Theory (mathematical logic)

<p>In <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a>, a theory (also called a formal theory) is a set of <a href="/facts/Sentence_(mathematical_logic)/lnsLzvTx">sentences</a> in a <a href="/facts/Formal_language/crDTyP8q">formal language</a>. In most scenarios a <a href="/facts/Deductive_system/KaeDAjK7">deductive system</a> is first understood from context, giving rise to a <a href="/facts/Formal_system/KaeDAjK7">formal system</a> that combines the language with deduction rules. An element 
  
    
      
        ϕ
        ∈
        T
      
    
    {\displaystyle \phi \in T}
  
 of a <a href="/facts/Deductively_closed/mSIIFmlE">deductively closed</a> theory 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is then called a <a href="/facts/Theorem/f2Pe1FMD">theorem</a> of the theory. In many deductive systems there is usually a subset 
  
    
      
        Σ
        ⊆
        T
      
    
    {\displaystyle \Sigma \subseteq T}
  
 that is called "the set of <a href="/facts/Axiom/mLjP14Mc">axioms</a>" of the theory <a href="/facts/Axiom/mLjP14Mc">
  
    
      
        T
      
    
    {\displaystyle T}
  
</a>, in which case the deductive system is also called an "<a href="/facts/Axiomatic_system/dwMz70d5">axiomatic system</a>". By definition, every axiom is automatically a theorem. A first-order theory is a set of <a href="/facts/First-order_logic/lcISqWIg">first-order</a> sentences (theorems) <a href="/facts/Recursion/CxSKxJoW">recursively</a> obtained by the <a href="/facts/Rule_of_inference/QV5NklQ9">inference rules</a> of the system applied to the set of axioms.
</p>

Theory (mathematical logic) open-in-new

Theory (mathematical logic)