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| sci.logic | Logic -- math, philosophy & computationa | 262,912 messages |
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| Message 262,731 of 262,912 |
| olcott to Richard Damon |
| Re: The Halting Problem asks for too muc |
| 25 Jan 26 14:09:17 |
XPost: sci.math, comp.theory From: polcott333@gmail.com On 1/25/2026 1:57 PM, Richard Damon wrote: > On 1/25/26 2:10 PM, olcott wrote: >> 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. > > Yes, the Natural Numbers are object created within the formal system of > Peano Arithmetic (as one way to define them) and in that system there > are a lot of properties of them that are True (or False). > > If there is a property of them that PA Created that it can't talk about, > that sounds very much like PA is just incomplete in its understanding of > what it does, just by the basic normal definition of incomplete. > > 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. -- Copyright 2026 Olcott |
(c) 1994, bbs@darkrealms.ca