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The Axiom of Choice
Can we choose an element from each set in a family of nonempty sets? Does every set have a well order? Is the axiom of choice arrogant overreach or does…
Jul 21
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Joel David Hamkins
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The Axiom of Choice
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Least upper bound principle and continuous induction
The least-upper-bound principle The subject of real analysis can be founded upon the least-upper-bound principle, a version of Dedekind completeness…
Jul 13
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Joel David Hamkins
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Least upper bound principle and continuous induction
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Quantifier elimination — Presburger arithmetic
The elementary theory of addition in the natural numbers admits a certain logical triviality—every assertion is equivalent to a simple combination of…
Jul 6
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Quantifier elimination — Presburger arithmetic
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June 2024
Quantifier elimination — theory of successor
Every assertion about the structure ⟨ℕ,S,0⟩ of the natural-number successor is trivial, being equivalent to a quantifier-free assertion in this…
Jun 29
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Quantifier elimination — theory of successor
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Continuity, instantaneous change, infinitesimals
How we faced the ghosts of departed quantities
Jun 22
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Joel David Hamkins
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Continuity, instantaneous change, infinitesimals
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Quantifier elimination — dense linear orders
In certain special theories, every assertion is equivalent to a trivial combination of atomic facts. Proving this for a given theory typically both…
Jun 15
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Joel David Hamkins
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Quantifier elimination — dense linear orders
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Equality as logical
Do we take the identity relation x = y as logically primitive? Must we?
Jun 8
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Joel David Hamkins
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Equality as logical
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Are the imaginary numbers real?
What are the complex numbers, really?
Jun 1
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Are the imaginary numbers real?
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May 2024
Reference de re versus de dicto
Subtle semantical issues arise when we refer to an object or individual directly, to the thing itself, or instead under a description, which may find a…
May 25
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Joel David Hamkins
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Reference de re versus de dicto
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Definite descriptions
Definite descriptions, such as "the badly juggling clown," succeed when the intended individual is indeed characterized by the property. But how does…
May 18
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Joel David Hamkins
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Definite descriptions
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What are the real numbers, really?
The real real numbers—what are they? Must we answer?
May 12
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Joel David Hamkins
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What are the real numbers, really?
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Stipulative definitions
The logic of expanding a formal language by introducing defined terms and relations
May 5
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Joel David Hamkins
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Stipulative definitions
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