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Substructures and chains of models
Truth propogates through an elementary chain of models to the limit.
Mar 25
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Joel David Hamkins
4
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Substructures and chains of models
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Theory theory
Let us explore the theory of theories, the theory of models, the theory of models of theories, and the theory of theories of models.
Mar 19
•
Joel David Hamkins
7
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Theory theory
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More applications of induction
An excerpt from Proof and the Art of Mathematics
Mar 13
•
Joel David Hamkins
2
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More applications of induction
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A compositional theory of truth
Tarski's disquotational theory of truth in a structure is a compositional theory of truth, by which the truth of a complex assertion reduces to the…
Mar 7
•
Joel David Hamkins
2
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A compositional theory of truth
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A formal language for first-order predicate logic
The basic syntax of first-order logic—the signature of a language, terms, atomic formulas, well-formed formulas, free and bound variables, interpreting…
Mar 2
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Joel David Hamkins
6
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A formal language for first-order predicate logic
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February 2024
Distinguishing mathematical structures by their theories
With first-order logic we can often express very subtle features that distinguish a favored structure from similar alternatives.
Feb 24
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Joel David Hamkins
6
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Distinguishing mathematical structures by their theories
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Chaos train and monkey madness—fun with quantifiers in first-order logic
Learn the language of first-order logic by immersion in some fun logic puzzles
Feb 19
•
Joel David Hamkins
5
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Chaos train and monkey madness—fun with quantifiers in first-order logic
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The countable random graph
Flip a coin to determine the edge relations between vertices in a countably infinite set. The result is a random graph, yet almost surely it is exactly…
Feb 10
•
Joel David Hamkins
4
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The countable random graph
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An algebra of orders
Did you know that we can add and multiply orders? There is a whole algebra of orders, with ℕ+1 and ℤ+ℤ, as well as ℕ+ℤ·ℤ and ℤ·ℚ, a truly fascinating…
Feb 4
•
Joel David Hamkins
7
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An algebra of orders
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January 2024
Mathematical induction
An excerpt from Proof and the Art of Mathematics
Jan 30
•
Joel David Hamkins
4
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Mathematical induction
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The lattice of all sets of natural numbers
Let us explore the vast and densely populated expanses of the lattice of all sets of natural numbers—the power set lattice ⟨P(ℕ),⊆⟩—seeking monsters and…
Jan 24
•
Joel David Hamkins
8
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The lattice of all sets of natural numbers
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Lattices
We explore the basic theory of lattices and investigate an intriguing instance, the lattice of partitions on a given set under refinement.
Jan 17
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Joel David Hamkins
13
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Lattices
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