In a different thread you claimed you understood Godel's theory. Here you merely prove you don't.

Suppose — Wallows

No need to suppose, unless you're talking aout something that has nothing to do with Godel.

we have a theory that is incomplete with axioms that are consistent. — Wallows

Not "with," but because - by definition.

Some of these axioms will remain unprovable as long as the theory is incomplete. — Wallows

Bassackwards and nonsensical ("bassackwards" is me just being charitable).

The theory — Wallows

Theory? What theory?

remains incomplete because there will always remain a possible world where some things, mathematical truths, physical laws, could have logically happened otherwise (invoking infinity here). — Wallows

Completely irrlelvant - that is, does not even arrise to the level of being wrong.

Inconsistencies will remain in the theory, and there's no hope of ascertaining how much longer the journey will continue. — Wallows

No temporal nor locational aspect to this.

As long as the theory is consistent, then, one can always add new axioms to the theory to expand its power and magnitude. — Wallows

Again, theory? And do you know what an axiom is?

Is that a coherent explanation for why we still go on or should we stop at some point and realize some deeper truth? — Wallows

I submit that the deeper truth here is that this thread is in terms of its topic incoherent and should stop here!

So, it seems you don't understand Gödel either. That's alright.

@andrewk, care to take a look here, if I went wrong?

Some people wonder to themselves, why did mathematics and science continue despite the findings of Gödel's Incompleteness Theorems. — Wallows

This is the absurd thing with people.

Anything that shows a possible limit or is seen as a limit is thought to be wrong, fatal, an end. As an obstacle.

As if a mathematical/logical theorem would doom science and scientific research.

The same way we have approached paradoxes or antinomies, people have desperately looked for a way to brush them under the carpet and declare them solved.

This is truly the problem.

As if a mathematical/logical theorem would doom science and scientific research. — ssu

It's not an "as if" here. One can always expand Gödel's alphabet to account for more than previously hoped for.

And this process, could, in theory, go on forever.

One can always expand Gödel's alphabet to account for more than previously hoped for.

And this process, could, in theory, go on forever. — Wallows

Honestly, there is something really incredible in the negative self reference, which you find in all incompleteness results. Gödel's incompleteness theorems, Turing's answer to the Entscheidungsproblem, at the most simple version in Cantor's diagonal argument. We simply cannot make the connection to the larger picture, but there surely is one.

We simply don't understand what it actually means. I think this is the most important thing to be discovered (if you can use that word) in mathematics and logic.

In my view, negative self reference shows what limitations subjectivity gives.

The normal response is quietism. I mean with the above logical preponderance, then what's the point of continuing research? Does it all boil down to psychologism?

You know, it's been a burning thought of mine as to why Wittgenstein called Gödel's Incompleteness Theorems as "logical tricks", and I believe the above is the answer why.

Supposedly, Gödel hated Wittgenstein when seeing him at the Vienna Circle.

In my view, negative self reference shows what limitations subjectivity gives. — ssu

Please elaborate.

Let's get "negative self reference" clarified sufficiently first. A set can cover itself infinitely and still have control of the procedure

Leibniz said contingent proofs take an eternity to prove, but not all truths are this way. Maybe there are things math can't touch

Leibniz said contingent proofs take an eternity to prove, but not all truths are this way. Maybe there are things math can't touch — Gregory

Ehh, that's an artifact of assuming that monads (logical atomism?) are logically the simplest possible things. But, that's off-topic, I think?

A set is merely a rationalized image in the imagination which represents one's intellect. It seems to me that we have freedom to do whatever we want with Godels theorem. What kind of cut can negative self reference do to a living consciousness?

It seems to me that we have freedom to do whatever we want with Godels theorem. — Gregory

Ehh, like what? It seems to me that the vast majority of the world decided to ignore it despite it cropping up in Tarski's, Turing's, and other work (diagonal lemma).

What kind of cut can negative self reference do to a living consciousness? — Gregory

Meinong's jungle.

Plato's beard.

I just want to get clear what Godel proved. You can use arithmetic without sets, and this theorem only applies to set theory.

Some people wonder to themselves, why did mathematics and science continue despite the findings of Gödel's Incompleteness Theorems. — Wallows

Because first, Gödelian incompleteness does not apply to physical theories. It applies (loosely speaking) to axiomatic systems of a particular logical structure, that support mathematical induction. No physical theory posits the existence of an infinite set of natural numbers. Incompleteness simply doesn't apply.

Secondly, incompleteness is not a statement about mathematical truth. It's a statement about axiomatic theories. Gödel himself was a Platonist. He was pointing out the limitations of axiomatic theories in discovering mathematical truth. But he did believe that "the truth is out there," as they used to say on the X-Files.

Because first, Gödelian incompleteness does not apply to physical theories. — fishfry

You almost caught me off guard there. I think, there's more to it than meets the eye... Ehem, Platonism? And if not, then why not?

It applies (loosely speaking) to axiomatic systems of a particular logical structure, that support mathematical induction. — fishfry

Yeah, that follows.

No physical theory posits the existence of an infinite set of natural numbers. Incompleteness simply doesn't apply. — fishfry

Uhh, don't you mean non-denumerable? Cantor's program could have been completed, he just assumed that the program would account for everything, where Godel just kinda dashed those hopes. Just saying.

Secondly, incompleteness is not a statement about mathematical truth. It's a statement about axiomatic theories. — fishfry

Which is?

He was pointing out the limitations of axiomatic theories in discovering mathematical truth. — fishfry

Like I said, there's nothing wrong with axioms if they don't assume everything to be a certain case. I'm pretty rough with set theory; but, it just morphed into something else. Or is it really true that his Incompleteness Theorems only apply to set theory?

No physical theory posits the existence of an infinite set of natural numbers. Incompleteness simply doesn't apply.
— fishfry

Uhh, don't you mean non-denumerable? Cantor's program could have been completed, he just assumed that the program would account for everything, where Godel just kinda dashed those hopes. Just saying. — Wallows

Cantor's work is not a physical theory.

Secondly, incompleteness is not a statement about mathematical truth. It's a statement about axiomatic theories.
— fishfry

Which is? — Wallows

That no formal axiomatic theory (that satisfies some key technical assumptions) can express all mathematical truth. Which is the answer to your original question. That's why we keep looking for mathematical truth. Because no formalism can capture it all. So we're never done just because we have a formalism.

That no formal axiomatic theory (that satisfies some key technical assumptions) can express all mathematical truth. Which is the answer to your original question. That's why we keep looking for mathematical truth. Because no formalism can capture it all. So we're never done just because we have a formalism. — fishfry

But, think of it this way. If there exists, a non-denumerably infinite alphabet, then we can enjoy everything there is to say about Cantor's work and program, ya?

But, think of it this way. If there exists, a non-denumerably infinite alphabet, ... — Wallows

In a physical theory?

then we can enjoy everything there is to say about Cantor's work and program, ya? — Wallows

I enjoy Cantor's beautiful work and everything there is to say about it even though it doesn't exist!

Math $\neq$ Physics.

In a physical theory?

Math ≠≠ Physics. — fishfry

I don't know where the certainty in that negation is stemming from. Care to clarify?

I don't know where the certainty in that negation is stemming from. Care to clarify? — Wallows

The discovery of non-Euclidean geometry in the 1840's by Bernhard Riemann and others.

The discovery of individually consistent but mutually inconsistent geometries was the moment we understood that Math $\neq$ Physics. Math alone can never tell us what's true.

Russell put it well.

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

https://www.goodreads.com/quotes/577891-pure-mathematics-consists-entirely-of-assertions-to-the-effect-that

Godel is talking about set theory, that has a physical counter part in eternal time. Aristotle said an infinity of days is only well behaved of there is a timeless person behind it. We are that person to set theory

The normal response is quietism. I mean with the above logical preponderance, then what's the point of continuing research? Does it all boil down to psychologism? — Wallows

When we don't have the answers, it can be so.

You know, it's been a burning thought of mine as to why Wittgenstein called Gödel's Incompleteness Theorems as "logical tricks", and I believe the above is the answer why. — Wallows

Sometimes brilliant minds don't get the point of the other. I don't remember where I read it, but I remember Wittgenstein accusing Gödel of simply finding again the paradox. Yet Russell's paradox is different and Gödel doesn't fall into it. Perhaps someone could view them as "logical tricks". And as one teacher in the university said to me over twenty years ago "from time to time someone attacks Gödel on the basis that it has 'circular reasoning'.

Please elaborate. — Wallows

We make models of reality, for example mathematical formulas that portray some aspect of the complex reality around us. Fine, but the problem of subjectivity comes with when that model itself has an impact on what it's modelling. Then it has to model itself into the model. Now you might argue that this can be still modeled and in many cases it surely can be, but not when the 'correct' answer is something that the model doesn't give.

This isn't a small difficulty. In the social sciences it has been understood for a long time that when studying economics or sociology, the theories themselves have an effect how we picture the World and thus effects our actions, hence theories are also subjective, not only objective. An needless to say how things change in physics when the measurement has an effect on what is measured.

Let's get "negative self reference" clarified sufficiently first. A set can cover itself infinitely and still have control of the procedure — Gregory

Negative self reference is different from ordinary self reference.

Example:

Try writing a response that you never will write.

Are there such responses? Yes. As we have a finite life we surely cannot write all responses, hence there exists those responses. I or Wallows of Fishfry can write responses that you don't write. But for you to write something that you don't write is impossible. Hope you notice the negative self reference.

Godel sets up an impossible scenario without proving it necessary to to go there. Math can do weird things. The jello of non Euclidean geometry is weird. What applies to the universe is impossible to ascertain with one hundred percent certainty.

Godel is talking about set theory, that has a physical counter part in eternal time. — Gregory

Do you have evidence that the universe is eternal? My understanding is that this is an open question in physics, the solution to which would make someone famous. Did Aristotle hold that time is eternal? How would he have known one way or the other?

The referent of the axiom of infinity is the abstract idea of the endless sequence of counting numbers 1, 2, 3, 4, ... but NOT any claim that such a thing has physical existence.

Aristotle thought the universe eternal for no other reason at least than that time seemed more sublime that way. How could time be infinite from counting? How could we be in the middle of time? These questions left Aristotle no choice than to posit a prime mover, and as many as around 60 equal prime movers possibly. This question has to to do wiith how our minds work with infinity. Did Godel prove that the infinity of math is unstable and that nihilism is true? Those are gray oceans

Aristotle to Aquinas to Scotus thought an eternity in the future to be impossible to complete, so how in the past then? Saying "God did it" doesn't explain how it works temporally. Imagine an eternal falling piano. In fact gravity alone seems sufficient to explain it without God, except the infinity part. Perhaps infinity in time is impossible to fathom, because breaking it up, or breaking it down, you have only intermediate moments and no first. Our minds want a well behaved infinity-etermity. I don't see how the liar paradox proves that math is unstable. Truth is unstable, but not math. Ye..

Sorry I just gasped at the comments about "Cantor's program". I meant Hilbert's program. Such a dumb and stupid mistake...

I knew what you meant lol

What's the different between logical atomism and Hilbert's program? I thought they were the same. It seems to me Godel threw the same monkey wrench into the latter the Russell through into the former, although Godel's theorem threw Russell while his own did not. I think it was just more techniquely drawn out, which is why Wittgenstein said it was the same