Vladimir Kanovei

A generic extension L[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{L}[x]}$$\end{document} by a real x is defined, in which the E0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{E}_0}$$\end{document}-class of x is a lightface Π21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\it \Pi}^1_2}$$\end{document} set containing no ordinal-definable reals.

If A ⊆ ω1, then there exists a cardinal preserving generic extension [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ][x ] of [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ] by a real x such that1) A ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ] and A is Δ1HC in [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ];2) x is minimal over [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ], that is, if a set Y belongs to [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ], then either x ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A, Y ] or Y ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ].The forcing we use implicitly provides reshaping of the given set A

A problem which enthusiasts of IST, Nelson's internal set theory, usually face is how to treat external sets in the internal universe which does not contain them directly. To solve this problem, we consider BST, bounded set theory, a modification of IST which is, briefly, a theory for the family of those IST sets which are members of standard sets. We show that BST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop the most advanced applications of nonstandard methods. In particular, we define in BST an enlargement of the BST universe which satisfies the axioms of HST, an external theory close to a theory introduced by Hrbaček. HST includes Replacement and Saturation for all formulas but contradicts the Power Set and Choice axioms, therefore to get an external universe which satisfies all of ZFC minus Regularity one has to pay by a restriction of Saturation. We prove that HST admits a system of subuniverses which model ZFC and Saturation in a form restricted by a fixed but arbitrary standard cardinal. Thus the proposed system of set theoretic foundations for nonstandard mathematics, based on the simple and natural axioms of the internal theory BST, provides the treatment of external sets sufficient to carry out elaborate external constructions.

We prove that if I is a partially ordered set in a countable transitive model $\mathfrak{M}$ of $\mathbf{ZFC}$ then $\mathfrak{M}$ can be extended by a generic sequence of reals $\mathbf{a}_i$, i $\in$ I, such that $\aleph^{\mathfrak{M}}_1$ is preserved and every $\mathbf{a}_i$ is Sacks generic over $\mathfrak{M}[\langle \mathbf{a}_j : j < i\rangle]$. The structure of the degrees of $\mathfrak{M}$-constructibility of reals in the extension is investigated. As applications of the methods involved, we define a cardinal invariant to distinguish product and iterated Sacks extensions, and give a short proof of a theorem that in $\omega_2$-iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds.

The parameter-free part $$\textbf{PA}_2^*$$ of $$\textbf{PA}_2$$, second order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $$\omega $$ -model of $$\textbf{PA}_2^*+ \textbf{CA}(\Sigma ^1_2)$$, in which an example of the full Comprehension schema $$\textbf{CA}$$ fails. Using Cohen’s forcing, we also define an $$\omega $$ -model of $$\textbf{PA}_2^*$$, in which not every set has its complement, and hence the full $$\textbf{CA}$$ fails in a rather elementary way.