Forums before death by AOL, social media and spammers... "We can't have nice things"
| sci.logic | Logic -- math, philosophy & computationa | 262,912 messages |
[ << oldest | < older | list | newer > | newest >> ]
| Message 262,730 of 262,912 |
| olcott to Richard Damon |
| Re: The Halting Problem asks for too muc |
| 25 Jan 26 13:10:56 |
XPost: sci.math, comp.theory From: polcott333@gmail.com On 1/25/2026 12:40 PM, Richard Damon wrote: > On 1/25/26 1:33 PM, olcott wrote: >> On 1/25/2026 12:27 PM, Richard Damon wrote: >>> On 1/25/26 8:24 AM, olcott wrote: >>>> On 1/25/2026 5:19 AM, Mikko wrote: >>>>> On 24/01/2026 16:01, olcott wrote: >>>>>> On 1/24/2026 2:20 AM, Mikko wrote: >>>>>>> On 23/01/2026 12:22, olcott wrote: >>>>>>>> On 1/23/2026 3:13 AM, Mikko wrote: >>>>>>>>> On 22/01/2026 18:40, olcott wrote: >>>>>>>>>> On 1/22/2026 2:21 AM, Mikko wrote: >>>>>>>>>>> On 21/01/2026 17:22, olcott wrote: >>>>>>>>>>>> On 1/21/2026 3:03 AM, Mikko wrote: >>>>>>>>>>>>> >>>>>>>>>>>>> No, it hasn't. In the way theories are usually discussed >>>>>>>>>>>>> nothing is >>>>>>>>>>>>> "ture in arithmetic". Every sentence of a first order >>>>>>>>>>>>> theory that >>>>>>>>>>>>> can be proven in the theory is true in every model theory. >>>>>>>>>>>>> Every >>>>>>>>>>>>> sentence of a theory that cannot be proven in the theory is >>>>>>>>>>>>> false >>>>>>>>>>>>> in some model of the theory. >>>>>>>>>>>>> >>>>>>>>>>>>>> only because back then proof theoretic semantics did >>>>>>>>>>>>>> not exist. >>>>>>>>>>>>> >>>>>>>>>>>>> Every interpretation of the theory is a definition of >>>>>>>>>>>>> semantics. >>>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> Meta‑math relations about numbers don’t exist in PA >>>>>>>>>>>> because PA only contains arithmetical relations—addition, >>>>>>>>>>>> multiplication, ordering, primitive‑recursive predicates >>>>>>>>>>>> about numbers themselves—while relations that talk about >>>>>>>>>>>> PA’s own proofs, syntax, or truth conditions live entirely >>>>>>>>>>>> in the meta‑theory; >>>>>>>>>>> >>>>>>>>>>> Methamathematics does not need any other relations between >>>>>>>>>>> numbers >>>>>>>>>>> than what PA has. But relations that map other things to numbers >>>>>>>>>>> can be useful for methamathematical purposes. >>>>>>>>>>> >>>>>>>>>>>> so when someone appeals to a Gödel‑style relation like >>>>>>>>>>>> “n encodes a proof of this very sentence,” they’re >>>>>>>>>>>> invoking a meta‑mathematical predicate that PA cannot >>>>>>>>>>>> internalize, which is exactly why your framework draws >>>>>>>>>>>> a clean boundary between internal proof‑theoretic truth >>>>>>>>>>>> and external model‑theoretic truth. >>>>>>>>>>> >>>>>>>>>>> Anyway, what can be provven that way is true aboout PA. You >>>>>>>>>>> can deny >>>>>>>>>>> the proof but you cannot perform what is meta-provably >>>>>>>>>>> impossible. >>>>>>>>>> >>>>>>>>>> Gödel’s sentence is not “true in arithmetic.” >>>>>>>>>> It is true only in the meta‑theory, under an >>>>>>>>>> external interpretation of PA (typically the >>>>>>>>>> standard model ℕ). Inside PA itself, the sentence >>>>>>>>>> is not a truth‑bearer at all. >>>>>>>>> >>>>>>>>> There is no concept of "truth-bearer" in an uninterpreted >>>>>>>>> theory because >>>>>>>>> there is not concept of "truth". The relevant concept is "sell- >>>>>>>>> formed- >>>>>>>>> formula" and Gödels sentence is one. It may be true or false in an >>>>>>>>> interpretation. >>>>>>> >>>>>>>> There is a >>>>>>>> "true on the basis of meaning expressed in language" >>>>>>>> and I figured out how to make it computable over the >>>>>>>> body of knowledge. >>>>>>> >>>>>>> Except that "true on the basis of meaning expressed in language" is >>>>>>> nmt computable and does not cover all of the body of knowldge. >>>>>> >>>>>> When the basis of "true" is proof theoretic semantics >>>>>> internal to the formal system relative to its own axioms >>>>>> and not truth conditional in a separate model outside >>>>>> of the system undecidability ceases to exist. >>>>> >>>>> No, it does not. It does not matter what you call it, a sentence >>>>> that cannot be neither proven nor disproven is undecidable because >>>>> that is what the word means. An example is Gödel's sentence in >>>>> Peano arithmetics. >>>>> >>>> >>>> When a truth predicate gets the input "What time is?" >>>> this input is rejected as not truth-apt. >>> >>> >>> That fine. >>>> >>>> When PA gets an expression that cannot be proven or >>>> refuted using its own axioms then this expression is >>>> not within its domain. >>>> >>> >>> Then most of Natural Number mathematics is isn't in its domain, >>> >> >> It is what it is. > > But PA was CREATED to allow us to define the Natural Numbers in an > axiomatic way. > Yet only within the actual axioms of PA. >> PA doesn't even know PA until you add a truth predicate. >> When you do add a truth predicate then PA knows PA. If >> you want more than that then meta-math can know "about" PA. >> This is one level of indirect reference away from knowing PA. > > In other words, you world is just inconsistant because it can't handle > itself. > > You just build your logic on equivocations and lies. > > But since PA doesn't have a truth predicate, you can't add it. > > What PA has, if you actually understand it, is that it was built on a > definition of logic that defines truth based on what flows out of the > possible infinite application of its axioms. > > When you try to build with a lessor logic, you don't get a PA that can > do what it needs to, and thus isn't actually an arithmatic. > >> >>> And, you can't KNOW if somehting is a valid question to ask until you >>> know the answer. >>> >>> This makes a fairly worthless domain to learn things in. >>> >>> By your definition, a question like can every even number, greater >>> than 2, be the sum of two prime numbers MIGHT not be within its >>> domain, even though it is purely a question about the capability of >>> numbers. >>> >> >> > -- Copyright 2026 Olcott |
(c) 1994, bbs@darkrealms.ca