Axiom-ness (Surface)
The intuitive, surface-level quality of being axiomatic. Expresses the property of a proposition as "self-evident" or "worthy of assertion" without formalization.
Axiom-ness · -ness suffix
Semantics
Semantic analysis of axiom-related terms. This page compares meaning domains and clarifies the distinction between surface-level (axiom-ness) and deep-level (axiomaticity) concepts.
Surface vs. Deep Semantics
Axiomaticity (Deep)
The degree to which a domain is formalized by axioms. A technical measure of how thoroughly a theory or system is axiomatized.
Axiomaticity · -ity suffix
Semantic Domains
Logic & Mathematics
Axioms as foundational propositions. Proof-theoretic and model-theoretic roles.
Philosophy
Axioms as self-evident truths. Epistemic and metaphysical aspects.
Natural Language
Axiomatic as "obvious" or "taken for granted." Colloquial usage.
Vocabulary for Understanding Axiom
Key terms and formal systems that provide context for understanding axioms. Each has an individual page with etymology and citations.
ZFCZermelo–Fraenkel set theory with the Axiom of Choice
The standard axiomatic foundation for mathematics. A first-order theory whose axioms formalize set membership and operations. The Axiom of Choice (AC) is one of its axioms.
ZKPZero-Knowledge Proof
A cryptographic protocol by which one party proves to another that a statement is true without revealing any information beyond the validity of the statement. Related to axiomatic reasoning in that both involve proving from assumptions.
FOLFirst-Order Logic
A formal logical system with quantifiers (∀, ∃) over individuals. The standard framework for axiomatizing mathematical theories (e.g., ZFC, Peano arithmetic).
Formal system
A system of axioms and inference rules. Axioms are the starting points; rules derive theorems. Examples: propositional logic, predicate logic, ZFC.
Inference rule
A rule that derives new propositions from existing ones. Modus ponens, universal instantiation, etc. Together with axioms, defines what is provable.
Model theory
Study of mathematical structures that satisfy given axioms. A model of a theory is a structure in which all axioms hold. Connects syntax (axioms) to semantics (interpretation).