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Ultraproducts of fields, II June 28, 2008

Posted by Alexandre Borovik in Uncategorized.
1 comment so far

I continue my post on ultraproducts. So, we want to understand in what sense an ultraproduct of finite fields F_i of unbounded order is a limit at infinity of finite fields. The answer now should be obvious: since ultraproducts are residue fields for maximal ideals in the cartesian product

R = \prod F_i,

the topology in question should be the canonical topology (Zariski topology) of the spectrum of the ring \mbox{Spec}(R). It instantly follows from the description of ideals and maximal ideals in R that this is the Stone topology on the set of ultrafilters on \mathbb{N}, or, what is th same, the Cech-Stone compactification \beta\mathbb{N} of the set \mathbb{N} with descrete topology. Therefore the answer is: an ultraproduct is the limit in the Cech-Stone compactification of a discrete countable set.

I have to admit that at this point I reached limits of my knowledge of set-theoretic topology and had to dip into Wikipedia. It happened that \beta\mathbb{N}, and even more so its non-principal part {\mathbb{N}}^* = \beta{\mathbb{N}} \setminus {\mathbb{N}} is characterised by some unique properties: If the continuum hypothesis holds then {\mathbb{N}}^* is the unique Parovicenko space, up to isomorphism.

According to Wikipedia, a Parovicenko space is a topological space X satisfying the following conditions:

  • X is compact Hausdorff
  • X has no isolated points
  • X has weight c, the cardinality of the continuum (this is the smallest cardinality of a base for the topology).
  • Every two disjoint open Fσ subsets of X have disjoint closures
  • Every nonempty Gδ of X has non-empty interior.

As you can see, {\mathbb{N}}^* is uniquely characterised by very natural properties.

It is yet another manifestation of of one of the most pardoxical properties of mathematical infinity: canonicity of workable constructions in the infinite domain.

NHS invented a new type of infinity… June 21, 2008

Posted by Alexandre Borovik in Uncategorized.
4 comments

My posts are likely to become shorter and sparser — as a result of a work trauma, I have developed a medical condition (see a photo) which makes typing very difficult, and I depend on the kind help of my wife with all my typing needs. On the bright side, my experience gave me a new understanding of infinity. From a mathematical viewpoint, indefinitely long waiting lists for treatment on NHS (National Health Service) were not something new, they were just a special case of potentially infinite natural series

Week 1, Week 2, Week 3, … etc.

going on and on in a very old-fashioned way. The real contribution of NHS to mathematics of infinity is that they make patients to wait (indefinitely again) to be included on a waiting list. It is an infinity of natural numbers enhanced by an additional constraint: you are not allowed to start counting.

A letter from a student June 17, 2008

Posted by Alexandre Borovik in Uncategorized.
27 comments

Hello Professor,

I hope everything is good and that you recovered from the accident in your
finger.

I just wanted to share some personal thoughts. Suppose we have a system that is discrete and finite. We have the natural numbers {1,2,3,…,T} where T is the symbol for the ”biggest natural number”. We will never have a value for T but we accept that it is a fixed natural number. We can also include 0 in our system. How much mathematics can we do?

We could define the usual addition and multiplication. But we will have a problem when the result is ”greater than T”. But nothing ”greater than T” exist…

Suppose for example that T=10. Then we have 2+3=5, 4+5=9, 5+5=10. But what about 5+7? We could just define 5+7= err (error). just like the small calculators do when they reach their limit. But err is not in our system… We could work modulo T OR we could just say that 5+7=10=T. and 8+9=T. and 7+8=10=T etc.

(If we choose the last option then obviously 1+T=T and T+T=T. So T has a similar behaviour as the infinite that we learned at school. But the big difference is that in our system T is a fixed natural number!)

I just can not see why we NEED infinite to make mathematics. Is it a matter of convenience? Is it just to make things simpler? If this is the case then we should accept that infinite and continuous entities are just tools, ideas, symbols that make our life easier. But we should remember that IDEAS themselves are FINITE since they live in the finite world of our brains!

I think the whole problem is more about aesthetics. I think that someone can accept that only discrete and finite things exist and, at the same time, that this belief does not destroy all the beautifull mathematics that have been created until now.

While I was searching the Internet I found this PDF document which I attached and I found amusing. I also found other people expressing similar views (against infinite) like for example Alexander Yessenin-Volpin.

Ultraproducts of fields, I June 16, 2008

Posted by Alexandre Borovik in Uncategorized.
4 comments

My immediate research interests more and more focus on an interplay between finite and infinite in algebra (at least this is where my chats with my PhD student drift to). In particular, I have to use frequently a specific construction, an ultrafilter product of fields. It is pretty sublime in the sense of David Corfield and leads to appearance of very interesting canonical objects.

We start with a family of fields F_i, i \in I . For simplicity assume that all fields are finite of unbounded order and that the index set I is just the set of natural numbers \mathbb{N} (actually this is the case most interesting to me). We form the Cartesian product R of F_i: this is just a set of infinite strings

\{ (f_1, f_2, \dots ) \mid f_i \in F_i \}

with componentwise operations of addition and multiplication. In what follows, we shall make frequent use of zero set of a string f = (f_1, f_2, \dots ), that is, the set of indices where the string components are zeroes:

zero(f) = \{ i \in {\mathbb{N}} \mid f_i =0\}.

Obviously, R is a commutative ring with unity. Let us look at its ideals. One can easily see that an ideal I in R is uniquely determined by the set

zero(I) = \{zero(f) \mid f \in I\};

indeed, non-zero components of a string f \in I can be arbitrarily changed without moving f outside of I by multiplying by appropriate string of invertible elements. One can also instantly see that zero(I) is a filter on \mathbb{N}, that is, it is a collection of non-empty subsets of \mathbb{N} closed under taking finite intersections and supersets, and, moreover, that the correspondence

I \mapsto zero(I)

is a one-to-one correspondence between proper ideals in R and filters of \mathbb{N} which preserves embedding of ideals and embedding of filters. Therefore, maximal ideals in R correspond to maximal filters on \mathbb{N}; the former and the latter exist by the Zorn Lemma, one of the equivalent formulations of the Choice Axiom. If now I is a maximal ideal in R, the fact that the factor ring R/I is a field and, in particular, has no zero divisors, translates in the fact that a maximal filter \mathcal{F} is an ultrafilter: it has the property that, for any subset X \subseteq \mathbb{N} either X or its complement {\mathbb{N}} \smallsetminus X belongs to \mathcal{F}.

Given an ultrafilter \mathcal{F} on \mathbb{N}, the ultraproduct F = \prod F_i/\mathcal{F} is nothing more than the corresponding residue field R/I. There are obvious principal (ultra)filters on \mathbb{N}, they consist of all subsets containing a given element i \in \mathbb{N}; obviously, the corresponding ultraproduct is just the original field F_i.

Non-principal filters do exists. One very interesting non-principal filter on \mathbb{N} is the Frechet filter consisting of all subsets with finite complements.

But if we take an ultrafilter containing a non-principal filter (it exists by the Zorn Lemma), the corresponding ultraproduct F has many marvelous properties. In particular, if all F_i have different characteristics, F turns out to be a field of characteristic zero (I leave the proof of this fact as an exercise to the reader).

In the next instalment of this post, I will discuss the meaning of a frequently used assertion that an ultraproduct F is a limit at infinity of finite fields F_i.

The Sublime June 11, 2008

Posted by dcorfield in Uncategorized.
7 comments

There’s an interesting post – Whatever Happened to Sublimity? – at the blog Siris. It includes a quotation from Edmund Burke

But let it be considered that hardly any thing can strike the mind with its greatness which does not make some sort of approach toward infinity; which nothing can do while we are able to perceive its bounds; but to see an object distinctly, and to perceive its bounds, are one and the same thing. A clear idea is, therefore, another name for a little idea. (A Philosophical Inquiry into the Origin of Our Ideas of the Sublime and the Beautiful, Part II, Section V.)

A natural question to ask, then, is where do we encounter the sublime in mathematics? And an obvious answer, you might think, would be the mathematical infinite.

Joseph Dauben has an interesting section in his book – Georg Cantor: his mathematics and philosophy of the infinite, Harvard University Press, 1979 – on how Cantor, receiving such discouragement from his mathematical colleagues, found an audience in certain thinkers within the Catholic church. Where earlier in the nineteenth century any attempt to describe a completed infinity was viewed as a sacrilegious attempt to circumscribe God, some theologians were open to Cantor’s new hierarchy of infinities, with its unreachable Absolute Infinite leaving room for the divine.

Personally, set theory has rarely invoked in me a sense of the sublime. On the other hand, the following comment by Daniel Davis does:

Behrens and Lawson use stacks, the theory of buildings, homotopy fixed points, the above model category, and other tools to make it possible to use the arithmetic of Shimura varieties to help with understanding the stable homotopy groups of spheres.

There’s plenty about the infinite in that statement, it’s true, but there’s so much more to it than that.

Brandon Watson, the blogger at Siris, writes

…what often strikes me when I look around at the philosophical scene today is how foreign this has all become. There are a few exceptions, but sublimity has vanished as a serious concern.

With regard to the philosophy of mathematics in particular I couldn’t agree more.

Axiom of Choice, quotes June 1, 2008

Posted by Alexandre Borovik in Uncategorized.
2 comments
  • The Axiom of Choice is obviously true, the Well-ordering theorem obviously false, and who can tell about Zorn’s lemma ?” ::— Jerry Bona -(This is a joke: although the axiom of choice, the well-ordering principle, and Zorn’s lemma are all mathematically equivalent, most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn’s lemma to be too complex for any intuition.)
  • “The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.” ::— Bertrand Russell -(The observation here is that one can define a function to select from an infinite number of pairs of shoes by stating for example, to choose the left shoe. Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) identical to each other.)
  • “The axiom gets its name not because mathematicians prefer it to other axioms.” ::— A. K. Dewdney -( This quote comes from the famous April Fool’s Day article in the computer recreations column of the Scientific American , April 1989.)

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