The position of constructive mathematics on the axiom of infinity is outright unsustainable

[Skepdick]

You placed something on the map that is NOT in the territory.
|N| mod 2 ∈ {0,1}

Out of the box, the mod operator is not defined for transfinite numbers. So, evaluating "ℵ₀ mod 2" first requires a legitimate, i.e. consistent, extension of the definition, which may or may not exist.

You are only misunderstanding because you want to misunderstand. Because your religion necessitates you misunderstanding.

No "legitimate", or "consistent" extension is required here.

For as long as a process is running you have 0, 1, 2, 3, ..., n+1, n+3, n+4, ...
For as long as the process is running the toggle is on, off, on, ..., n mod2, (n+1) mod 2, (n+2) mod 2, ...

When the infinite computation completes the process is no longer running (it's no longer IN time).
ALL the natural numbers are now there.And the toggle becomes frozen IN time.
No fancy mathematical extensions are needed - just common sense about what "completion" means.

When the process is running the toggle is toggling. Present continuous tense.
When the process completes the toggle has stopped toggling. Past perfect tense.

This is grammatically and necessarily true irrespective of your semantics.

The toggle is either 1; or it's 0.

[godelian]

For as long as the process is running the toggle is on, off, on, ..., n mod2, (n+1) mod 2, (n+2) mod 2, ...

What you are describing is a physical computation ("toggle"). It won't complete its run through the natural numbers.

[Skepdick]

For as long as the process is running the toggle is on, off, on, ..., n mod2, (n+1) mod 2, (n+2) mod 2, ...

What you are describing is a physical computation ("toggle")

You seem confused about who is describing; and what it is that is being described.
If you don't like the word "toggle" call it a "parity".

I am taking your axioms at face value. A completed (past tense!) infinity is no longer completing!

I am simply accepting that which you are committed to. N as a completed totality. With a parity bit.

it won't complete its run through the natural numbers.

I don't want it to "complete its run through the natural numbers".

Now that the natural numbers are completed (past tense!) the parity is no longer "running through the natural numbers"

[Skepdick]

What you are describing is a physical computation ("toggle"). It won't complete its run through the natural numbers.

You want me to get around your mental block of a "toggle"? Sure...

Lets use a clock. Or a circle! How about both?

0 -> 12 o'clock, 0 degrees, North
1 -> 6 o'clock, 180 degrees, South,
2 -> 12 o'clock, 0 degrees, North,
3 -> 6 o'clock, 180 degrees, South
....

When you have completed infinite increments around the clock does it show 12 o'clock or 6'o clock?

When you have completed infinite half-rotations around the circle are you at the 0 degrees/North; or the 180 degrees/South?

Nothing physical here. Just geometry.

[godelian]

What you are describing is a physical computation ("toggle"). It won't complete its run through the natural numbers.

You want me to get around your mental block of a "toggle"? Sure...

Lets use a clock. Or a circle! How about both?

0 -> 12 o'clock, 0 degrees, North
1 -> 6 o'clock, 180 degrees, South,
2 -> 12 o'clock, 0 degrees, North,
3 -> 6 o'clock, 180 degrees, South
....

When you have completed infinite increments around the clock does it show 12 o'clock or 6'o clock?

When you have completed infinite half-rotations around the circle are you at the 0 degrees/North; or the 180 degrees/South?

Nothing physical here. Just geometry.

By creating an analogy with physical objects, you are engaging in physicalism. When I detect traces of physicalism, I revert to the formalist take on mathematics:

https://en.wikipedia.org/wiki/Formalism ... thematics)

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism is that mathematics is not a body of propositions representing an abstract sector of reality.

According to formalism, mathematical statements are not "about" numbers, sets, triangles, or any other mathematical objects in the way that physical statements are about material objects. Instead, they are purely syntactic expressions—formal strings of symbols manipulated according to explicit rules without inherent meaning.

Davis and Hersh already pointed out that Platonists may retreat to formalism. Whenever any form of physicalism seems to be involved, that is indeed what I do:

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

Philip J. Davis and Reuben Hersh have suggested in their 1999 book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism.

The fact that we believe in the existence of a Platonic realm does not detract from the fact that mathematics must be conducted exclusively by means of meaningless string manipulations only.

Say that lim(x→∞) 1/x = 0. In this case, the physical computations never end, but the platonic ones do. Say that seqfinal expresses for the natural numbers what lim(x→∞) expresses for the real or complex numbers. We can determine by completing the Platonic induction over the natural numbers that seqfinal(1/n) = 0.

There are many functions f(n) for which seqfinal( f(n) ) is decidable. In the case of n mod 2, however, seqfinal(n mod 2) is not decidable.

Just like lim(x→∞)(f(x)) is not decidable for every function f(x) over the real/complex numbers, seqfinal(g(n)) is not decidable for every function g(n) over the natural numbers.

Furthermore, I completely reject your example of a clock because mathematics is not a body of propositions representing an abstract sector of (physical) reality.

[Skepdick]

What you are describing is a physical computation ("toggle"). It won't complete its run through the natural numbers.

You want me to get around your mental block of a "toggle"? Sure...

Lets use a clock. Or a circle! How about both?

0 -> 12 o'clock, 0 degrees, North
1 -> 6 o'clock, 180 degrees, South,
2 -> 12 o'clock, 0 degrees, North,
3 -> 6 o'clock, 180 degrees, South
....

When you have completed infinite increments around the clock does it show 12 o'clock or 6'o clock?

When you have completed infinite half-rotations around the circle are you at the 0 degrees/North; or the 180 degrees/South?

Nothing physical here. Just geometry.

By creating an analogy with physical objects, you are engaging in physicalism. When I detect traces of physicalism, I revert to the formalist take on mathematics:

Can you even smell your own bullshit?

I created an analogy with a geometric object - a circle.
I just gave you different coordinate systems for the circle. Clock, angular and compass.

Are circles physical now?

[godelian]

Can you even smell your own bullshit?

I created an analogy with a geometric object - a circle.
I just gave you different coordinate systems for the circle. Clock, angular and compass.

Are circles physical now?

There is no need for geometry. It can be completely handled by meaningless symbol manipulation. Therefore, it should. I repeat, seqfinal(n mod 2) is not decidable.

[Skepdick]

Can you even smell your own bullshit?

I created an analogy with a geometric object - a circle.
I just gave you different coordinate systems for the circle. Clock, angular and compass.

Are circles physical now?

There is no need for geometry. It can be completely handled by meaningless symbol manipulation. Therefore, it should. I repeat, seqfinal(n mod 2) is not decidable.

Great! Now all you have to explain is the discontinuity.

How did the decidable become undecidable when the process completed?

Why does completion destroy decidability?

If something is decidable at every finite step, what magical transformation makes it undecidable at the limit?

[godelian]

Great! Now all you have to explain is the discontinuity.

How did the decidable become undecidable when the process completed?

Why does completion destroy decidability?

ChatGPT: Why is f(n)=n mod 2 undecidable at infinity?

The function f(n)=n mod  2 is not undecidable at infinity — it's just not defined in the usual way when you try to take the limit as n→∞. Let's clarify what’s going on.

1. What is f(n)=n mod  2?

This is the remainder when n is divided by 2:
f(n)=0 if n is even,
f(n)=1 if n is odd.

So the function simply alternates between 0 and 1 as n increases:
f(1)=1, f(2)=0, f(3)=1, f(4)=0, …

2. What happens as n→∞?

If you're asking:
lim ⁡n→∞(n mod  2),
then this limit does not exist in the classical sense. Here's why:
The values do not converge to a single number.
Instead, they oscillate between 0 and 1 forever.
In technical terms, the sequence f(n)=n mod 2 diverges.

So it's not undecidable, but undefined as a limit, due to oscillation.

3. What do you mean by “undecidable at infinity”?

If you're using “undecidable” in a computability or logic sense (like Gödel or Turing), that’s a separate concept.
A function like f(n)=n mod  2 is computable.
It’s not undecidable in that sense.
The issue is only about limit behavior, which is undefined, not undecidable.

4. If you're asking about limits modulo 2…

You might be trying to define:
lim ⁡n→∞(n mod  2)∈{0,1}.
But this doesn't converge, so you can't assign it a limit in standard analysis or modular arithmetic.
However, there is a set-theoretic analogy (if you're talking about something like ∣N∣mod  2∣N∣mod2), where people ask: is the cardinality of the natural numbers even or odd?
That question is ill-defined — cardinal numbers are not arithmetic quantities in the usual sense, so modulo 2 doesn't apply in a straightforward way.

Summary
f(n)=n mod  2 is well-defined for every n∈N.
Its limit as n→∞ does not exist, due to infinite oscillation.
It's not undecidable, just non-convergent.

It is technically not undecidable. It is also not uncomputable. The reason why the answer is indeterminate is because it is non-convergent and therefore undefined. Undecidable, uncomputable, undefined, inexistent, and non-convergent are considered to be indeterminate error states that are incompatible with each other.

I sometimes use these error levels interchangeably, but in fact, they are not. I should not say "undecidable" when it is "undefined", because they represent a different reason why the evaluation of the expression goes haywire.

It's like when getting a result code $? after running a shell script. Zero means successful execution, while any value between 1 and 255 represents another reason why it went wrong. Sometimes, they collapse the error codes into just a few nondescript ones that reflect the gravity of the issue. Example:

$ man ls

NAME
ls - list directory contents

SYNOPSIS
ls [OPTION]... [FILE]...

DESCRIPTION
List information about the FILEs (the current directory by default).
Sort entries alphabetically if none of -cftuvSUX nor --sort is speci‐
fied.

Mandatory arguments to long options are mandatory for short options
too.

-a, --all
do not ignore entries starting with .

-A, --almost-all
do not list implied . and ..

...

Exit status:
0 if OK,

1 if minor problems (e.g., cannot access subdirectory),

2 if serious trouble (e.g., cannot access command-line argument).

AUTHOR
Written by Richard M. Stallman and David MacKenzie.

REPORTING BUGS
GNU coreutils online help: <https://www.gnu.org/software/coreutils/>
Report any translation bugs to <https://translationproject.org/team/>

So, 0 means "Okay", 1 means "minor problems", and 2 means "serious trouble". So, they collapsed the error codes into two levels of gravity: "not good" and "very bad". Technically, you should not say "very bad" when the problem is merely "not good".

[Skepdick]

Great! Now all you have to explain is the discontinuity.

How did the decidable become undecidable when the process completed?

Why does completion destroy decidability?

ChatGPT: Why is f(n)=n mod 2 undecidable at infinity?

The function f(n)=n mod  2 is not undecidable at infinity — it's just not defined in the usual way when you try to take the limit as n→∞. Let's clarify what’s going on.

1. What is f(n)=n mod  2?

This is the remainder when n is divided by 2:
f(n)=0 if n is even,
f(n)=1 if n is odd.

So the function simply alternates between 0 and 1 as n increases:
f(1)=1, f(2)=0, f(3)=1, f(4)=0, …

2. What happens as n→∞?

If you're asking:
lim ⁡n→∞(n mod  2),
then this limit does not exist in the classical sense. Here's why:
The values do not converge to a single number.
Instead, they oscillate between 0 and 1 forever.
In technical terms, the sequence f(n)=n mod 2 diverges.

So it's not undecidable, but undefined as a limit, due to oscillation.

3. What do you mean by “undecidable at infinity”?

If you're using “undecidable” in a computability or logic sense (like Gödel or Turing), that’s a separate concept.
A function like f(n)=n mod  2 is computable.
It’s not undecidable in that sense.
The issue is only about limit behavior, which is undefined, not undecidable.

4. If you're asking about limits modulo 2…

You might be trying to define:
lim ⁡n→∞(n mod  2)∈{0,1}.
But this doesn't converge, so you can't assign it a limit in standard analysis or modular arithmetic.
However, there is a set-theoretic analogy (if you're talking about something like ∣N∣mod  2∣N∣mod2), where people ask: is the cardinality of the natural numbers even or odd?
That question is ill-defined — cardinal numbers are not arithmetic quantities in the usual sense, so modulo 2 doesn't apply in a straightforward way.

Summary
f(n)=n mod  2 is well-defined for every n∈N.
Its limit as n→∞ does not exist, due to infinite oscillation.
It's not undecidable, just non-convergent.

It is technically not undecidable. It is also not uncomputable. The reason why the answer is indeterminate is because it is non-convergent and therefore undefined. Undecidable, uncomputable, undefined, inexistent, and non-convergent are considered to be indeterminate error states that are incompatible with each other.

I sometimes use these error levels interchangeably, but in fact, they are not. I should not say "undecidable" when it is "undefined", because they represent a different reason why the evaluation of the expression goes haywire.

It's like when getting a result code $? after running a shell script. Zero means successful execution, while any value between 1 and 255 represents another reason why it went wrong. Sometimes, they collapse the error codes into just a few nondescript ones that reflect the gravity of the issue. Example:

$ man ls

NAME
ls - list directory contents

SYNOPSIS
ls [OPTION]... [FILE]...

DESCRIPTION
List information about the FILEs (the current directory by default).
Sort entries alphabetically if none of -cftuvSUX nor --sort is speci‐
fied.

Mandatory arguments to long options are mandatory for short options
too.

-a, --all
do not ignore entries starting with .

-A, --almost-all
do not list implied . and ..

...

Exit status:
0 if OK,

1 if minor problems (e.g., cannot access subdirectory),

2 if serious trouble (e.g., cannot access command-line argument).

AUTHOR
Written by Richard M. Stallman and David MacKenzie.

REPORTING BUGS
GNU coreutils online help: <https://www.gnu.org/software/coreutils/>
Report any translation bugs to <https://translationproject.org/team/>

So, 0 means "Okay", 1 means "minor problems", and 2 means "serious trouble". So, they collapsed the error codes into two levels of gravity: "not good" and "very bad". Technically, you should not say "very bad" when the problem is merely "not good".

Complete non-sequitur.

The series is neither divergent nor convergent.

It is either 0; or 1. At any given point.
It is always-decidable.
Until it isn’t.

I didn’t ask you to calculate the limit.
I asked you why decidability gets lost AT the limit.

any finite subset of N has parity. Why doesn’t N have parity?

What happens at the transition from finite -> infinite that makes parity disappear ?

The formalists are claiming there's a region where fundamental properties just... disappear.

Cardinality 0 - has parity.
Cardinality 1 - has parity.
Cardinality 2 - has parity.
…
Cardinality n - has parity
<WHAT HAPPENED HERE?!?>
Cardinality aleph ℵ₀ - NO parity

[godelian]

I asked you why decidability gets lost AT the limit.

Technically, it is not that there is no algorithm to compute it. So, in terms of hair splitting, it is decidable. The fact that it is not defined in ∞ does not render it undecidable because technically, ∞ is not part of the domain of the function x mod y. ChatGPT is very peculiar about the use of terminology covering "error reasons", i.e. undefined, uncomputable, undecidable, and so on.

any finite subset of N has parity. Why doesn’t N have parity?

Well, 5/x always has a defined result. Why doesn't 5/0 have a defined result? Same question.

What happens at the transition from finite -> infinite that makes parity disappear ?

In the example of 5/x. What happens at the transition to 0 that makes the result disappear? Same question.

[Skepdick]

I asked you why decidability gets lost AT the limit.

Technically, it is not that there is no algorithm to compute it. So, in terms of hair splitting, it is decidable. The fact that it is not defined in ∞ does not render it undecidable because technically, ∞ is not part of the domain of the function x mod y.

Contradiction.

In classical logic decidable and defined are related properties.
if X is decidable then X is defined.

1. |x| is defined/decidable for all sets
2. x mod 2 is defined/decidable for all numbers.
3. Under set theory |ℕ| is defined/decidable as ℵ₀
4. Therefore |ℕ| mod 2 should be defined/decidable by function composition.

Why isn't ℵ₀ in the domain of x mod 2 ?

Well, 5/x always has a defined result. Why doesn't 5/0 have a defined result? Same question.

It's not the same question, BUT the answer to "Why doesn't 5/0 have a defined result?" is because your premise is false.

In the example of 5/x. What happens at the transition to 0 that makes the result disappear? Same question.

The questions are not the same. You've switched from talking about x in ℕ to talking about x in ℝ.

For |x| mod 2 we're in the domain of cardinal arithmetic.
For 5/x we're in the domain of real number arithmetic.

The question is not domain-agnostic.

There's no principled reasons why ℵ₀ mod 2 is be undefined, unlike division by zero.
In fact, if you were a real Platonist you could trivially resolve this by pointing out that ℵ₀ mod 2 is the quintessential Boolean.
In the most literal sense possible: ℵ₀ mod 2 is a boolean with NO value.

ℵ₀ mod 2 ≡ bool x

On Platonic metaphysic a Boolean that is neither 0 nor 1 is a constructed witness to the negation of LEM.

When you bullshit, why do you bullshit?

[godelian]

Why isn't ℵ₀ in the domain of x mod 2 ?

DeepSeek

Cardinal numbers like ℵ₀ do not support arithmetic operations in the same way as integers. For example, division with remainder (as in modulo) is not defined for infinite cardinals.

.

[Skepdick]

Why isn't ℵ₀ in the domain of x mod 2 ?

DeepSeek

Cardinal numbers like ℵ₀ do not support arithmetic operations in the same way as integers.

Really? Please, do tell what kind of 1 is |{0}| if it's not an integer.

For example, division with remainder (as in modulo) is not defined for infinite cardinals.

Rinse. Repeat.

Cardinality 0 - modulo is defined
Cardinality 1 - modulo is defined
Cardinality 2 - modulo is defined
…
Cardinality n - modulo is defined
<WHAT HAPPENED HERE?!?>
Cardinality aleph ℵ₀ - modulo is NOT defined

[godelian]

Why isn't ℵ₀ in the domain of x mod 2 ?

DeepSeek

Cardinal numbers like ℵ₀ do not support arithmetic operations in the same way as integers.

Really? Please, do tell what kind of 1 is |{0}| if it's not an integer.

For example, division with remainder (as in modulo) is not defined for infinite cardinals.

Rinse. Repeat.

Cardinality 0 - modulo is defined
Cardinality 1 - modulo is defined
Cardinality 2 - modulo is defined
…
Cardinality n - modulo is defined
<WHAT HAPPENED HERE?!?>
Cardinality aleph ℵ₀ - modulo is NOT defined

I have no clue. I don't know more about this particular error message than what the various AI agents have compiled from their databases. They merely insist that the other error messages and error reasons such as "undecidable" or "uncomputable" are not applicable to this corner case.