Set Turing Machines
We define a generalization of the Turing machine that computes on general sets. Our main theorem states that the class of generalized Turing machine computable functions and the class of Set Recursive functions coincide.
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arXiv subjects
Garvin Melles
Publications and source records attributed to Garvin Melles.
A saturated model of an unsuperstable theory of cardinality greater than its theory has the small index property
A model M of cardinality lambda is said to have the small index property if for every G subseteq Aut(M) such that [Aut(M):G] <= lambda there is an A subseteq M with |A|< lambda such that Aut_A(M) subseteq G. We show that if M^* is a saturated model of an unsuperstable theory of cardinality > Th(M), then M^* has the small index property.
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The Consistency of $ZFC+CIFS$
This paper is a technical continuation of ``Natural Axiom Schemata Extending ZFC. Truth in the Universe?'' In that paper we argue that $CIFS$ is a natural axiom schema for the universe of sets. In particular it is a natural closure condition on $V$ and a natural generalization of $IFS(L).$ Here we shall prove the consistency of $ZFC\ +\ CIFS$ relative to the existence of a transitive model of $ZFC$ using the compactness theorem together with a class forcing.
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The automorphism group of a saturated model has a large dense free subgroup
We prove that for a stable theory $T,$ if $M$ is a saturated model of $T$ of cardinality $λ$ where $λ> \big|T\big|,$ then $Aut(M)$ has a dense free subgroup on $2^λ$ generators. This affirms a conjecture of Hodges.
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Natural Internal Forcing Schemata Extending ZFC
Let V be the universe of sets and V_α the sets of rank \leqα. We develop some axiom schemata for set theory based on the following three assumptions: 1. V \models ZFC 2. V is large with respect to the class of ordinals 3. V is large with respect to each of the V_α
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Some Natural Internal Forcing Schemata Extending ZFC
We give arguments for and prove the consistency of some internal forcing axioms.
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$^*$Forcing
Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete Boolean algebra in $M.$ (In general a proper class.) We define a generalized notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing extension of $M$ is a transitive set of the form $M[{\bf G}]$ where ${\bf G}$ is an $M$-complete ultrafilter on ${\bf B}.$) We prove that 1. If ${\bf G}$ is a $^*$forcing complete ultrafilter on ${\bf B},$ then $M[{\bf G}]\models ZFC.$ 2. Let $H\sub M.$ If there is a least transitive model $N$ such that $H\in M,$ $Ord^M=Ord^N,$ and $N\models ZFC,$ then we denote $N$ by $M[H].$ We show that all models of $ZFC$ of the form $M[H]$ are $^*$forcing extensions of $M.$ As an immediate corollary we get that $L[0^{\#}]$ is a $^*$forcing extension of $L.$
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