On Gödel's second incompleteness theorem
A very short proof of Gödel's second incompleteness theorem (for set theory, second order arithmetic etc.)
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arXiv subjects
Thomas Jech
Publications and source records attributed to Thomas Jech.
Full reflection of stationary sets at regular cardinals
A stationary subset S of a regular uncountable cardinal kappa reflects fully at regular cardinals if for every stationary set T subseteq kappa of higher order consisting of regular cardinals there exists an alpha in T such that S cap alpha is a stationary subset of alpha. We prove that the Axiom of Full Reflection which states that every stationary set reflects fully at regular cardinals, together with the existence of n-Mahlo cardinals is equiconsistent with the existence of Pi^1_n-indescribable cardinals. We also state the appropriate generalization for greatly Mahlo cardinals.
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On a conjecture of Tarski on products of cardinals
We look at an old conjecture of A. Tarski on cardinal arithmetic and show that if a counterexample exists, then there exists one of length omega_1 + omega .
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A partition theorem for pairs of finite sets
Every partition of [[omega_1]^{< omega}]^2 into finitely many pieces has a cofinal homogeneous set. Furthermore, it is consistent that every directed partially ordered set satisfies the partition property if and only if it has finite character.
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Full reflection of stationary sets below aleph_omega
It is consistent that for every n >= 2, every stationary subset of omega_n consisting of ordinals of cofinality omega_k where k = 0 or k <= n-3 reflects fully in the set of ordinals of cofinality omega_{n-1}. We also show that this result is best possible.
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A note on canonical functions
We construct a generic extension in which the aleph_2 nd canonical function on aleph_1 exists.
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