April | 2009 | A kind of library

580 -Partition calculus (6)

April 24, 2009

1. The ${\mbox{Erd\H os}}$ -Rado theorem

Large homogeneous sets (of size ${\omega_1}$ or larger) can be ensured, at the cost of starting with a larger domain. The following famous result was originally shown by ${\mbox{Erd\H os}}$ and Rado using tree arguments (with ${\kappa+1}$ lowered to ${\kappa}$ in the conclusion). We give instead an elementary substructures argument due to Baumgartner, Hajnal and ${\mbox{Todor\v cevi\'c},}$ which proves the stronger version. For ${\kappa}$ a cardinal let ${2^{<\kappa}=\sup_{\lambda<\kappa}2^\lambda.}$

Theorem 1 ( ${\mbox{Erd\H os}}$ -Rado) Let ${\kappa}$ be a regular cardinal and let ${\lambda=(2^{<\kappa})^+.}$ Then

$\displaystyle \lambda\rightarrow(\kappa+1)^2_\mu$

for all ${\mu<\kappa.}$

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305 -Homework set 8

April 24, 2009

This set is due May 1 at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

1. Solve exercises 11, 14, and 24 from Chapter 14 of the textbook.

2. Solve exercises 1, 9, and 17 from Chapter 15 of the textbook.

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305 -9. Groups

April 24, 2009

We want to define the notion of group that will be fundamental to determine which polynomials are solvable by radicals. This notion is very important and appears in every area of mathematics.

We motivate the definition through the example that most concerns us: automorphisms of fields. They are particular class of isomorphisms, so we begin with them.

Some of the arguments below have been discussed in previous lectures.

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Leave a Comment » | 305: Abstract Algebra I | Tagged: automorphism, isomorphism | Permalink
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580 -Partition calculus (5)

April 21, 2009

1. Larger cardinalities

We have seen that ${\omega\rightarrow(\omega)^n_m}$ (Ramsey) and ${\omega\rightarrow[\omega]^n_\omega}$ ( ${\mbox{Erd\H os}}$ -Rado) for any ${n,m<\omega.}$ On the other hand, we also have that ${2^\kappa\not\rightarrow(3)^2_\kappa}$ ( ${\mbox{Sierpi\'nski}}$ ) and ${2^\kappa\not\rightarrow(\kappa^+)^2}$ ( ${\mbox{Erd\H os}}$ -Kakutani) for any infinite ${\kappa.}$

Positive results can be obtained for larger cardinals than ${\omega}$ if we relax the requirements in some of the colors. A different extension, the ${\mbox{Erd\H os}}$ -Rado theorem, will be discussed later.

Theorem 1 ( ${\mbox{Erd\H os}}$ -Dushnik-Miller) For all infinite cardinals ${\lambda,}$ ${\lambda\rightarrow(\lambda,\omega)^2.}$

This was originally shown by Dushnik and Miller in 1941 for ${\lambda}$ regular, with ${\mbox{Erd\H os}}$ providing the singular case. For ${\lambda}$ regular one can in fact show something stronger:

Theorem 2 ( ${\mbox{Erd\H os}}$ -Rado) Suppose ${\kappa}$ is regular and uncountable. Then
$\displaystyle \kappa\rightarrow_{top}(\mbox{Stationary},\omega+1)^2,$ which means: If ${f:[\kappa]^2\rightarrow2}$ then either there is a stationary ${H\subseteq\kappa}$ that is ${0}$ -homogeneous for ${f}$ , or else there is a closed subset of ${\kappa}$ of order type ${\omega+1}$ that is ${1}$ -homogeneous for ${f}$ .

(Above, top stands for “topological.”)

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6 Comments | 580: Topics in set theory | Tagged: elementary embedding, Grigor Sargsyan, Jan Tryba, Jonsson cardinal, measurable cardinal, Ramsey cardinal, Reinhardt cardinal, Rowbottom cardinal, W. Hugh Woodin, weakly compact cardinal | Permalink
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305 -8. Irreducibility

April 20, 2009

In this lecture I want to present a couple of short results that are nevertheless very useful in practice when trying to show that a given polynomial in ${{mathbb Q}[x]}$ is irreducible. Of course, we may assume that the polynomial actually has integer coefficients. In this case, it turns out that analyzing whether the polynomial factors over ${{mathbb Z}[x]}$ suffices.

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Leave a Comment » | 305: Abstract Algebra I | Tagged: Eisenstein criterion, Gauss lemma, irreducible polynomial | Permalink
Posted by andrescaicedo

580 -Cardinal Arithmetic

April 20, 2009

Here is a pdf file with the contents of the lectures on cardinal arithmetic. As with the previous chapter, it follows closely the style of the notes. There are fewer typos than in the posts, and once again I made a minuscule tidying up. Please let me know of comments, corrections, and suggestions.

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580 -Some choiceless results

April 16, 2009

I reformatted the posts for the lectures on choiceless results. Here they are as a pdf file. It follows closely the style of the notes. I fixed a couple of typos, and made a minuscule tidying up. Please let me know of comments, corrections, and suggestions.

Update (Dec. 3, 2009). The pdf file still contains an error that was pointed out to me recently. In the third lecture on choiceless results, the correct argument is given.

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Mathoverflow

Relative Densities September 12, 2013

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 […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Solutions to the Continuum Hypothesis May 18, 2010

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Are there any non-linear solutions of Cauchy's equation ($f(x+y)=f(x)+f(y)$) without assuming the Axiom of Choice? March 6, 2011

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Andrés E. Caicedo

David Gale's subset take-away game October 12, 2010

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Andrés E. Caicedo

A translation of the Cantor set contained in the irrationals February 22, 2014

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Andrés E. Caicedo

Math.StackExchange

Answer by Andrés E. Caicedo for Well-ordering of the reals in ZF with constructibility? April 19, 2018

$L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZFC}$) well-orders all of $L$, so that its restriction to any specific set $A$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction to $\ […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Assuming ZF and the axiom of constructibility, $\alpha > \omega, V_\alpha = L_\alpha \iff \alpha = \aleph_\alpha$ February 28, 2018

Define the sequence of beth numbers by $\beth_0=\aleph_0$, $\beth_{\alpha+1}=2^{\beth_\alpha}$ and $\beth_\lambda=\sup_{\beta

Andrés E. Caicedo

Answer by Andrés E. Caicedo for On the definition of Inductive set. Equivalences February 18, 2018

The two concepts are different. For example, $\omega$, the first infinite ordinal, is the standard example of an inductive set according to the first definition, but is not inductive in the second sense. In fact, no set can be inductive in both senses (any such putative set would contain all ordinals). In the context of set theory, the usual use of the term […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Find a contradiction to: If $t^n < (\sup \{t \in \mathbb{R}_{>0}: t^n < x\})^n$ then $t^n < x$ when this supremum is greater than $x$. February 16, 2018

I will show that for any positive integers $n,\ell,k$ there is an $M$ so large that for all positive integers $i$, if $i/M\le \ell$, then the difference $$ \left(\frac iM\right)^n-\left(\frac{i-1}M\right)^n $$ is less than $1/k$. Let's prove this first, and then argue that the result follows from it. Note that $$ (i+1)^n-i^n=\sum_{k=0}^{n-1}\binom nk i^ […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Prove by induction that $n^2>7n+1$ for all $n \ge 8$ February 15, 2018

I think it is cleaner to argue without induction. If $n$ is a positive integer and $n\ge 8$, then $7n$ is both less than $n^2$ and a multiple of $n$, so at most $n^2-n$ and therefore $7n+1$ is at most $n^2-n+1

Andrés E. Caicedo

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A kind of library

580 -Partition calculus (6)

305 -Homework set 8

305 -9. Groups

580 -Partition calculus (5)

305 -8. Irreducibility

580 -Cardinal Arithmetic

580 -Some choiceless results

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