Regarding Russell's response to Gödel's Incompleteness Theorems, there are at least two different opinions. For example in this book chapter it is argued by Francisco Rodriguez-Consuegra that Russell fully understood the Incompleteness Theorems.
user131753
However, the general opinion regarding the issue is that Russell didn't understood the Incompleteness Theorems or at least thought that the Incompleteness Theorems implies that Peano Arithmetic is inconsistent rather than incomplete.
user131753
But historically, this interpretation of Russell's quote (the one @user21820 quotes above) is due to Leon Henkin . Recently I had an extensive discussion regarding this issue with one of my professor. If you want to know about this issue in more detail, I will be glad to tell you about this @DavidReed.
user131753
3:50 AM
Feel free to contribute anything substantial to the ongoing discussion @JosephWeissman.
user131753
4:26 AM
@DavidReed What is the name of book that you bought?
@HWalters I am a logician, so I'm pretty sure I know much more about cardinals and ordinals than that video will explain. Thanks anyway. See the following post for a brief concise sketch of how ordinals and cardinals are built in ZFC, which is the current conventional foundation for modern mathematics, though I myself do not believe ZFC is meaningful.
Counting has two purposes, namely for specifying sizes and indices. These are directly related for finite quantities, because the number of natural numbers (including $0$) less than $n$ (before the position $n$) is $n$. But in set theory, when generalizing to infinite sets these two notions becom...
