On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <jasonre...@gmail.com> wrote:
> On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett <bhkellet...@gmail.com> > 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". There are syntactically correct statements in the system that are not theorems, and neither are their negation theorems. Godel's theorem merely shows that some of these statements may be true in a more general system. 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. Bruce > For example: > https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems > > "*Gödel's incompleteness theorems* are two theorems > <https://en.wikipedia.org/wiki/Theorem> of mathematical logic > <https://en.wikipedia.org/wiki/Mathematical_logic> that demonstrate the > inherent limitations of every formal axiomatic system > <https://en.wikipedia.org/wiki/Axiomatic_system> capable of modelling > basic arithmetic <https://en.wikipedia.org/wiki/Arithmetic>. These > results, published by Kurt Gödel > <https://en.wikipedia.org/wiki/Kurt_G%C3%B6del> in 1931, are important > both in mathematical logic and in the philosophy of mathematics > <https://en.wikipedia.org/wiki/Philosophy_of_mathematics>. The theorems > are widely, but not universally, interpreted as showing that Hilbert's > program <https://en.wikipedia.org/wiki/Hilbert%27s_program> to find a > complete and consistent set of axioms > <https://en.wikipedia.org/wiki/Axiom> for all mathematics > <https://en.wikipedia.org/wiki/Mathematics> is impossible." > > > And > https://en.wikipedia.org/wiki/Halting_problem#G%C3%B6del's_incompleteness_theorems > > "Since we know that there cannot be such an algorithm, it follows that > the assumption that there is a consistent and complete axiomatization of > all true first-order logic statements about natural numbers must be false." > > > 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
