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

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.

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

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.

The only other source of expressions of BOAK are deductions from these axioms. — PL Olcott

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

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.

Yet the way that truth actually works — PL Olcott

Great! You know what truth is; I do not (nor anyone else that I know of). What is truth?

And you might consider paying a little more attention to Godel - as well to the definition of any axiom. I.e., axioms are unprovable: does that mean that they're untrue? And are you confusing finite with infinite?

And you might consider paying a little more attention to Godel - as well to the definition of any axiom. I.e., axioms are unprovable: does that mean that they're untrue? And are you confusing finite with infinite? — tim wood

I have been very diligently studying these things for two decades. I have written many papers.
What I mean by axiom is any expression of language that has been stipulated to be true. That cats are animals is stipulated to be true. Infinite proofs do not derive knowledge because they never reach their conclusion.

(a) The BOAK can prove every instance of... pair that cannot be proved making the BOAK complete.
(b) The BOAK cannot prove some instances of... pairs cannot be proved, thus humans have no way to know that they cannot be proved. — PL Olcott

Pair or pairs of what? This is not English.

I wrote that per Godel there are an uncountable infinity of axioms, not that the proofs were of infinite length.

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. — PL Olcott

So, what is truth? A definition please. And how and in what way is "the whole notion of undecidability" a misconception?

The BOAK cannot possibly be incomplete in the Gödel sense. It is either
complete in the Gödel sense or its incompleteness cannot be shown. — PL Olcott

What is it that you imagine completeness/incompleteness to be? Godel demonstrated that for systems at least as strong as arithmetic, complete implies inconsistent, with the consequence that every expression in that system is provable. By constructing his peculiar expression, he showed there were expressions that were unprovable but true in the system, therefore the system being incomplete.

(Demonstrating in passing that if "truth" were definable, then he could create an expression that asserted its own untruth, being then both true and false at the same time.)

In as much as your BOAK is related/limited to what is computable, it seems to me it does not encompass what Godel's proofs encompass, being merely a universe of "safe" expressions. Within such a "sanitized" system I suppose you can say that incompleteness is a misconception, but that seems, then, trivial.

Pair or pairs of what? This is not English. — tim wood

Every expression of the language of BOAK can be proved in BOAK, or it is simply untrue within BOAK.
This is the (expression of language / BOAK formal system) pair.

I wrote that per Godel there are an uncountable infinity of axioms, not that the proofs were of infinite length. — tim wood

This does not apply to BOAK.

What is it that you imagine completeness/incompleteness to be? Godel demonstrated that for systems at least as strong as arithmetic, complete implies inconsistent, with the consequence that every expression in that system is provable. By constructing his peculiar expression, he showed there were expressions that were unprovable but true in the system, therefore the system being incomplete. — tim wood

He never showed that there are expressions that are unprovable but true in the system.
He showed that G is unprovable in F and provable (thus true) in meta-mathematics.

This <is> that way that truth really works:
If there is no truth-maker in formal system F making G true in F then G is simply untrue in F.
We never misconstrue this as F is incomplete.

(Demonstrating in passing that if "truth" were definable, then he could create an expression that asserted its own untruth, being then both true and false at the same time.) — tim wood

It seems utterly ridiculous to me that people still fail to understand that the liar paradox is simply not a truth bearer. Is this sentence true or false: "What time is it?"
(Obviously neither because it is a question and not a statement).

He never showed that there are expressions that are unprovable but true in the system. He showed that G is unprovable in F and provable (thus true) in meta-mathematics. — PL Olcott

My bad, correction accepted.

Every expression of the language of BOAK can be proved in BOAK, or it is simply untrue within BOAK. — PL Olcott

I accept this as a definition. But what is the pair?

You stipulate axioms are true - but obviously not provable. Either there are no axioms in BOAK, or there are unprovable expressions in BOAK. Or perhaps you meant that every expression in BOAK is provable except the axioms. I take it then that every statement in BOAK is either an axiom or the conclusion of a proof in BOAK. Is there a method in BOAK for deciding whether, given an unproved expression, it is true?

Godel's G has nothing to do with the liar paradox beyond a distant resemblance. G, interpreted, says that a certain proposition in his system P is undecidable. And as you correctly point out, the truth of G is demonstrated meta-mathematically. But G is about provability and not about truth. Were truth definable as decidability is, then Godel's G would have said that a certain proposition is not true, and meta-mathematically (maybe) that it is true. The conclusion being that truth is not definable as provability is. That is why I asked you what truth is. I'd like to read what you have to say about it.

All of the basic facts of the model of the current world are stipulated to be necessarily true, thus are the axioms of BOAK. — PL Olcott

OK, this means an uncountable collection of "axioms". How could you organize these axioms in such a fashion they represent a data set in CS? What is axiom #1 ?

Some time back we had a promising theory of everything that started with the premise all facts could be catalogued within a program. But when asked "how?", things began to fade.

I accept this as a definition. But what is the pair? — tim wood

(a) Formal system BOAK (b) Every expression of the language of BOAK can be proved in
(a) and (b) is the pair.

(a) The BOAK can prove every instance of (formal system /expression)
pair that cannot be proved making the BOAK complete. — PL Olcott

You stipulate axioms are true - but obviously not provable. Either there are no axioms in BOAK, or there are unprovable expressions in BOAK. Or perhaps you meant that every expression in BOAK is provable except the axioms. I take it then that every statement in BOAK is either an axiom or the conclusion of a proof in BOAK. Is there a method in BOAK for deciding whether, given an unproved expression, it is true? — tim wood

(1) Cats are animals is an axiom of BOAK.
(2) Any unprovable expressions in the language of BOAK are simply untrue.
Because it is the {body of analytical knowledge} every expression is BOAK is true.
(3) Every axiom is stipulated to be true and every expression deduced from axioms is true.

Mendelson represents theorems this way ⊢C, in other words C is provable from axioms, thus according to Haskell Curry that makes it true. https://www.liarparadox.org/Haskell_Curry_45.pdf

But G is about provability and not about truth. — tim wood

That is its mistake. Every analytic truth must have an analytic truthmaker connection to the formalized set of semantic meanings that make it true. When this provability semantic connection is missing then the expression is simply untrue. That G is unprovable in F merely means that G is untrue in F it does not actually means that F is incomplete.

Since this is generically the way that analytic truth really works mathematics is not free to override this.

OK, this means an uncountable collection of "axioms". How could you organize these axioms in such a fashion they represent a data set in CS? What is axiom #1 ? — jgill

The body of all current analytical general knowledge is not only countable it is finite.

They would be organized as a knowledge ontology inheritance hierarchy.
https://en.wikipedia.org/wiki/Ontology_(information_science)

The Cyc project has already done this on a massive scale.
https://en.wikipedia.org/wiki/Cyc
The Cyc project has {Thing} at the root of its knowledge tree.

Some time back we had a promising theory of everything that started with the premise all facts could be catalogued within a program. But when asked "how?", things began to fade. — jgill

It is basically the model of the current world plugged into a knowledge ontology inheritance hierarchy.

Some time back we had a promising theory of everything that started with the premise all facts could be catalogued within a program. But when asked "how?", things began to fade. — jgill

That’s because somewhere along the line you have things you know without knowing how you know them. We can’t explain explaining. The epistemic buck has to stop somewhere.

The BOAK can prove every instance of (formal system /expression)
pair that cannot be proved making the BOAK complete. — PL Olcott

The BOAK can prove every instance that cannot be proved?

It is possible the BOAK is just a kind of encyclopedia - is that what you're trying to say?

The Cyc project has {Thing} at the root of its knowledge tree — PL Olcott

The objective of the Cyc project was to codify, in machine-usable form, the millions of pieces of knowledge that compose human common sense.

A far cry from listing all facts (axioms).

The body of all current analytical general knowledge is not only countable it is finite. — PL Olcott

So, the system of axioms is constantly increasing. Proof it is finite at a particular time?

What I mean by axiom is any expression of language that has been stipulated to be true — PL Olcott

"True" by what measures? What of potential inferences not realized?

Season's Greetings

The BOAK can prove every instance that cannot be proved?

It is possible the BOAK is just a kind of encyclopedia - is that what you're trying to say? — tim wood

I don't know how I came up with that wording.
Every expression of the BOAK is either stipulated to be true (AKA axioms) or deduced from these axioms. Unprovable simply means untrue, thus undecidable cannot possibly exist.

What I mean by axiom is any expression of language that has been stipulated to be true
— PL Olcott
"True" by what measures? What of potential inferences not realized? — jgill

We know that dogs are animals and thus not fifteen story office buildings only because of the meanings of the words {dog} and {fifteen story office buildings} that have been stipulated. Unless these meanings are stipulated the words {dog} and {fifteen story office buildings} remain meaningless.

The body of analytic truth is only true on the basis of the connection of terms to the meanings that make the expression true. The lack of such a connection simply makes the expression untrue. Undecidability cannot possibly occur.

It is possible the BOAK is just a kind of encyclopedia - is that what you're trying to say? — tim wood

No the BOAK is a knowledge ontology inheritance hierarchy having the same structure as the Cyc project. Unlike a mere dictionary the requires a human mind, the knowledge ontology connections create a human mind. The term {human} may have as many connections as there are millimeters to the nearest star.

What I'm trying to get to is understanding whether the propositions of BOAK are there because they're provable or there only because they have been proved in the sense that a proof of them has been given. E.g. - and possibly not the best example - of "sheep can live on Mars," and, "sheep cannot live on Mars," one of these is true and in theory and in principle provable (maybe depending on what is meant by "live on"). Is either of these in BOAK? If not, why not?

What I'm trying to get to is understanding whether the propositions of BOAK are there because they're provable or there only because they have been proved in the sense that a proof of them has been given. E.g. - and possibly not the best example - of "sheep can live on Mars," and, "sheep cannot live on Mars," — tim wood

All of the general knowledge known to humans derives that {sheep cannot live on Mars} on the basis that sheep must have an atmosphere that Mars does not provide. This same knowledgebase would also know that the atmosphere that sheep require could be artificially provided, thus the question: "Can sheep live on Mars?" is a different question than "Can sheep possibly live on Mars?"

Ok. Is that in the BOAK?

↪PL Olcott Ok. Is that in the BOAK? — tim wood

Every single detail of all of the general knowledge known to humans is in the BOAK. Can sheep live on Mars? is not general knowledge thus must be derived on the basis of general knowledge.

I think that's induction, the logic of which starts with "If.... That is, not proved but granted. Yes? No?

I think that's induction, the logic of which starts with "If.... That is, not proved but granted. Yes? No? — tim wood

It is deduction from true premises. The formal system required must have a human level of comprehension directly hard-wired into it. It must know every slight nuance of the meaning of every word. The word {human} may have a full definition that makes a book light years tall.

Some {humans} go to Universities. {Universities} teach from books. {Books} have knowledge. The knowledge that {Books} have is {the text of every book ever written}.

Nah, induction, or else you're begging the question. That is, it's not an axiom and it's not proved, though it may be generally accepted. But that breaks your BOAK, or seems to.

induction, or else you're begging the question. — tim wood

It in not probable that sheep need air to breath therefore it is not inductive inference. Inductive inference only deals with probabilities it never deals with certainties. The meaning of the words {Sheep} and {Mars} conclusively proves that sheep cannot survive in the Mars atmosphere.

"the conclusion of a deductive argument is certain given the premises are correct; in contrast, the truth of the conclusion of an inductive argument is at best probable, based upon the evidence given."
https://en.wikipedia.org/wiki/Inductive_reasoning

Undecidability cannot possibly occur. — PL Olcott

Not undecidability. Rather potential facts. I know my Corgi could not understand analytic geometry, and that is a general assumption for Corgis. But there is the faint possibility that one will come along and understand the math. Then that fact becomes a member of the set of "axioms" in your system. But it is not at present. Thus your system continues to grow, and with each new axiom there is the possibility of contradicting a previously established axiom. So, when you ponder Godel what system are you talking about?

My question remains: show how exactly all axioms can be listed for reference. What is axiom #1?, #2?, . . .

My question remains: show how exactly all axioms can be listed for reference. What is axiom #1?, #2?, . . — jgill

It is currently known that humans are the only life on the Earth that can understand analytic geometry.
Listing all the axioms make fill a book light years deep. The Cyc project spent 1000 labor years on this.