Torn between classical and constructive mathematics

[Skepdick]

Check the wikipedia page for more details: https://en.wikipedia.org/wiki/True_arithmetic

Surely, the wikipedia page has no "ineffable theorem"?
Surely I am asking the person who claims to have such a theorem?

Are you operating within a structure in which x-x is ALWAYS defined; or a structure in which x-x is NEVER defined?

Which of these two distinct structures is "true arithmetic"?

[godelian]

Check the wikipedia page for more details: https://en.wikipedia.org/wiki/True_arithmetic

Surely, the wikipedia page has no "ineffable theorem"?
Surely I am asking the person who claims to have such a theorem?

Yanovsky. Read his paper:

http://www.sci.brooklyn.cuny.edu/~noson ... ovable.pdf

Because the set of subsets of natural numbers is uncountably infinite, and there are many mathematical facts about each subset, the set all mathematical facts is uncountably infinite. In stark contrast, the collection of all properties that can be expressed or described by language is only countably infinite because there is only a countably infinite collection of expressions. All the other properties of natural numbers cannot even be expressed. There are not enough expressions for them. They are beyond language. Let us call facts that we can describe with mathematical expressions “expressible.” The rest of the facts are “inexpressible” (see Figure 2). Some true mathematical facts are expressible while the vast, vast majority of mathematical facts are inexpressible.

Yanovsky uses the term "inexpressible". It means the same as "ineffable".

[Skepdick]

Check the wikipedia page for more details: https://en.wikipedia.org/wiki/True_arithmetic

Surely, the wikipedia page has no "ineffable theorem"?
Surely I am asking the person who claims to have such a theorem?

Yanovsky. Read his paper:

http://www.sci.brooklyn.cuny.edu/~noson ... ovable.pdf

Because the set of subsets of natural numbers is uncountably infinite, and there are many mathematical facts about each subset, the set all mathematical facts is uncountably infinite. In stark contrast, the collection of all properties that can be expressed or described by language is only countably infinite because there is only a countably infinite collection of expressions. All the other properties of natural numbers cannot even be expressed. There are not enough expressions for them. They are beyond language. Let us call facts that we can describe with mathematical expressions “expressible.” The rest of the facts are “inexpressible” (see Figure 2). Some true mathematical facts are expressible while the vast, vast majority of mathematical facts are inexpressible.

Yanovsky uses the term "inexpressible". It means the same as "ineffable".

I don't care what Yanovsky has to say.

To speak of "THE set" (of natural numbers) carries the grammatical implication of there being only one such set; because "THE" is the definite article in English.

So which definite set is THE set of natural numbers?

Is it THE set which contains an additive identity element; or THE set without an additive identity element?

[godelian]

I don't care what Yanovsky has to say.

I am only repeating what he said. I am not going to liberally invent stuff, because then I cannot back it up with an existing publication.

[Skepdick]

I don't care what Yanovsky has to say.

I am only repeating what he said. I am not going to liberally invent stuff, because then I cannot back it up with an existing publication.

So back it up with your own words.

I am not asking you to invent anything - I am asking you to make a choice.

Given your "ineffable theorem" (your intuitive understanding)... which definite set is THE set of natural numbers?

Is it THE set which contains an additive identity element; or THE set without an additive identity element?

As an instance of LEM: Either ∃x,y ∈ ℕ : x + y = y; or ¬∃x,y ∈ ℕ : x + y = y

[godelian]

I don't care what Yanovsky has to say.

I am only repeating what he said. I am not going to liberally invent stuff, because then I cannot back it up with an existing publication.

So back it up with your own words.

I am not asking you to invent anything - I am asking you to make a choice.

Given your "ineffable theorem" (your intuitive understanding)... which definite set is THE set of natural numbers?

Is it THE set which contains an additive identity element; or THE set without an additive identity element?

As an instance of LEM: Either ∃x,y ∈ ℕ : x + y = y; or ¬∃x,y ∈ ℕ : x + y = y

In my opinion, PA could axiomatize one as its first natural number instead of zero. It would impact part of its truth but it would not lead to contradictions.

The algebraic structures that would emerge would definitely be poorer.

[Impenitent]

"Torn between classical and constructive mathematics"

Torn?

Divided...

-Imp

[godelian]

"Torn between classical and constructive mathematics"

Torn?

Divided...

-Imp

Ha! Maybe also a bit multiplied?

[Skepdick]

In my opinion, PA could axiomatize one as its first natural number instead of zero. It would impact part of its truth but it would not lead to contradictions.

The algebraic structures that would emerge would definitely be poorer.

If you want richer structures - why not define arithmetic on Z; or Q; or R?; or C?

[godelian]

If you want richer structures - why not define arithmetic on Z; or Q; or R?; or C?

You can bring back (Z,Q) to N using equivalence classes of (nested) ordered pairs and then somehow fill in the remaining gaps to get (R,C). At the basis, you just have N, which is ultimately the foundation for all computability and of how computers work. With richer structures being existence-dependent on N, they are not the correct starting point. It is PA interpreted by N that is at the core of things.

For example, model theory is mostly about PA (and N). I guess it should also be possible on Z,Q,R, or C, but I have never run into an interesting publication that does that.

[Skepdick]

You can bring back (Z,Q) to N using equivalence classes of (nested) ordered pairs and then somehow fill in the remaining gaps to get (R,C). At the basis, you just have N, which is ultimately the foundation for all computability and of how computers work. With richer structures being existence-dependent on N, they are not the correct starting point. It is PA interpreted by N that is at the core of things.

Yeah... you don't get to jump back into your song&dance.

Which N is "at the basis"? The one indexed at 0; or the one indexed at 1?

If you can't decide...let me tell you about Nullable types...

https://en.wikipedia.org/wiki/Nullable_type

You can't solve the semi-predicate problem in PA. What do you return when I throw 2-3 at your realization (e.g model) of PA?

https://en.wikipedia.org/wiki/Semipredicate_problem

You are stuck in the Platonic latrine of pure mathematics. Realizability is where all this "pure" nonsense falls apart.

https://math.andrej.com/asset/data/c2c.pdf

[godelian]

You are stuck in the Platonic latrine of pure mathematics. Realizability is where all this "pure" nonsense falls apart.

Not all mathematical objects are realizable. In model theory, most are actually not. That does not mean that model theory is not an interesting subject. For example, how do you do compactness theorem or Löwenheim-Skolem theory constructively?

[Skepdick]

Not all mathematical objects are realizable.

Yes, but all constructed mathematical objects are realizable.

In model theory, most are actually not. That does not mean that model theory is not an interesting subject. For example, how do you do compactness theorem or Löwenheim-Skolem theory constructively?

You are completely missing the point. A set of first-order sentences is satisfiable if and only if every finite subset of it has a model.

You can't even tell me if 0 is a subset of of N; or not. And if it is a subset - what is its model?

[godelian]

You can't even tell me if 0 is a subset of of N; or not.

In modern formalizations of PA, 0 is considered an element of N.

[Skepdick]

You can't even tell me if 0 is a subset of of N; or not.

In modern formalizations of PA, 0 is considered an element of N.

Yes, but you were talking about "ineffable theorems", not formalized axioms (such as PA).

Frankly, I have no idea what your point is. You can formalize N as {0,1,2,3,...}; or as {1,2,3,4,...} or as {3,4,5,6, ...}; or as any {n, n+1, n+2, n+3, ...}.
You can even define it as {n-t,..., n-2, n-1, n, n + 1, n+2, ...}.

Those are all isomorphic, with an arbitrary starting element. So I still have no idea what you mean by "the set of natural numbers".

is the initial element different/important in some significant way than all other elements?