Non-standard model

Definition

A non-standard model is a structure that satisfies the axioms of a given theory but differs from the standard or intended interpretation of that theory. These models often introduce elements that are not typically present in standard interpretations, allowing for a broader understanding of the axioms and the relationships between them. They play a significant role in demonstrating that certain properties or truths can vary significantly based on the chosen model, illustrating the diversity in interpretations within first-order logic.

5 Must Know Facts For Your Next Test

1. Non-standard models can contain 'non-standard' elements, such as infinitesimals or other unusual objects, which are not found in standard interpretations.
2. These models are crucial in demonstrating the independence of certain properties from specific theories, revealing how various models can still satisfy the same axioms.
3. In first-order logic, non-standard models can exist for many theories, particularly those concerning natural numbers and real numbers, leading to unexpected conclusions.
4. Non-standard models help highlight the limitations of first-order logic by showing that some truths may hold in one model but not in another.
5. The existence of non-standard models raises important questions about the nature of mathematical truth and interpretation, prompting further exploration into how axioms relate to different structures.

Review Questions

• How do non-standard models illustrate the differences in interpretations of axioms within first-order logic?
  • Non-standard models showcase how various interpretations can satisfy the same set of axioms while incorporating different elements that deviate from standard expectations. For example, in arithmetic, a non-standard model might include infinitesimal quantities that do not appear in the standard model of natural numbers. This divergence illustrates that the same logical framework can support multiple understandings, revealing complexities and nuances in mathematical logic.
• Discuss the implications of non-standard models on our understanding of completeness and consistency within theories.
  • Non-standard models challenge traditional notions of completeness and consistency by demonstrating that some axioms can be satisfied in unexpected ways. For instance, while a theory might be complete in its standard model, non-standard models can show that there are other interpretations where certain statements may not hold. This reveals potential gaps in our understanding of what it means for a theory to be consistent and complete since non-standard interpretations may lead to different conclusions about truth and provability.
• Evaluate how non-standard models contribute to ongoing discussions about mathematical truth and interpretation in formal logic.
  • Non-standard models significantly influence discussions about mathematical truth by suggesting that what is considered 'true' can vary based on the chosen model. This variability prompts deeper inquiry into how we define truth within formal systems and challenges the idea of a singular, absolute interpretation. As these models reveal new structures and relationships within logical frameworks, they encourage a broader view of mathematics as an interpretative discipline rather than merely a collection of objective truths, fostering an environment where diverse perspectives are valued.

Related terms

standard model: A standard model is the typical interpretation of a theory in which all axioms are satisfied without introducing any unusual or unexpected elements.

model theory: Model theory is a branch of mathematical logic that deals with the relationship between formal languages and their interpretations, or models.

completeness theorem: The completeness theorem states that if a formula is true in every model of a theory, then there is a proof of that formula from the axioms of the theory.