> On 15 May 2019, at 02:29, 'Brent Meeker' via Everything List > <[email protected]> wrote: > > > > On 5/14/2019 9:49 AM, Jason Resch wrote: >> 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. > > That's not true. There are axiomatic systems that are complete.
OK. Here are seven one: any RA theory with one axiom deleted can be extended into a complete and decidable theory. Once you have the seven axioms(*), that is no more possible. You can add as many axioms as you want, as long as they are recognisable as axioms, the theory will be incomplete. I guess Jason meant “enough rich theory” or “Turing universal theory”. > >> In other words truth =/= proof, truth is always greater that what can be >> proved. > > That's because you have recourse to an idea of "true" that is outside of > logical inference...such as "empirically true”. Or such as true in the standard model of arithmetic. Bruno > > Brent > > -- > 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] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/b6356661-df48-36eb-2ac6-70430b5b4a5f%40verizon.net > > <https://groups.google.com/d/msgid/everything-list/b6356661-df48-36eb-2ac6-70430b5b4a5f%40verizon.net?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/AF1E11CF-E8D4-4778-8205-F18641109ACB%40ulb.ac.be.
