When we define formal systems as a finite list of basic facts and allow semantic logical entailment as the only rule of inference we have
systems that can express any truth that can be expressed in language.
Also with such systems Undecidability is impossible. The only
incompleteness are things that are unknown or unknowable.
[ Followup-To: set ]
In comp.theory olcott <[email protected]> wrote:
On 5/5/2025 3:12 PM, Alan Mackenzie wrote:
olcott <[email protected]> wrote:
On 5/5/2025 2:34 PM, Alan Mackenzie wrote:
olcott <[email protected]> wrote:
On 5/5/2025 1:52 PM, Alan Mackenzie wrote:
olcott <[email protected]> wrote:
On 5/5/2025 1:19 PM, Alan Mackenzie wrote:
olcott <[email protected]> wrote:
On 5/5/2025 11:05 AM, Alan Mackenzie wrote:
[ .... ]
Follow the details of the proof of Gödel's Incompleteness >>>>>>>>>>> Theorem, and apply them to your "system". That will give you >>>>>>>>>>> your counter example.
My system does not do "provable" instead it does "provably true".
I don't know anything about your "system" and I don't care. If >>>>>>>>> it's a formal system with anything above minimal capabilities, >>>>>>>>> Gödel's Theorem applies to it, and the "system" will be incomplete >>>>>>>>> (in Gödel's sense).
I reformulate the entire notion of "formal system"
so that undecidability ceases to be possible.
Liar. That is impossible.
[ Irrelevant nonsense snipped. ]
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
Is that too difficult for you?
Not at all. One of the truths you inescapably end up with is Gödel's >>>>> Theorem. Either that, or the system is self-contradictory or too weak >>>>> to do anything at all.
Gödel's theorem cannot possibly be recreated when
True(x) is defined to apply truth preserving
operations to basic facts.
On the contrary, whether or not True(x) can be so defined, Gödel's
theorem cannot be avoided.
[ .... ]
That would appear to be well beyond your level of understanding. You >>>>> ought to show some respect towards those who do understand these things.
I have spent 22 years focusing on pathological self-reference.
My understanding really is deeper.
It might be a little deeper than it was, but that's not saying very much. >>> The concept of proof by contradiction, for example, is way beyond you.
Even the very idea of a mathematical proof, its status, its significance >>> is beyond you. You don't even understand what it is you're lacking.
Those 22 years have been suboptimally spent.
As I said, you ought to show a bit of respect to those who understand
these mathematical things.
So you don't understand that when True(x) is
defined to only apply truth preserving operations
to basic facts that are stipulated to be true
that every input including random gibberish
and self-contradiction IS DECIDABLE AS TRUE OR ~TRUE.
That's like being challenged by a young child to understand some detail
of his newest fantasy. Except you're not a child, and ought to have an adult's sense of proportion and reality, and a sense of your own
limitations. You're lacking these.
On 5/5/2025 4:50 AM, Mikko wrote:
On 2025-05-05 02:23:56 +0000, olcott said:
When we define formal systems as a finite list of basic facts and allow
semantic logical entailment as the only rule of inference we have
systems that can express any truth that can be expressed in language.
Also with such systems Undecidability is impossible. The only
incompleteness are things that are unknown or unknowable.
A formal system has a formal language. Unless the language is too
restricted for most interesting purposes the negation of every
sentence is another sentence. In a consistent system some sentence
is unprovable. If the negation of that system is also unprovable
then the system is incomplete.
My system skips merely "provable" and goes directly to "provably true". Expressions such as "This sentence is not true" and its negation
are not provably true, thus rejected as semantically unsound.
| Sysop: | Keyop |
|---|---|
| Location: | Huddersfield, West Yorkshire, UK |
| Users: | 715 |
| Nodes: | 16 (2 / 14) |
| Uptime: | 47:02:33 |
| Calls: | 12,112 |
| Calls today: | 3 |
| Files: | 15,010 |
| Messages: | 6,518,497 |