I concluded my series of talks by showing the following theorem of Viale:
Theorem (Viale). Assume and let be an inner model where is regular and such that Then .
This allows us to conclude, via the results shown last time, that if holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .
An elaboration of this argument is expected to show that, at least if we strengthen the assumption of to , then computes correctly ordinals of cofinality .
Under an additional assumption, Viale has shown this: If holds in , is a strong limit cardinal, , and in we have that is regular, then in the cofinality of cannot be . The new assumption on allows us to use a result of Dzamonja and Shelah, On squares, outside guessing of clubs and , Fund. Math. 148 (1995), 165-198, in place of the structure imposed by . It is still open if the corresponding covering statement follows from , which would eliminate the need for this the strong limit requirement.
Go to the intermezzo for a discussion of consistency strengths.
This entry was posted on Friday, October 24th, 2008 at 4:52 pm and is filed under math.LO, Set theory seminar. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.
4 Responses to Set theory seminar -Forcing axioms and inner models VII
I saw a while ago in a book by Clifford Pickover, that whether the Flint Hills series $\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open. I would think that the question of its convergence is really about the density in $\mathbb N$ of the sequence of numerators of the standard convergent approximations to $\pi$ (which, in itself, seems li […]
I am not sure whether this qualifies as "well-known". Anyway, in set theory, in the study of the partition calculus (transfinite generalizations of Ramsey's theorem), effort centered for a while in studying relations of the form $$ \omega^m\to(\omega^n,k)^2 $$ for $m,n,k$ positive integers. Here, exponentiation is in the ordinal sense. This re […]
This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR0009444 (5,151d). (Anyway, it is probably easier to read a more modern pre […]
One of the best places to track these things down is The mathematical coloring book, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about. As far as I can tell, the question is sometimes called the Erdős-Turán con […]
Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]
As we learned in this question, provably in $\mathsf{ZF}$, if $X$ is infinite then $|X|^n\le(n+1)^{|X|}$ for any natural number $n$, and this is optimal in the sense that, in general, $n+1$ cannot be replaced with $n$. A naïve inductive attempt to show the inequality suggests that perhaps we can actually prove that $|X|\cdot n^{|X|}\le(n+1)^{|X|}$, although […]
In Ralph P. Boas's A primer of real functions, page 118, this is discussed in the following way: The derivative of infinite order of $f$ is defined on an interval $I$ iff the sequence $(f^{(n)})$ converges uniformly on $I$ (it is enough to require uniform convergence on compact subsets of $I$). Call $L$ the limit of this sequence, so $L$ is continuous a […]
Recall that the beth ($\beth$) numbers are defined by transfinite recursion as $\beth_0=0$, $\beth_{\alpha+1}=2^{\beth_\alpha}$ and $\beth_\lambda=\sup_{\alpha
The point here is that two functions are close iff they agree on an initial segment, that is, $d(f,g)\le 1/(n+1)$ iff $f(0)=g(0),f(1)=g(1),\dots,f(n-1)=g(n-1)$. Now, if $(f_n)_n$ is a Cauchy sequence, then, for each $n$, there is $N_n$ such that for all $m,k>N_n$ we have $d(f_m,f_k)\le1/(n+1)$. That is, all functions $f_m$ with $m>N_n$ agree on their f […]
No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]
and let
be an inner model where
is regular and such that
Then
.
![I_{<f} [\lambda]](https://s0.wp.com/latex.php?latex=I_%7B%3Cf%7D+%5B%5Clambda%5D&bg=ffffff&fg=333333&s=0&c=20201002)
A kind of library 
[…] Go to next talk. […]
[…] Seventh Talk, October 24, 2008. […]
[…] Seventh talk, October 24, 2008. […]
[…] Go to next talk. […]