete, then its set of provably true statements will encompass all truths expressible within that framework. However, Gödel's Incompleteness Theorems show that for many systems, particularly those strong enough to encapsulate arithmetic, there are always truths that cannot be proven, highlighting the limitations inherent in our formal systems.
• Discuss how Gödel's Incompleteness Theorems challenge our understanding of provably true statements within mathematical systems.
  • Gödel's Incompleteness Theorems reveal profound limitations regarding provably true statements in formal mathematical systems. They demonstrate that there are always true propositions about natural numbers that cannot be derived from the axioms of the system. This challenges our understanding by showing that no single system can capture all mathematical truths, suggesting that while we may have a set of provably true statements, there are always more truths lying outside our ability to prove them within any given formal structure.
• Evaluate the implications of recursively enumerable sets on the classification of sets of provably true statements.
  • Recursively enumerable sets play a significant role in classifying sets of provably true statements by illustrating the boundaries between what can be computed versus what can be proven. A recursively enumerable set consists of elements for which there is a procedure to list them, yet it may not allow for decision procedures regarding membership. This highlights a critical distinction where some truths about numbers may be recognizable through computation but remain unprovable within a given set of axioms. This relationship emphasizes both the power and limitations of formal systems in representing all mathematical truths.

Related terms

Axioms: Basic statements or propositions assumed to be true without proof, serving as the foundational building blocks for a logical system.

Gödel's Incompleteness Theorems:

A pair of results demonstrating that within any sufficiently powerful logical system, there exist true statements that cannot be proven within that system.

Recursively Enumerable Sets:

A class of sets for which there exists a Turing machine that will list all the members, meaning every member can be generated by some computation, even if not all members can be proven true.