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Register a new userA model with everything except for a well-ordering of the reals
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We construct a model of $mathsf{ZF} + mathsf{DC}$ containing a Luzin set, a Sierpi{n}ski set, as well as a Burstin basis but in which there is no a well ordering of the continuum.
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N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $mathbb R$ so that no infinite sumset $X+X={x+y:x,yin X}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:mathbb Rto r$ with $r$ finite there is an infinite $Xsubseteq mathbb R$ so that $c$ is constant on $X+X$.
Let $M^sharp_n(mathbb{R})$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n(mathbb{R})$ the class-sized model obtained by iterating the topmost measure of $M_n(mathbb{R})$ class-many times. We characterize the sets of reals which are $Sigma_1$-definable from $mathbb{R}$ over $M_n(mathbb{R})$, under the assumption that projective games on reals are determined: (1) for even $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Pi^1_{n+1}$; (2) for odd $n$, $Sigma_1^{M_n(mathbb{R})} = Game^mathbb{R}Sigma^1_{n+1}$. This generalizes a theorem of Martin and Steel for $L(mathbb{R})$, i.e., the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $mathbb{R}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists for all $ninmathbb{N}$, and that determinacy of all projective games of length $omega^2$ with moves in $mathbb{N}$ is equivalent to the statement that $M^sharp_n(mathbb{R})$ exists and satisfies $mathsf{AD}$ for all $ninmathbb{N}$.
We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees.
We will prove that there exists a model of ZFC+``c= omega_2 in which every M subseteq R of cardinality less than continuum c is meager, and such that for every X subseteq R of cardinality c there exists a continuous function f:R-> R with f[X]=[0,1]. In particular in this model there is no magic set, i.e., a set M subseteq R such that the equation f[M]=g[M] implies f=g for every continuous nowhere constant functions f,g:R-> R .
We prove that there exists a countable $beta$-model in which, for all reals $X$ and $Y$, $X$ is definable from $Y$ if and only $X$ is hyperarithmetical in $Y$. We also obtain some related results and pose some related questions.
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