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WHATS_ZFC_DONE_FOR_ME

                                                  September 14, 2020

In a relatively recent issue of Scientific American
(from 2019, I think) I remember seeing a mathematician
comment on ZFC (Zermelo-Fraenkel + Choice) stating
that it was "amazing" what it could do.


So, what has it done?  Has there been any new developments
in math developed starting with ZFC as the "foundations"?

                     
               Some bits from wikipedia on       
               "Zermelo-Fraenkel set theory":      (With my usual idiosyncratic
                                                   paragraphing added.)
               
               [link]   

                        "Gödel's second incompleteness theorem
                        says that a recursively axiomatizable
                        system that can interpret Robinson
                        arithmetic can prove its own consistency
                        only if it is inconsistent."

                        "Moreover, Robinson arithmetic can be
                        interpreted in general set theory, a small
                        fragment of ZFC. Hence the consistency of
                        ZFC cannot be proved within ZFC itself
                        (unless it is actually inconsistent)."

                        "The consistency of ZFC does follow from
                        the existence of a weakly inaccessible
                        cardinal, which is unprovable in ZFC if
                        ZFC is consistent."

                        "Nevertheless, it is deemed unlikely that
                        ZFC harbors an unsuspected contradiction;
                        it is widely believed that if ZFC were
                        inconsistent, that fact would have been
                        uncovered by now."

    I like that bit:

    In place of Russell's dream of an edifice of
    certainty built on a few incontrovertible
    principles with unassailable logic, we have
    this Popperian truth critereon applied to
    math: if it resists falsification after
    sustained inquiry, then we'll assume it's
    probably true.


                        "This much is certain -- ZFC is immune to
                        the classic paradoxes of naive set theory:
                        Russell's paradox, the Burali-Forti paradox,
                        and Cantor's paradox."

                           You'd certainly hope so: it was designed
                           around the need to dodge them.







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