> 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:
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.
Ok, I wrote a long reply trying to figure out what exactly 2 is for him in terms of modern set theory notation, but I got really muddled. German’s not my mother-tongue/the pdf isn’t searchable, and when he says “Anzahl”/“Zahl” I’m never sure if he’s talking about size or abstract property, and would have to read a lot more of the book to figure out!
[but naively: in §76 “n folgt in der natürlichen Zahlenreihe unmittelbar auf m.” is defined as “es giebt einen Begriff F und einen unter ihn fallenden Gegenstand x der Art, dass die Anzahl, welche dem Begriffe F zukommt, n ist, und dass die Anzahl, welche dem Begriffe ““unter F fallend aber nicht gleich x” zukommt, m ist””. “ is ‘n follows directly after m’ is defined as: there’s a preposition F and an object satisfying F such that the number of objects satisfying F is n and the number of objects satisfying “satisfying F but not x” is m. This really looks von Neumann like!]
He also talks about numbers in §56 (but Zahlen, not Anzahlen)...with the problem of deciding in Julius Caesar is an (ordinal?) number or not, or whether anything has Julius Caesar as a number.
ALL of this is confusing with what wikipedia says on the matter of defining natural number “Gottlob Frege and Bertrand Russell each proposed defining a natural number n as the collection of all sets with n elements.” - which may also be the case, but lacks a citation alas. https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
I don’t know if I have it in me to straighten this all out (that is, to RTFM).
Correct. Frege (and then following him Bertrand Russell in Principia Mathematica) used a definition when a natural number n was the set of all sets with n elements. More info and contrast with the (later chronologically) Von Neumann definition: https://en.m.wikipedia.org/wiki/Set-theoretic_definition_of_...
To me it read as though the author cleverly worded it as 'he could do something like this' to separate Frege's idea (theory of sets) from the specific example.
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...