MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

# Ashutosh

 1,861 Reputation 1754 views

## Registered User

 Name Ashutosh Member for 3 years Seen yesterday Website Location Age
 May10 revised Restrictions of null/meager idealtypos corrected May10 revised Restrictions of null/meager idealadded 31 characters in body; added 1 characters in body May10 asked Restrictions of null/meager ideal May6 revised Construction of a maximal idealadded 111 characters in body May6 accepted Borel ideals on $\omega$ are meager? May6 comment Borel ideals on $\omega$ are meager?You are right. No need to extend the ideal. May6 answered Borel ideals on $\omega$ are meager? May3 accepted extensions of lebesgue measure May2 comment Partition relation, almost a Ramsey cardinal?Define a new coloring $d:[\kappa]^{< \omega} \rightarrow \omega^{< \omega}$ such that for $s \in [\kappa]^{< \omega}$, $d(s)$ encodes the $c$-colors of all finite increasing sequences in $s$. Get an $H \in [\kappa]^{\kappa}$ such that, $(\forall n \in \omega)|d([H]^n)| \leq 16$. Let $s_1^{0}, s_1^{1}, s_2^{0}, s_2^1, \dots , s_5^0, s_5^1 \in [H]^m$ be a sequence of sets such that max of each set is less than the min of next one and $c(s_i^0) \neq c(s_i^1)$, for $1 \leq i \leq 5$. But this meanss $|d([H]^{5m}| \geq 32$. May2 comment Partition relation, almost a Ramsey cardinal?Suppose (for example), that for every coloring $c:[\kappa]^{< \omega} \rightarrow \omega$, there is a set $H \in [\kappa]^\kappa$ such that $(\forall n \in \omega)|c([H]^n)| \leq 16$. Note that this implies that cofinality of $\kappa$ is at least $\omega_1$. Let us try to bring $16$ down to $1$. Assume that this is impossible for some coloring $c$. Then for every $\kappa$ sized subset $A \subseteq \kappa$, there is some $m$ and there are arbitrarily long sequences $s_1, s_2, \dots s_k \in [A]^{m}$ such that max$(s_i) <$ min$(s_{i+1})$ and $c(s_i) \neq c(s_{i+1})$. Apr29 answered extensions of lebesgue measure Feb17 comment totally ordered chain in the powerset with big cardinalityHere's a reference: William Mitchell - Aronszajn trees and the independence of the transfer property, Annals of Math. Logic., Vol. 5, Issue 1, 1972, pp. 21-46 Feb17 comment totally ordered chain in the powerset with big cardinalityIf I remember correctly, William Mitchell has shown in his thesis that if you add $\omega_{\omega_1}$ Cohen reals to the universe then in the resulting model there is no subset of $\mathcal{P}(\omega_1)$ of size $2^{\omega_1}$ which is linearly ordered under inclusion. Feb12 comment A question about definable non-empty sets containing no definable elements.One can avoid inaccessibles by appealing to Shelah's model where a similar result holds for sets with Baire property. Feb12 comment A question about definable non-empty sets containing no definable elements.Doesn't Solovay's model of ZFC + "All HOD(R) sets of reals are Lebesgue measurable" witness that, for any definable set (without parameters), ZFC cannot prove that such a set is Lebesgue non measurable? Jan31 awarded ● Enthusiast Jan20 comment Even XOR Odd Infinities?If $x + x = 0$ and $y + y = 1$, then $x = x(y + y) = xy + xy = (x + x)y = 0$.