2 results
Finitely generated free Heyting algebras: the well-founded initial segment
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- Journal:
- The Journal of Symbolic Logic / Volume 77 / Issue 4 / December 2012
- Published online by Cambridge University Press:
- 12 March 2014, pp. 1291-1307
- Print publication:
- December 2012
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In this paper we describe the well-founded initial segment of the free Heyting algebra
α on finitely many, α, generators. We give a complete classification of initial sublattices of 2 isomorphic to 1 (called ‘low ladders’), and prove that for 2 ≤ α < ω, the height of the well-founded initial segment of α is ω2.
Non-well-foundedness of well-orderable power sets
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- Journal:
- The Journal of Symbolic Logic / Volume 68 / Issue 3 / September 2003
- Published online by Cambridge University Press:
- 12 March 2014, pp. 879-884
- Print publication:
- September 2003
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Tarski [5] showed that for any set X, its set ω(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation ∣ω(X)∣ = ∣Y∣. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation
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