On 1/17/2026 3:46 AM, Mikko wrote:
On 15/01/2026 22:37, olcott wrote:
On 1/15/2026 4:02 AM, Mikko wrote:
On 15/01/2026 07:30, olcott wrote:
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>> and proof depends on truth-conditional semantics; once this is
replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of G||delrCOs proof the notion of >>>>>>> such a gap becomes unfounded.
But that isn't what Incompleteness is about, so you are just
showing your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
G||del constructed a sentence that is correct by the rules of first
order Peano arithmetic
within truth conditional semantics and non-well-founded
in proof theoretic semantics. All of PA can be fully
expressed in proof theoretic semantics. Even G can be
expressed, yet rejected as semantically non-well-founded.
G||del's sentence is a sentence of Peano arithmetic so its primary
meaning is its arithmetic meaning. Peano's postulates fail to
capture all of its arithmetic meaning but it is possible to add
other postulates without introducing inconsistencies to make
G||del's sentence provable in a stronger theory of natural numbers.
Plain PA has no internal notion of truth; any truth
talk is metarCatheoretic.
On 17/01/2026 17:54, olcott wrote:
On 1/17/2026 3:46 AM, Mikko wrote:
On 15/01/2026 22:37, olcott wrote:
On 1/15/2026 4:02 AM, Mikko wrote:
On 15/01/2026 07:30, olcott wrote:
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>>> and proof depends on truth-conditional semantics; once this is >>>>>>>> replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of G||delrCOs proof the notion >>>>>>>> of such a gap becomes unfounded.
But that isn't what Incompleteness is about, so you are just
showing your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis
from truth-conditional semantics to proof-theoretic semantics.
G||del constructed a sentence that is correct by the rules of first
order Peano arithmetic
within truth conditional semantics and non-well-founded
in proof theoretic semantics. All of PA can be fully
expressed in proof theoretic semantics. Even G can be
expressed, yet rejected as semantically non-well-founded.
G||del's sentence is a sentence of Peano arithmetic so its primary
meaning is its arithmetic meaning. Peano's postulates fail to
capture all of its arithmetic meaning but it is possible to add
other postulates without introducing inconsistencies to make
G||del's sentence provable in a stronger theory of natural numbers.
Plain PA has no internal notion of truth; any truth
talk is metarCatheoretic.
Of course. Truth is a meta-theoretic concept. The corresponding concept
about an uninterpreted theory is theorem.
The statement that there is a sentence that is neither provable nor the negation of a provable sentence does not refer to truth.
On 1/18/2026 5:18 AM, Mikko wrote:
On 17/01/2026 17:54, olcott wrote:
On 1/17/2026 3:46 AM, Mikko wrote:
On 15/01/2026 22:37, olcott wrote:
On 1/15/2026 4:02 AM, Mikko wrote:
On 15/01/2026 07:30, olcott wrote:
On 1/14/2026 9:44 PM, Richard Damon wrote:
On 1/14/26 4:36 PM, olcott wrote:
Interpreting incompleteness as a gap between mathematical truth >>>>>>>>> and proof depends on truth-conditional semantics; once this is >>>>>>>>> replaced by proof-theoretic semantics a framework not yet
sufficiently developed at the time of G||delrCOs proof the notion >>>>>>>>> of such a gap becomes unfounded.
But that isn't what Incompleteness is about, so you are just
showing your ignorance of the meaning of words.
You can't just "change" the meaning of truth in a system.
Yet that is what happens when you replace the foundational basis >>>>>>> from truth-conditional semantics to proof-theoretic semantics.
G||del constructed a sentence that is correct by the rules of first >>>>>> order Peano arithmetic
within truth conditional semantics and non-well-founded
in proof theoretic semantics. All of PA can be fully
expressed in proof theoretic semantics. Even G can be
expressed, yet rejected as semantically non-well-founded.
G||del's sentence is a sentence of Peano arithmetic so its primary
meaning is its arithmetic meaning. Peano's postulates fail to
capture all of its arithmetic meaning but it is possible to add
other postulates without introducing inconsistencies to make
G||del's sentence provable in a stronger theory of natural numbers.
Plain PA has no internal notion of truth; any truth
talk is metarCatheoretic.
Of course. Truth is a meta-theoretic concept. The corresponding concept
about an uninterpreted theory is theorem.
The statement that there is a sentence that is neither provable nor the
negation of a provable sentence does not refer to truth.
For nearly a century, discussions of arithmetic have quietly
relied on a fundamental conflation: the idea that
rCLtrue in arithmeticrCY meant rCLtrue in the standard model of rao.rCY
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