Philosophy Now Forum

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Stipulating this specification of True and False:
Axiom(1) True(F, x) ↔ (F ⊢ x).
Axiom(2) False(F, x) ↔ (F ⊢ ¬x).

Stipulating that formal systems are Boolean:
Axiom(3) ∀F ∈ Formal_System ∀x ∈ Closed_WFF(F) (True(F,x) ∨ False(F,x))

Within the above stipulations formal proofs to theorem consequences
now express the sound deductive inference model eliminating incompleteness,
undecidability and inconsistency from the notion of formal systems.

Within the above stipulations the following logic sentence:
∃F∃G (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
(1) Asserts there exists expressions G of formal system F that are neither true nor false.
(2) Is decided to be false on the basis of Axiom(3).

Making the following paragraph false:
The first incompleteness theorem states that in any consistent formal system
F within which a certain amount of arithmetic can be carried out, there are
statements of the language of F which can neither be proved nor disproved
in F. (Raatikainen 2018)

You appear to be trying to REPEAT your beliefs regardless of logical scrutiny against your position. While repeating your asserted thesis suffices for some to believe you, it is not an appropriate LOGICAL argument. It is a rhetorical trick that politicians use.

Ignore Tarski or Godel because these require as much depth as their original efforts. If you want to appeal to something specific about some error in reasoning of these people, you need an audience of those already speaking their language. Instead of expecting us to understand you by a prerequisite external investment, you require teaching us specifically your own process of reasoning first with respect to a general audience.

That is, if you must rely on linking us to arguments elsewhere, I question whether you are not using this for some deceptive purposes...like how hackers might lure one to their site to drop some malware.

So I warn you ahead of time that if you are sincere, you should be able to try to relate to people here without necessity to link to external homework. You are repeating these axioms but not explaining them HERE.

What is "true" is merely an agreement between at least two factors. What is "false" is anything else. You can DEFINE what is 'false' to be a symbolic word that refers to "an agreement between at least two factors," in contrary form to how we normally use it. It is not illegal. But it doesn't solve the actual problem of the theorems involved.

The incompleteness theorems all summarily say that you cannot find one machine of reasoning that can (1) solve all problems, and (2) that you can't use any machine of reasoning to PROVE itself. If you are denying this, you are saying that (1) there IS a universal machine that can solve all problems and/or (2) that you CAN have a machine that can prove itself true by insiders without having to go outside!

The problem relates to the same problem in kind of having a real "perpetual machine" that can give us unlimited energy. Such a machine CAN exist but cannot inform us outside of it. Any atom, for instance, is a kind of perpetual machine. But any effort to 'tap' into it requires "poking" it; any energy we get out of it is due to that poke. So if we "peek" some resulting information out of it, it is due to the fact that we "poked" it in the first place.

If you are INSIDE that machine, like our Universe, we are stuck to being unable to proving anything external to it because we DEFINE anything we experience AS what is INSIDE. To determine what is 'from OUTSIDE' would require knowing ALL of the infinite facts inside to eliminate our faith that some information came from outside of our Universe.

You are either being deceptive to us, yourself, or possibly both in some way. I'd give you charity if you weren't continuing to repeat your cause in separate threads and insisting on us to require looking up outside sources for something that you are expected to prove in your efforts here alone.

So, PROVE

(1) that you can prove to us INSIDE this forum ONLY that we can prove everything outside of this forum using some logic you can formally devise in this forum here alone, and

(2)that you can prove your own reasoning 'true' without resort to outside evidence.

THAT would be the true power of proof you require to show the incompleteness theorems are wrong.

I Dare you to try.

Scott Mayers wrote: ↑Fri Apr 19, 2019 5:15 am
What is "true" is merely an agreement between at least two factors. What is "false" is anything else. You can DEFINE what is 'false' to be a symbolic word that refers to "an agreement between at least two factors," in contrary form to how we normally use it. It is not illegal. But it doesn't solve the actual problem of the theorems involved.

My two key reviewers refute my work on the basis that True ↔ False.
Does that seem reasonable to you?

I can't even move on to the very first step of explaining any of
my material when people refuse to even agree that True IS NOT False.

When we stipulate this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Then we can redefine formal proof to theorem consequences such that it
expresses the sound deductive inference model thereby making incompleteness
and inconsistency impossible.

In the sound deductive inference model:
If its not provable from true premises then its not true.
If its not refutable from true premises then its not false.

Undecidable sentences such as this one: G ↔ (F ⊬ G)
are unsound and nothing more.

Scott Mayers wrote: ↑Fri Apr 19, 2019 5:28 am So, PROVE

(1) that you can prove to us INSIDE this forum ONLY that we can prove everything outside of this forum using some logic you can formally devise in this forum here alone, and

(2)that you can prove your own reasoning 'true' without resort to outside evidence.

THAT would be the true power of proof you require to show the incompleteness theorems are wrong.

I Dare you to try.

I just did that.

In the sound deductive inference model
provability** and truth are the same making it impossible for provability
and truth to diverge. **(a connected chain of deductive inference steps
from true premises to a true conclusion).

I show how to transform formal proof into the sound deductive inference model.

PeteOlcott wrote: ↑Fri Apr 19, 2019 5:32 am

Scott Mayers wrote: ↑Fri Apr 19, 2019 5:15 am
What is "true" is merely an agreement between at least two factors. What is "false" is anything else. You can DEFINE what is 'false' to be a symbolic word that refers to "an agreement between at least two factors," in contrary form to how we normally use it. It is not illegal. But it doesn't solve the actual problem of the theorems involved.

My two key reviewers refute my work on the basis that True ↔ False.
Does that seem reasonable to you?

First off, complements are exclusive but negations alone are only a PART of the domain of the complement, not the whole of it with respect to reality and why the logic USING only the binary truth values are limited.

So, what is "not true" is certainly "some of what is non-true" but not necessarily any specific part of what is "some of all that is non-true". So something 'false' is the whole domain of the complement of what is 'not-true' only when or where you define the binary operations of not-true == false.

The incompleteness theorems are speaking about what a SINGLE CLOSED system of logic can 'confirm' about itself or other systems of 'logic' based upon the binary truth value systems. But not all systems of reasoning are "logically" restricted to reality without being inclusive of all that is also relatively "illogical" because reasoning is a broader category of what 'reasoning' involves.

Since absolutely everything in Totality may be 'true', what is 'false' would be a sub-domain of what is 'true'. I share this interpretation but necessarily makes nothing 'false' of the whole.

The reasons the incompleteness theorems hold are due to them attempting to speak about absolute truth when expecting what is 'not-true' to lie outside of the domain of any reasoning.

Another way to think of this is that when we 'posit' something as true or false, we are pointing specifically to some concept. If the domain is larger than the binary possibilities, then what is 'not-true' is not necessarily 'false'. The logicians of the past recognized this and treat logical evaluations greater than the strict binary realities of 'true' xor 'false' as "trivial" (Tri-vial), meaning something with three or more values literally.

So you ARE correct that in totality there is completeness and consistency. But it is of a system of rationale that is inclusive of all possibilities in Totality as a whole. As such, any system that is absolutely universal is necessarily "trivial"....has more than the mere logical binary truth values in an absolute sense.

In essence, what is 'true' about Totality == what is 'true' AND what is 'false' which is summarized by the simplistic definition of the Boolean constants:

0 = 0 and 1

which is equivalent to

Nothing == Nothing and Anything.

When extended to something greater than some functioning logic system, this is a random system with respect to the whole of reality because we could always find everything 'true'. It is inclusive but something we are unable to call, 'logical' because it would be like having some more complete calculator such that for any input gives you an infinite set of outputs.

I can't even move on to the very first step of explaining any of
my material when people refuse to even agree that True IS NOT False.

What is true is only "not-false" when we look at exclusive realities of something posited. What you likely mean is that when something is 'posited', whatever value you posited is not-(not-posited). If you posit something though, it doesn't mean that what is not-posited is not-able-to-be-posited.

The words relating to this is the root term, "pose". If I pose something, I am presenting something very real. But the act of doing so doesn't mean that what I have not-posed is not real. So what is real about what is not-posed is NOT equal to what is real about what IS posed.

I think you are confusing the domain of the logical theorems of the incompleteness theorems. They are based on assigning a complement of what some premise is 'posed' as being exclusive of what is not 'posed'.

The root also relates to the terms, "possible". So what is posed is obviously 'possible'. But it cannot impose what impossible without literally 'imposing' the position of what is not true.

I know this is confusing language but it means that what is outside of what we present as 'true', cannot speak of what is 'false'. If this were the case, if I said it is

TRUE 'that I am alive',

then this would require being 'true' at all times and all places in the whole universe if 'true' always means 'not false'.

PeteOlcott wrote: ↑Fri Apr 19, 2019 6:08 am

Scott Mayers wrote: ↑Fri Apr 19, 2019 5:28 am So, PROVE

(1) that you can prove to us INSIDE this forum ONLY that we can prove everything outside of this forum using some logic you can formally devise in this forum here alone, and

(2)that you can prove your own reasoning 'true' without resort to outside evidence.

THAT would be the true power of proof you require to show the incompleteness theorems are wrong.

I Dare you to try.

I just did that.

In the sound deductive inference model
provability** and truth are the same making it impossible for provability
and truth to diverge. **(a connected chain of deductive inference steps
from true premises to a true conclusion).

I show how to transform formal proof into the sound deductive inference model.

I don't even understand what you are saying here. I too have studied a lot on logic and though I know I lack sharing the same metalanguage studies you have, I find it difficult to grasp what you are countering about those theorems.

I expressed the challenge to not use external reference because this is how the original incompleteness proofs had to do step by step. First, you need to literally PROVE the completeness of your own thinking without reference to outside logic to us as the audience, otherwise you cheat if you expect others to default to some 'trust' about reasoning beyond our own capacity to judge it internally.

You need the audience to accept your reasoning capacity as complete and consistent here first, something I think requires either a lot of set-up depth about some specific logic you can teach here. Otherwise you are left to beg us to agree or disagree on a colloquial level only.

I first recommend that you try to explain what you think the incompleteness theorems are stating first.

In common language, I expressed that the Incompleteness theorems are all summarily expressed as meaning that:

(1) Any machine/mechanical-process that can exhaust some large domain of reality sufficiently, is still not able to solve all problems of a given range of problems that the domain can be extended to cover.

I can have a generalized computer that can solve all kinds of problems based on any set of inputs and outputs finitely. But let's say I want it to also solve the problem of the computer's own electronic faults. I might imagine having a computer that is hooked up with arms and periphery sufficient to repair itself should something fail in its hardware. But what if the failure is in the part of the circuit that controls the repairing periphery itself? Then it requires something external to reboot it. We could then devise an extended machine that tests to see if the machine is 'hanging'. But then this part of its hardware too might have a problem and we'd then need another extended machine to test the hardware of the hardware that tests whether it too is hanging. This logic goes on without CLOSURE. So with respect to the expected ideal machine that can completely solve all of its own problems is never complete. THIS is what this 'theorem' part means to me. It cannot solve all problems of a domain larger than itself.

(2) That no machine can reflexively reconstruct (== prove) itself given only itself.

If, for instance, we had this ideal peripheral system attempt to take its own chips out to observe it, it requires a distinct part of it that can be shut down while it is 'observing' another part of itself. This can be done with parallel co-existing machines -- one that pulls out components of the shut-down part, then have the other do the same for it. These require two machine systems that are distinct though. Thus, if the whole 'logic' of the system has complementary logic systems, it can 'prove' part of itself at one time, but cannot prove its whole system with both parts running simultaneously.

----
Now maybe I'm the one not understanding the theorems. Then I'd need you to explain to me what I'm in error of here.

Scott Mayers wrote: ↑Fri Apr 19, 2019 5:28 am So, PROVE

(1) that you can prove to us INSIDE this forum ONLY that we can prove everything outside of this forum using some logic you can formally devise in this forum here alone, and

(2)that you can prove your own reasoning 'true' without resort to outside evidence.

THAT would be the true power of proof you require to show the incompleteness theorems are wrong.

I Dare you to try.

I totally rewrote this to make it much simpler.