Paradoxes, paradoxically

So I was able – fortunately – to wake up early and to do some legit reading, despite having only a handful of sleep hours (4-ish?). That’s a definite positive. Right now, I’m about 30 minutes away from a forced obligation (that’s a definite negative), but I wanted to use the 30 minutes I have to still do something constructive. Rather than spend this time wracking my brain with really difficult, hard-to-understand reading that would leave me mentally exhausted for the aforementioned obligation, I decided to come here and write a little exposition regarding something mathematical.

In particular, I’m going to talk about the so-called Richard’s Paradox (see here).

Of course, the fact that I’m avoiding theoretical math to postpone mental exhaustion while using the time to come here and talk about theoretical math is a bit of a paradox as well, so I’ll basically be expositing, paradoxically, about paradoxes.

You have no idea how much I crack myself up.

The ideology that birthed Richard’s paradox is intimately tied to the idea of metamathematics, that is, the study of metatheories – theories about mathematical theories – using mathematical ideas and quantification. I’m not going to get too deeply involved in the discussion on that particular topic; the interested reader, of course, can scope out more here.

To begin, we let \mathbb{N} denote the set of nonzero positive integers (aka, the natural numbers) and we investigate the collection of all “formal English language statements of finite length” which define a number n of \mathbb{N}. For example, The first prime number, The smallest perfect number, and The cube of the first odd number larger than five are such statements, as they verbally describe the numbers 2, 6, and 73=343, respectively. On the other hand, statements like The number larger than all other numbers and Scotland is a place I’d like to visit fail to make the list due to the fact that the first doesn’t describe a number in \mathbb{N} and the second doesn’t describe a number at all. Let \mathcal{A}_n denote the collection of all so-called qualifying statements, that is, statements that do describe elements n\in\mathbb{N}.

Note, first, that the collection \mathcal{A}_n is infinite due to the fact that the statements The ith natural number is a qualifying statement for all i=1,2,\ldots. It’s also countable: Only a countable number of words exist in the English language, and each statement in \mathcal{A}_n consists of a finite union of these countably many words. This fact, along with obvious language considerations, says that \mathcal{A}_n can actually be given an ordering.

Indeed, consider a two-part ordering: First, organize the statements in \mathcal{A}_n by length so that the shortest statements appear first, and then organize statements of the same length by standard lexicographical (dictionary) ordering. The result is an ordered version of the countably infinite collection \mathcal{A}_n which we’ll again denote by \mathcal{A}_n.

As of now, almost nothing has been done. Continue reading

Online reading seminar for Zhang’s “bounded gaps between primes”

Dr. Terence Tao has arranged for an online reading seminar to go through Dr. Yitang Zhang’s recent proof of the “Bounded Gaps Conjecture.” To say that this is a wonderful opportunity to pick something valuable up regarding a field that’s very hot right now in the research community would be the ultimate understatement.

What's new

In a recent paper, Yitang Zhang has proven the following theorem:

Theorem 1 (Bounded gaps between primes) There exists a natural number $latex {H}&fg=000000$ such that there are infinitely many pairs of distinct primes $latex {p,q}&fg=000000$ with $latex {|p-q| \leq H}&fg=000000$.

Zhang obtained the explicit value of $latex {70,000,000}&fg=000000$ for $latex {H}&fg=000000$. A polymath project has been proposed to lower this value and also to improve the understanding of Zhang’s results; as of this time of writing, the current “world record” is $latex {H = 4,802,222}&fg=000000$ (and the link given should stay updated with the most recent progress.

Zhang’s argument naturally divides into three steps, which we describe in reverse order. The last step, which is the most elementary, is to deduce the above theorem from the following weak version of the Dickson-Hardy-Littlewood (DHL) conjecture for some $latex {k_0}&fg=000000$:

Theorem 2 ($latex {DHL[k_0,2]}&fg=000000$) Let $latex {{\mathcal H}}&fg=000000$ be an…

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Being (re)born(-again)

I wasn’t around these parts much yesterday for a number of reasons: I spent the better part of the midday on campus, splitting time between campus errands, meeting with one professor about the stuff he’s working on, and parsing through another professor‘s latest research paper; the times around that were being occupied by birthday things because yesterday was the twenty-seventh annual celebration of my being birthed.

That’s right: I’m 27 now, which means – among other things – that I have fewer than 365 days (I’m not sure exactly how many) to supplant Jean-Pierre Serre as the youngest Fields Medal winner. Not a good feeling for a just-blossoming mathematician. Le sigh.

I guess the good news is that I have 13 years to maybe squeeze one in there. Thirteen years…that’s 4,747 days from today. That makes it seem like I have plenty of time to work!

</forced optimism>

In any case…

The paper I’m reading is on the behavior of M-conformal mappings taking values in a Clifford algebra \mathcal{C}\ell_{m,n} over \mathbb{R}. I’m gonna take a second to discuss some of the preliminaries of that topic and to maybe outline the proof of a basic assertion that’s generally considered general knowledge among those in the know. Worth noting is that the background of Clifford Algebras (and hence of the analysis of functions thereon) extends far more deeply than discussed here and so this should in no way be taken as an actual, worthwhile exposition on the ease/difficulty of the topic.

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