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This is remarkably well done. I'm reading Hofstadter's Godel, Escher, Bach and found this at the perfect time. Thanks for the write up!
Man, I just searched about that book, and it is like there is a world I don't know anything about. Would you mind sharing the names of your favorite books?
Something I've always been curious about: does Gödel's theorem imply an infinity of inconsistent statements, or is the "this statement cannot be proven..." statement the only one? If it's the latter, then of what practical significance is the singularity? If a system is incomplete only in that regard, couldn't one redefine incompleteness to exclude it, and render the system complete for all practical purposes?
I don’t know Frege too well, but believe this is due to von Neumann, not Frege:
https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_def...
Something I've always been curious about: does Gödel's theorem imply an infinity of inconsistent statements, or is the "this statement cannot be proven..." statement the only one? If it's the latter, then of what practical significance is the singularity? If a system is incomplete only in that regard, couldn't one redefine incompleteness to exclude it, and render the system complete for all practical purposes?
> There is no set whose cardinality is strictly between that of the integers and the real numbers.
Or to put it another way, there exists no intermediate type of infinity between countable infinities (the set of integers) and uncountable infinities (the set of real numbers).
The CH is independent of ZFC -- both CH and its negation can be included as new axioms to ZFC and both versions are logically consistent if and only if ZFC is -- meaning that being able to prove the CH is an incompleteness in ZFC.
> This proof showed that “1 + 1”, does indeed equal “2”. It took 2 volumes to get here. I know this seems logical to mathematicians, but it feels to me like having to take 2 volumes to prove something than any child knows intuitively is... I don't know what word I am looking for... obsessive?
Something I've always been curious about: does Gödel's theorem imply an infinity of inconsistent statements, or is the "this statement cannot be proven..." statement the only one? If it's the latter, then of what practical significance is the singularity? If a system is incomplete only in that regard, couldn't one redefine incompleteness to exclude it, and render the system complete for all practical purposes?
About avoiding singularities, the GEB book (Godel,Escher,Bach) mentions at the beginning: mathematicians like Russel tried to avoid paradoxes by moving them out of the system, but Godel shown that it doesn't work if you want a complete and consistent system (complete and consistent are technical terms, Wikipedia does a good explanation: https://en.wikipedia.org/wiki/Kurt_Gödel#Incompleteness_theo...).
Also related to your question: https://en.wikipedia.org/wiki/Chaitin%27s_constant
Sorry if I don't get the technical terms right, hopefully someone else in HN can explain this better.
> For example, a gentleman called Frege discovered that he could craft a theory of sets, which could represent just about everything. For numbers, for example, he could do something like this: [ 0 is {}, 1 is {{}}, 2 is { {{}} {} }, etc. ] I don’t know Frege too well, but believe this is due to von Neumann, not Frege: https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_def...
> For example, a gentleman called Frege discovered that he could craft a theory of sets, which could represent just about everything. For numbers, for example, he could do something like this: [ 0 is {}, 1 is {{}}, 2 is { {{}} {} }, etc. ] I don’t know Frege too well, but believe this is due to von Neumann, not Frege: https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_def...
https://ia800207.us.archive.org/22/items/diegrundlagende00fr...
It starts on book page 87, or PDF page 125.
Basically, what he seems to be doing is to define 0 as the number of "everything that is not equal to itself" ("die Anzahl, welche dem Begriffe 'sich selbst ungleich' zukommt"), and 1 to be the number of "everything that is equal to 0" ("die Anzahl, welche dem Begriffe 'gleich 0' zukommt"), etc.