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
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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

Answer by Andrés E. Caicedo for Can we have an infinite sequence of decreasing cardinality all terms of which have equal sized power sets? August 12, 2018

This is consistent, at least under a rather tame large cardinal assumption. (One can also produce examples by manipulating Dedekind finite sets, but Asaf's answer addresses this. The answer here works even in the context of $\mathsf{DC}$.) For instance, see MR3612001. Conley, Clinton T.; Miller, Benjamin D. Measure reducibility of countable Borel equiva […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice? August 11, 2018

Let me add some comments to Asaf's answer. We are looking for situations with $|X|

Andrés E. Caicedo

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The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Does constructing non-measurable sets require the axiom of choice? October 14, 2010

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

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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 […]

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Answer by Andrés E. Caicedo for Clarification about Infinite Sums in Jech's Set Theory 3rd Edition August 12, 2018

Using the regularity of $\kappa$ and the fact that the infinite cardinal $\lambda$ is less than $\kappa$, Jech notes that $$ \kappa^\lambda=\bigcup_{\alpha

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A set $x$ is hereditarily $\varphi$ if and only if every member of its transitive closure (including $x$ itself) has property $\varphi$. For instance, being hereditarily finite means not just that $x$ is finite, but also every member of $x$ is finite, and every member of every member of $x$, and so on. (Note that in the absence of foundation, this is a bit p […]

Andrés E. Caicedo

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Let $s$ be the supremum of the $\mu$-measures of members of $\mathcal G$. By definition of supremum, for each $n$, there is $G_n\in\mathcal G$ with $\mu(G_n)>s-1/n$. Letting $G=\bigcup_n G_n$, then $G\in \mathcal G$ since $\mathcal G$ is closed under countable unions, and $\mu(G)=s$, since it is at least $\sup_n\mu(G_n)$ but it is at most $s$ (by definiti […]

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Set theory ordinals problem Prove that if $b$ is a limit ordinal and $a June 27, 2018

The result you are trying to prove is false. For example, if $a=\omega+1$ and $b=\omega+\omega$, then $a+b=\omega\cdot 3>b$. Here is what is true: first, the key result you should establish (by induction) is that An ordinal $\alpha>0$ has the property that for all $\beta

Andrés E. Caicedo

Answer by Andrés E. Caicedo for Are there any new axioms being developed? June 5, 2018

Very briefly: Yes, there are several programs being developed that can be understood as pursuing new axioms for set theory. For the question itself of whether pursuing new axioms is a reasonably line of inquiry, see the following (in particular, the paper by John Steel): MR1814122 (2002a:03007). Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, […]

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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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