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> When Russell was working on this, set theory was an abstract hobby, not the basis of industrial-strength systems I don’t think that’s true is it? People were

> When Russell was working on this, set theory was an abstract hobby, not the basis of industrial-strength systems I don’t think that’s true is it? People were doing very serious mathematics with sets before Russell. For example Richard Dedekind (finally) provided the rigorous construction of the real numbers using Dedekind cuts (specially defined subsets of the Rationals) in 1858. Russell’s paradox required some rethinking of the axioms of set theory which brought about ZF and ZFC set theory but those weren’t really very different from what came before - just better defined.

seanhunter 3y ago
Just to explain myself perhaps a little better. Set theory since at least the mid 19th century has been fundamental to the basis of mathematics. The natural numbers are defined starting fromthe empty set and a successor operation that given a set a returns a U {a}, then the integers are defined as infinite sets of pairs from the cartesian product of the natural numbers with a particular common property[1], and rationals as infinite sets of pairs of integers[2]. Then Dedekind constructed the reals from the rationals and proved the least upper bound property. His construction is essentially to cut the numberline of the Rationals into two sets with specific properties (but essentially one less than and one greater than or equal to a particular real number, with the lower set having no largest element). There is an excellent pair of youtube videos explaining how number systems are defined using sets https://youtu.be/dKtsjQtigag?si=3JFgz2YTUqB7EAv3 https://youtu.be/dKtsjQtigag?si=3JFgz2YTUqB7EAv3 and https://youtu.be/IzUw53h12wU?si=D0lx1vCiy5FwjlwC https://youtu.be/IzUw53h12wU?si=D0lx1vCiy5FwjlwC [1] Which you could think of as being that everything in the set [(a,b)] has a-b equal to some common integer which is what we think of as that particular integer. So the set [(2,0)] contains (2,0), (4,2) (-4,-6) and so on to infinitiy in both directions and that entire set of pairs we think of as the integer “2” [2] where the common property is a division rather than a subtraction
oglop 3y ago
Dedekinds paper is I believe 1872 actually and this is more in line with the set theory research explosion as cantor had published his results by this point. Plus him and Dedekind were close friends so they would have known of eachother’s research. Russell’s paradox is not until like 1900 in response to I believe Frege’s own publication. This is why Hilbert then includes questions around the foundations of mathematics in his list because of these issues with sets and logic, while also offering up the famous phrase “we shall not be chased from the paradise cantor has created for us”.

seanhunter 3y ago

Just to explain myself perhaps a little better. Set theory since at least the mid 19th century has been fundamental to the basis of mathematics. The natural numbers are defined starting fromthe empty set and a successor operation that given a set a returns a U {a}, then the integers are defined as infinite sets of pairs from the cartesian product of the natural numbers with a particular common property[1], and rationals as infinite sets of pairs of integers[2]. Then Dedekind constructed the reals from the rationals and proved the least upper bound property. His construction is essentially to cut the numberline of the Rationals into two sets with specific properties (but essentially one less than and one greater than or equal to a particular real number, with the lower set having no largest element). There is an excellent pair of youtube videos explaining how number systems are defined using sets https://youtu.be/dKtsjQtigag?si=3JFgz2YTUqB7EAv3 https://youtu.be/dKtsjQtigag?si=3JFgz2YTUqB7EAv3 and https://youtu.be/IzUw53h12wU?si=D0lx1vCiy5FwjlwC https://youtu.be/IzUw53h12wU?si=D0lx1vCiy5FwjlwC [1] Which you could think of as being that everything in the set [(a,b)] has a-b equal to some common integer which is what we think of as that particular integer. So the set [(2,0)] contains (2,0), (4,2) (-4,-6) and so on to infinitiy in both directions and that entire set of pairs we think of as the integer “2” [2] where the common property is a division rather than a subtraction

oglop 3y ago

Dedekinds paper is I believe 1872 actually and this is more in line with the set theory research explosion as cantor had published his results by this point. Plus him and Dedekind were close friends so they would have known of eachother’s research. Russell’s paradox is not until like 1900 in response to I believe Frege’s own publication. This is why Hilbert then includes questions around the foundations of mathematics in his list because of these issues with sets and logic, while also offering up the famous phrase “we shall not be chased from the paradise cantor has created for us”.