# 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;