On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett <[email protected]> wrote:
> On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <[email protected]> wrote: > >> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <[email protected]> >> wrote: >> >> >>> Are you claiming that there is an objective arithmetical realm that is >>> independent of any set of axioms? >>> >> >> Yes. This is partly why Gödel's result was so shocking, and so important. >> >> >>> And our axiomatisations are attempts to provide a theory of this realm? >>> In which case any particular set of axioms might not be true of "real" >>> mathematics? >>> >> >> It will be either incomplete or inconsistent. >> >> >> >>> Sorry, but that is silly. The realm of integers is completely defined by >>> a set of simple axioms -- there is no arithmetic "reality" beyond this. >>> >>> >> The integers can be defined, but no axiomatic system can prove everything >> that happens to be true about them. This fact is not commonly known and >> appreciated outside of some esoteric branches of mathematics, but it is the >> case. >> > > All that this means is that theorems do not encapsulate all "truth". > Where does truth come from, if not the formalism of the axioms? Do you agree that arithmetical truth has an existence independent of the axiomatic system? > There are syntactically correct statements in the system that are not > theorems, and neither are their negation theorems. > Yes. > Godel's theorem merely shows that some of these statements may be true in > a more general system. > So isn't this like scientific theories attempting to better describe the physical world, with ever more general and more powerful theories? > That does not mean that the integers are not completely defined by some > simple axioms. It means no more than that 'truth' and 'theorem' are not > synonyms. > > I agree with this. Jason -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
