But I think it is a fair representation of Godel's arguments that there are an uncountably infinite number of axioms. — tim wood

Yet the way that truth actually works is that unprovable literally means untrue within any finite formal system such as the BOAK. The whole notion of undecidability is a misconception.

What does the BOAK say about axioms, or absolute presuppositions (aka hinge propositions)? — tim wood

All of the basic facts of the model of the current world are stipulated to be necessarily true, thus are the axioms of BOAK. The only other source of expressions of BOAK are deductions from these axioms.

Possibly false assumptions are not allowed. Only facts and deductions from facts are allowed. If a person disagrees that a cat is an animal then they are wrong in the same sense as disagreeing with arithmetic. 2 + 3 = 5 if you disagree then you are necessarily incorrect.

According to the Curry-Howard model - with which I admit I am completely unfamiliar - what takes the place of the axioms in mathematical science? — alan1000

https://en.wikipedia.org/wiki/Ontology_(information_science) is an inheritance hierarchy {tree of knowledge} model of the current world using Richard Montague meaning postulates of formalized natural language.

↪PL Olcott Yes, I got that, and I concur. — Wayfarer

This is by far the greatest philosophy group that I have ever been in.

↪PL Olcott I believe the first documented instance of Pi is from Babylonian sources, but never mind, the basic point stands. — Wayfarer

My example was to show that mathematical truths are discovered thus not created.

Do you think a further distinction can be made between real and unreal abstractions? — Wayfarer

Coherent versus incoherent.

there are also abstractions that are unreal, meaning they don't refer to anything over and above the content of speech or thought - for example, fictional characters or imaginary numbers. — Wayfarer

Good example now I know exactly what you mean.

I resist the idea that abstractions are the constructions of the mind. — Wayfarer

The way that I address this is that the value of PI was entailed by the concept of round at least at the point in time that the first caveman looked up and saw a round full Moon.

That's not the kind of world that the OP is asking about. It's clearly talking about something like a world of mind-independent material objects. — Michael

You can presume that, yet that was not stated.
I took the question to mean: Of every possibility that can exist are their
any of them where the world can be definitely proven to exist?

If "the world" is construed to include projections from one's own mind
then yes, otherwise no.

It is impossible to detect the difference between a perfect simulation of
reality and reality itself because "perfect simulation" means that no discernable
difference exists.

There are other variations of this same theme. Brain-in-a-vat, et cetera.
In the Matrix Deja Vu indicated a glitch in the matrix, thus not a perfect
simulation. In Hindu Maya the glitch seems to be detectable on the basis
of discovering that things in the world are too closely correlated to one's
thoughts held silently within the mind. In other words one finds that one
is subconsciously controlling aspects of the world.

The pain is conclusive proof that the fist exists.
— PL Olcott

Pain can be caused by things other than fists. — Michael

There are only two aspects to reality
(a) Abstract ideas kept in the mind.
(b) What appears to be physical sensations from the sense organs.
If there are any examples of (b) then this proves that the world exists
even if the world is mere a projection from one's own mind. If your
fist hurts this proves that your fist exists at least as a projection from
your own mind.

If we keep seeing the guy that changes the light bulb of the Sun changing its light bulb then how would we know that we are experiencing reality and not a poor simulation (or vice versa)? — Michael

Not at all. We know that the simulation of a giant star millions of miles away is a very terrible simulation.

The Truman Show
https://www.imdb.com/title/tt14675964/?ref_=nm_flmg_t_36_act
I think that at the end he saw them turn the lights off that were the stars in the sky.

If it was a poor simulation we would never be having this conversation because it would be common knowledge that everyone would know.
— PL Olcott

I don't see how this follows. — Michael

If we keep seeing the guy that changes the light bulb of the Sun changing its light bulb then we would know that the Sun is not a giant star millions of miles away.

We can tell that it is not a poor simulation.
— PL Olcott

How so? Maybe this is exactly what a poor simulation is like. Perhaps in reality grass is red and the Earth has two moons. — Michael

If it was a poor simulation we would never be having this conversation because it would be common knowledge that everyone would know.

Sure. But if one had only ever experienced a poor simulation of reality and never experienced reality then one wouldn't know that one was experiencing a poor simulation of reality and not experiencing reality. — Michael

We can tell that it is not a poor simulation.
Detecting the subtle difference between a very excellent simulation and a perfect one might prove very difficult. If my understanding of Zen Buddhism is correct then this is the primary focus of Zen.

I believe the topic is physical things in the real world.
— Patterner

How do I know that I am perceiving a physical thing in a real world and not just dreaming or hallucinating or being tricked by an evil scientist who has my brain in a vat and is stimulating my visual cortex with nanomachines? — Michael

It is by definition impossible to detect the difference between reality and a perfect simulation of reality.
If the simulation is less than perfect then there may be tell-tale signs.
If (for example) reality is a projection from one's own mind, then one might see signs of this.

When I think about why is it called "apple", how to describe it, what is its nature, why apples exist, etc. I'm getting into concepts. These are not facts. They are subject to interpretation. So, we cannot call them "truth". — Alkis Piskas

Whenever we are dealing with phonetic or symbolic encodings of semantic meanings we are dealing with abstractions. When we are looking directly at an apple the visual sensation of this apple is not an abstraction. The entirety of reality is sensations and abstractions.

The two theories of truth: correspondence deals with sensations and and coherence deals with abstractions. AKA the synthetic versus analytic divide.

That the world exists (an abstract concept) is verified to be true (also an abstract concept) on the basis of anything that appears to be any physical sensation (not merely an abstract concept).
— PL Olcott

But what isn't verified is that there is more to the world than those physical sensations. — Michael

That is outside of the scope of the original question.
In one sense or another the world <is> proved to definitely exist.

The pain is conclusive proof that the fist exists.
— PL Olcott

Pain can be caused by things other than fists. — Michael

That the world exists (an abstract concept) is verified to be true (also an abstract concept) on the basis of anything that appears to be any physical sensation (not merely an abstract concept).

Truth itself <is> purely conceptual.

The relevance of the ‘reality’ of the existence of such a fist does naught to reduce sensation of pain. — I like sushi

The pain is conclusive proof that the fist exists.

OK. But I think you are stretching the issue or digging into it too much and that you are getting too conceptual about it, — Alkis Piskas

Categorically exhaustive reasoning
The only correct path to truth is to consider every possibility categorically. By doing this categorically we compress an infinite list of possibilities into a finite sequence of short lists of categories. — PL Olcott

I’ve adjusted my response: you are correct in that there is no reason to believe in the existence of the world when not perceived, — Mww

Yes because within the hypothesis that the world is a projection from one's own mind it does actually cease to exist while no longer perceived.

But isn't that a case of solipsism? Does it mean that someone who lost sensibility in his sense organ has no world? Therefore he doesn't have the world, but also without the world, he doesn't exist anymore in the world? — Corvus

[Reason for believing in the existence of the world]
When we look at the most extreme of all possibilities: AKA solipsism, and we confirm that even in this case the world does exist, then we know that the world does definitely exist.

Can you define your concept of the world? For instance, what colour is the world? — Corvus

"The world" is simply every direct experience of what appears to be any physical sensation from any sense organ. This is opposed to and contrast with purely analytical knowledge held within the mind.

The question should be rather posed the other way around: Is there a reason why not to believe in the existence of the cup anymore? — Alkis Piskas

If all of what seems to be physically manifest reality is actually merely a projection from one's own mind then when the perception of an object ceases to exist the object also ceases to exist because its only existence was one's perception of it.

Categorically exhaustive reasoning
The only correct path to truth is to consider every possibility categorically. By doing this categorically we compress an infinite list of possibilities into a finite sequence of short lists of categories.

The world definitely exists at least as a projection (of what at least appears physical sensations) from one's own mind. The world may have never existed physically. It may be the case that when you close your eyes everything that you were "seeing" ceases to exist until you open your eyes again.
8 hours ago
— PL Olcott
Sounds like a case of Immaterial idealism. Could it be a Berkelean? — Corvus

I always come up with all of these things on my own from scratch. I am merely using my own system of categorically exhaustive reasoning to examine the boundary conditions of the problem.

Of every category that can possibly be there are no categories where the world does not exist.

The world definitely exists at least as a projection (of what at least appears physical sensations) from one's own mind. The world may have never existed physically. It may be the case that when you close your eyes everything that you were "seeing" ceases to exist until you open your eyes again.

↪Corvus Go back and look again. The Earth has been shown to rotate even when you are asleep. Therefore the earth exists even when you are asleep.

Frankly this thread is a manifestation of ↪Ciceronianus's question concerning affectation. — Banno

This is only true when one assumes that reality is not simply a projection from one's own mind.

It turns out that Heinlein's "fair witness" is the only actually correct way of doing this. While one is perceiving the existence of the world one has complete proof that the world exists at least in the sense of a set of (what at least appears to be) sensory perceptions.

This remains true even if the world never physically existed. When one no longer is perceiving objects, then it would be the case that these objects have utterly ceased to exist in every sense (besides memories of them) when these objects are mere projections from one's own mind.
— PL Olcott

Yeah, this sounds interesting. I will do some reading and search on Heinlein's Fair Witness (never heard of the name before), and have some contemplation on it. Will get back to you if I have any points to discuss or ask.

The only path to the actual truth is to continue to hypothesize possibilities until they are conclusively proven to be definitely false. Both belief and disbelief tend to short-circuit this.
— PL Olcott

Wow, yeah, this is what I believe too. :up: — Corvus

I don't know how to simply upvote your reply.

I don't know how to erase a comment.

It turns out that Heinlein's "fair witness" is the only actually correct way of doing this. While one is perceiving the existence of the world one has complete proof that the world exists at least in the sense of a set of (what at least appears to be) sensory perceptions.

This remains true even if the world never physically existed. When one no longer is perceiving objects, then it would be the case that these objects have utterly ceased to exist in every sense (besides memories of them) when these objects are mere projections from one's own mind.

The only path to the actual truth is to continue to hypothesize possibilities until they are conclusively proven to be definitely false. Both belief and disbelief tend to short-circuit this.

Yes, because there is simply nothing that a round square could be. I think this is the main point after all. — javi2541997

No the actual main point is that the halting problem proofs are incorrect because they require a computer program to provide a correct answer to a self-contradictory question that has no correct answer because it is a self-contradictory question.

When we correct the halting problem so that it is no longer asking a self-contradictory question we get a different answer.

Termination Analyzer H is Not Fooled by Pathological Input D
https://www.researchgate.net/publication/369971402_Termination_Analyzer_H_is_Not_Fooled_by_Pathological_Input_D

Nonetheless, to bake a cake using only house bricks is something which is logically impossible but actually possible. Because depending on the concepts of my - or your - reality, that cake can eventually be cooked using only house bricks. Maybe it is an impossible task for you, but not for me. Agree? — javi2541997

If one can rearrange the molecules of a house brick to become an angel food cake then it is not logically impossible to make an angel food cake from house bricks.

There is nothing that anyone can do to make an object that has four equal length sides and simultaneously has zero equal length sides.

My original example of an impossible task was to bake a perfect angel food cake using only house bricks for ingredients. Someone pointed out the rearranging the molecules of the bricks could make this possible. He suggested that I use the term logically impossible. Then people here complained that they never heard of this and had no idea what it means. The key example that I knew was the existence of a thing that required simultaneous mutually exclusive properties.

Myself included, I am not going to lie to you. What you explain and write in your threads is very interesting, but I admit that I don't usually understand what it really means. — javi2541997

Logically impossible is the maximum of all impossibilities.

Things that the creator of the universe cannot do are logically impossible things. God could make
a real live two-dimensional Bugs Bunny, this only require rewriting the laws of nature. God
cannot make a square circle, because it is contradictory, it must have (mutually exclusive)
properties that it cannot have.

According to this premise, why should we demand from 'God' to make a single geometric object that is entirely a square and, simultaneously, is entirely a circle on the same two-dimensional plane then? — javi2541997

That is not what I meant. I want to define a task that is logically impossible. Most people don't know what logically impossible means.

For people trying to feel smart about arguing that you can square a circle — Vaskane

I have never been talking about that. I am talking about making single geometric object that <is> entirely a square (and thus not a circle) and simultaneously <is> entirely a circle (thus not a square) in the same two-dimensional plane. I wanted to define a task that even God could not do.

You're applying something like Gödel's theorem to something like modal logic. No wonder we can't understand each other. Logic uses a lot of propositions that aren't theorems. The "logical status" of a statement does not need a "complete theorem" in order to be .. a logical conclusion. — L'éléphant

Mathematical Incompleteness determines that a formal system <is> incomplete when-so-ever
WFF x of the language L of a formal system F can neither be proved nor refuted in F.
Tarski even uses the actual Liar Paradox as the key basis of his whole Undefinability Theorem:
(3) x ∉ Provable if and only if x ∈ True. https://liarparadox.org/Tarski_275_276.pdf

Here is the Liar Paradox as a WFF of Minimal Type Theory LP := ~True(LP)
The ":=" operator is like macro substitution and provides the means for an expression to directly refer to its actual self.

Prolog rejects the same expression when encoded in Prolog:
?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

Yet mathematical incompleteness still blames the formal system and not the semantically unsound expression.

Self-contradictory statements are not truth bearers.

If that happens, we don't judge it as incomplete -- we judge it as contingently false in this system, but not in all possible worlds. A proposition is non-contingent only if, necessarily, it cannot be the case (that is, in all possible worlds, it is false). — L'éléphant

That is factually incorrect. As soon as any WFF of any formal system is determined to neither be provable nor refutable in that formal system then that formal system <is> determined to be incomplete.

Gödel himself said that this does include self-contradictory expressions.

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

Antinomy
...term often used in logic and epistemology, when describing a paradox or unresolvable contradiction. https://www.newworldencyclopedia.org/entry/Antinomy

Thank god that "incompleteness" is not accepted as one of the logical status of a statement. — L'éléphant

Incompleteness <is> accepted when any WFF cannot be either proved or refuted within a formal system EVEN IF it cannot be proved or refuted in this formal system because it <is> self-contradictory in this formal system. That seems to be its huge error.

↪PL Olcott The Liar is a bit more involved than just that. There are a wide range of formalisations. — Banno

I created Minimal Type Theory that spits out the directed graph of its own WFF.

This is the only system that I know of where the Liar Paradox can be formalized correctly,
every other system cheats and knowingly formalizes self-reference incorrectly.
https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF

LP := ~True(LP)

Gödel does not use the liar. The sentence of interest is not "This sentence is not true" but "This sentence cannot be proved". — Banno

G := (F ⊬ G)

Prolog rejects both of the above expressions as semantically unsound.
It detects that same cycle in their directed graph that I call pathological
self reference.

Because the Liar Paradox is self-contradictory it cannot be included in formal systems
that require all expressions to be either satisfiable or their negation satisfiable.

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

Antinomy
...term often used in logic and epistemology, when describing a paradox or unresolvable contradiction. https://www.newworldencyclopedia.org/entry/Antinomy

Thus when we plug the formalized {epistemological antinomy} of the Liar Paradox into
a similar undecidability proof, we find that this semantically unsound expression "proves"
that the formal system that contains it is incomplete.