Joel David Hamkins’ 1000th MO Answer is Coming

Update (May 2014): The second MO contributor to answer 1000 questions is another distinguished mathematician (and a firend) Igor Rivin.

Joel David Hamkins’ profile over MathOverflow reads: “My main research interest lies in mathematical logic, particularly set theory, focusing on the mathematics and philosophy of the infinite. A principal concern has been the interaction of forcing and large cardinals, two central concepts in set theory. I have worked in group theory and its interaction with set theory in the automorphism tower problem, and in computability theory, particularly the infinitary theory of infinite time Turing machines. Recently, I am preoccupied with the set-theoretic multiverse, engaging with the emerging field known as the philosophy of set theory.”

Joel is a wonderful MO contributor, one of those distinguished mathematicians whose arrays of MO answers in their areas of interest draw coherent deep pictures for these areas that you probably cannot find anywhere else. And Joel is also a very highly decorated and prolific MO contributor, whose 999th answer appeared today!!

Here is a very short selection of Joel’s answers. To (MO founder) Anton Geraschenko’s question What are some reasonable-sounding statements that are independent of ZFC? Joel answered; “If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y.” Joel gave a very thorough answer to  my question on Solutions to the Continuum Hypothesis; His 999th answer is on the question Can an ultraproduct be infinite countable? (the answer is yes! but this is a large cardinal assumption.) Update: Joel’s 1000th answer on a question about logic in mathematics and philosophy was just posted.

Joel also wrote a short assay, the use and value of MathOverflow over his blog. Here it is:

The principal draw of mathoverflow for me is the unending supply of extremely interesting mathematics, an eternal fountain of fascinating questions and answers. The mathematics here is simply compelling.

Mathoverflow has also taught me a lot about good mathematical exposition, both by the example of other’s high quality writing and by the immediate feedback we all get on our posts. This feedback reveals what kind of mathematical explanation is valued by the general mathematical community, in a direct way that one does not usually get so well when writing a paper or giving a conference talk. This kind of knowledge has helped me to improve my mathematical writing in general.

So, thanks very much mathoverflow! I am grateful.

A Theorem About Infinite Cardinals Everybody Should Know

Cantor proved and we all know that for every cardinal  $\kappa$ we have

$2^{\kappa}>{\kappa}.$

This is a very basic fact about cardinal arithmetic and it is nice that the proof works for finite and infinite cardinals equally well. (For the finite case it looks that Cantor’s proof is genuinly different than the ordinary proof by induction.)

Do you know some other results about the arithmetic of infinite cardinals? We know that there are many statement that are independent from ZFC the axioms of sets theory but are there some results which can be proved unconditionally?

Here is a theorem of Shelah. For simplicity we will assume that the special continuum hypothesis $2^{\aleph_0}=\aleph_1$.

Theorem: $\prod_{i-0}^{\infty}\aleph_i$ $<\aleph_{\omega_4}.$

Here $\omega_4$ is the first ordinal which corresponds to $\aleph_4$.

Remark: without assuming the special continuum hypothesis if $2^{\aleph_0}=\aleph_{\alpha}$ then the theorem asserts that $\prod_{i-0}^{\infty}\aleph_{\alpha+i}<\aleph_{\alpha+\omega_4}.$

Want to know more? Read Uri Avraham and Menachem Magidor chapter on Cardinal Arithmetics;