Undergrad About the existence of Hamel basis for vector spaces

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A vector space always admits an algebraic (Hamel) basis, a theorem reliant on the Axiom of Choice (AC). Without AC, it is possible to construct vector spaces, such as ##\mathbb R_\mathbb Q##, that do not have a Hamel basis, although this does not imply that such spaces lack a basis entirely. The existence of a Hamel basis is equivalent to the truth of the Axiom of Choice, meaning that if AC is false, the statement regarding the existence of a basis is also false. The discussion highlights the complexities surrounding the existence of bases in infinite-dimensional spaces and the implications of set theory axioms on these mathematical truths. Ultimately, the relationship between the Axiom of Choice and Hamel bases remains a nuanced topic in the realm of vector spaces.
  • #31
A well-formed formula (wff) is (syntactically) provable if and only if it is valid. This defines the relation between (Semantic) truth and (Syntactic) provability. I'm not aware of how the term 'hold' is used, by there are wff's in Sentence Logic that are contingent. And, yes, every provable statement in FOL os Sentence Logic is a Tautology; true in every model.
 
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  • #32
WWGD said:
A well-formed formula (wff) is (syntactically) provable if and only if it is valid. This defines the relation between (Semantic) truth and (Syntactic) provability. I'm not aware of how the term 'hold' is used, by there are wff's in Sentence Logic that are contingent. And, yes, every provable statement in FOL or Sentence Logic is a Tautology; true in every model.
Ok, a well-formed formula (wff) is (syntactically) provable if and only if it is valid since we're assuming a sound and (semantically) complete logic system (like FOL or Sentential (or propositional) logic are).

What does it mean that in Sentential logic there are contingent wffs ?
 
  • #33
cianfa72 said:
Ok, a well-formed formula (wff) is (syntactically) provable if and only if it is valid since we're assuming a sound and (semantically) complete logic system (like FOL or Sentential (or propositional) logic are).

What does it mean that in Sentential logic there are contingent wffs ?
Statements that aren't tautologies, like ##A \rightarrow B ##, which is not true when A is true and B is false.
 

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