In Robert Stoll's "Set Theory and Logic", there is the following passage on effectiveness of theorems (p. 375) : Mathematical logicians have shown that for many interesting axiomatic theories ...
Consider the following theorem of Koepke: 'A set x of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of the constructible universe L". Taking ...
You can see how mathematical notation evolved during the last centuries here. I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just ...
For me, facts like the independence of the continuum hypotheses from ZFC cast a doubt on the "law of the excluded middle". (In this context, the doubt is that there might be no "final set theory" such ...
I want to write a philosophical essay centered about Goedel's incompleteness theorem. However I cannot find any real philosophical consequences that I can write more than half a page about. I read the ...
What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math? Would there be any mathematical problems left to solve? What ...
In the comment thread of an answer, I said: The computable numbers are based on the intuitionistic continuum, and are not finitary. To which T.. replied: Computable numbers are not based on ...