On Tue, May 14, 2019 at 6:14 PM Bruce Kellett <[email protected]> wrote:
> On Wed, May 15, 2019 at 2:49 AM Jason Resch <[email protected]> wrote: > >> On Mon, May 13, 2019 at 6:06 PM Bruce Kellett <[email protected]> >> wrote: >> >>> On Tue, May 14, 2019 at 1:50 AM Jason Resch <[email protected]> >>> wrote: >>> >>>> >>>> But then what is arithmetical truth? We have no label for it. It cannot >>>> be derived from or defined by labels. >>>> >>> >>> What is truth? (Pontus Pilate). Arithmetical statements are true if they >>> are theorems derived from the axioms. >>> >> >> This is false. In every consistent system of axioms there are statements >> that are true but cannot be derived from the axioms. In other words truth >> =/= proof, truth is always greater that what can be proved. >> > > Read what I said. "Arithmetical statements are true if they are theorems > derived from the axioms." I think that you might agree with this. > My apologies, I took your "if" for an "if and only if". > But this does not mean that there are not true things that are not > theorems of the given axiom system (if you extend the notion of 'truth') -- > but they are theorems in a more powerful (extended) axiom system. > Incompleteness is only a problem if you think that the axiom system you > started from covers everything. This is not unlike physics -- we start from > some theory and find things that a true but which cannot be derived within > the theory. So we go to a new, extended, better theory. > Yes exactly. > > Incompleteness is not particularly powerful or mysterious. > I think it's a headshot for nominalism, or any theory that holds math is human invented. As you say, we keep having to come up with better systems of axioms. Why would we need to if math is human invented? What drives the selection of new axioms if arithmetical truth is not objective and an object of reality that we study? > > In arithmetic we define the integers. We find that these have properties > that are not fully axiomatizable. So what. We have created a world (model) > which we explore and discover certain things. But we created this world. We > could create any number of worlds -- the world of sets, Riemannian > manifolds, or whatever. These would all have their own incompleteness > results. But they are all created worlds. Once created, they have an > independent existence. But the physical world differs in that we did not > create it -- it is the ultimate given. > It's the most obvious given, after consciousness. But as for ultimate, I'm not so sure. Jason > > Bruce > > -- > 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 view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/CAFxXSLTQ4Y%3DP0R%3D3rb08A%3Di3a9tfO7duHrmE%3DzJf5mXboU_-3w%40mail.gmail.com > <https://groups.google.com/d/msgid/everything-list/CAFxXSLTQ4Y%3DP0R%3D3rb08A%3Di3a9tfO7duHrmE%3DzJf5mXboU_-3w%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > -- 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 view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CA%2BBCJUjJjoT-zMKPh1bJ21Mm-8A3v%3D_MyVh%3DyzKjnC9RNF_eUw%40mail.gmail.com.
