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Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 11:45 am
by godelian
Skepdick wrote: ↑Thu Apr 10, 2025 10:49 am
Yes, but you were talking about "ineffable theorems", not formalized axioms (such as PA).
I have already quoted Yanovsky's take on the matter. He certainly has a provable point of view, which is simply unobjectionable in terms of classical logic. Of course, he cannot construct such true but ineffable sentence. That is why constructive mathematics is not suitable for dealing with the issue. I have also pointed out that I do not believe that the axiom of choice is sustainable in this context either.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 12:25 pm
by Skepdick
godelian wrote: ↑Thu Apr 10, 2025 11:45 am
I have already quoted Yanovsky's take on the matter. He certainly has a provable point of view, which is simply unobjectionable in terms of classical logic. Of course, he cannot construct such true but ineffable sentence. That is why constructive mathematics is not suitable for dealing with the issue. I have also pointed out that I do not believe that the axiom of choice is sustainable in this context either.
OK, but what makes his English sentences "true" ? To say that an axiomatic system so and so cannot do so and so; is as true as saying that a chicken can't fly to Mars.
Negative statements are constructively provable. That's what constructive negation IS.
A constructive proof for the negation of P you simply have to derive an absurdity from P. Your system can't instantiate absurdities? Too bad!
Limitations in expressing contradictions reflect weaknesses in the expressivity of your formal system, not evidence of ineffable truths.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 12:59 pm
by godelian
Skepdick wrote: ↑Thu Apr 10, 2025 12:25 pm
Limitations in expressing contradictions reflect weaknesses in the expressivity of your formal system, not evidence of ineffable truths.
Imagine that sentence S says that subset K is a subset of the natural numbers ℕ. There are 2^ℵ₀ such true sentences, which is uncountable, while language can only represent ℵ₀ many sentences, which is countable. Hence, most sentences S are true but ineffable. That is the very simple argument that Yanovsky makes.
There is no formal system that can express more than ℵ0 many truths, simply because language cannot do that. So, as soon as there are more truths than ℵ0, the formal system will be unable to express them. Of course, constructive mathematics will claim that a true but ineffable sentence does not exist because we cannot construct it. I reject that point of view. You may be able to count these objects. That is enough proof of their existence, regardless of whether you can also construct them or not.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 1:10 pm
by Skepdick
godelian wrote: ↑Thu Apr 10, 2025 12:59 pm
Skepdick wrote: ↑Thu Apr 10, 2025 12:25 pm
Limitations in expressing contradictions reflect weaknesses in the expressivity of your formal system, not evidence of ineffable truths.
Imagine that sentence S says that subset K is a subset of the natural numbers ℕ. There are 2^ℵ₀ such true sentences, which is uncountable, while language can only represent ℵ₀ many sentences, which is countable. Hence, most sentences S are true but ineffable. That is the very simple argument that Yanovsky makes.
Language can represent whatever you want it to represent. Did that point go over your head?
I am yet to grasp what you mean by an "ineffable sentence". It's not a sentence until it's expressed.
godelian wrote: ↑Thu Apr 10, 2025 12:59 pm
There is no formal system that can express more than ℵ0 many truths, simply because language cannot do that.
So use a better encoding/compression schema.
godelian wrote: ↑Thu Apr 10, 2025 12:59 pm
So, as soon as there are more truths than ℵ0, the formal system will be unable to express them.
That's not true.... use better encoding/compression schema.
godelian wrote: ↑Thu Apr 10, 2025 12:59 pm
Of course, constructive mathematics will claim that a true but ineffable sentence does not exist because we cannot construct it. I reject that point of view. You may be able to count these objects. That is enough proof of their existence, regardless of whether you can also construct them or not.
So you have sentences floating in your head? Unexpressed?
What a weird conception of how language works...
The whole point that para-consistent systems are more expressive than consistent one really is foreign to you.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 1:19 pm
by godelian
Skepdick wrote: ↑Thu Apr 10, 2025 1:10 pm
That's not true.... use better encoding/compression.
Compression does not reduce the number of sentences. At best, it makes a number of these sentences shorter. The maximum number of sentences still remains uncountable infinite.
Compression won't allow you to enumerate the real numbers either. It doesn't matter that some of these real numbers would become shorter.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 1:28 pm
by Skepdick
godelian wrote: ↑Thu Apr 10, 2025 1:19 pm
Compression does not reduce the number of sentences. At best, it makes a number of these sentences shorter. The maximum number of sentences still remains uncountable infinite.
Yes. And? Which one can't you express?
godelian wrote: ↑Thu Apr 10, 2025 1:19 pm
Compression won't allow you to enumerate the real numbers either. It doesn't matter that some of these real numbers would become shorter.
OK... produce a number you can't express.
Ineffability is a self-limiting belief. And the principle of explosion is your prison guard.
https://projecteuclid.org/journals/notr ... 10-005.pdf
The reason why the concept of ‘natural number’ is inherently vague is that a
central feature of it, which would be involved in any characterization of the
concept, is the validity of induction with respect to any well-defined prop-
erty; and the concept of a well-defined property in turn exhibits a particular
variety of inherent vagueness, namely indefinite extensibility. A concept is
indefinitely extensible if, for any definite characterization of it, there is a nat-
ural extension of this characterization, which yields a more inclusive concept;
this extension will be made according to some general principle for gener-
ating such extensions, and, typically, the extended characterization will be
formulated by reference to the previous, unextended characterization. . . . An
example is the concept of ‘ordinal number’. Given any precise specification
of a totality of ordinal numbers, we can always form a conception of an or-
dinal number which is the upper bound of that totality, and hence of a more
extensive totality.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 4:30 pm
by godelian
Skepdick wrote: ↑Thu Apr 10, 2025 1:28 pm
OK... produce a number you can't express.
I can't. Does that mean that uncomputable numbers do not exist?
ChatGPT: Does constructive mathematics deny that uncomputable numbers exist?
In Classical Mathematics:
Uncomputable numbers do exist — for example, by diagonalization, we know that there are real numbers that cannot be computed by any algorithm. The set of computable numbers is countable, while the set of real numbers is uncountable.
In Constructive Mathematics:
The situation is more subtle, because constructive mathematics only accepts the existence of a mathematical object if you can construct it explicitly — typically via an algorithm or a rule that produces it. So, uncomputable numbers do not "exist" in the constructive sense, because you cannot explicitly describe or compute them.
I reject the constructivist take on the matter.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 4:46 pm
by Skepdick
godelian wrote: ↑Thu Apr 10, 2025 4:30 pm
Skepdick wrote: ↑Thu Apr 10, 2025 1:28 pm
OK... produce a number you can't express.
I can't. Does that mean that uncomputable numbers do not exist?
No, it just means their existence is vacuous. Let x be an uncomputable numbers.
Now do something with it.
godelian wrote: ↑Thu Apr 10, 2025 4:30 pm
Uncomputable numbers do exist — for example, by diagonalization, we know that there are real numbers that cannot be computed by any algorithm. The set of computable numbers is countable, while the set of real numbers is uncountable.
That's nonsense. If you can produce a real number via diagonalization (which is an infantry compuational process) you've produced a computable number.
Diagonalization is
precisely how you've computed it.
godelian wrote: ↑Thu Apr 10, 2025 4:30 pm
So, uncomputable numbers do not "exist" in the constructive sense, because you cannot explicitly describe or compute them.
I reject the constructivist take on the matter.
No, you don't. Here's x:UncomputableNumber. It's defined so axiomatically. Like 0 is defined as a number - axiomatically.
Now DO something with it.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 5:00 pm
by godelian
Skepdick wrote: ↑Thu Apr 10, 2025 4:46 pm
No, you don't. Here's x:UncomputableNumber
DO something with it.
I am not interested in doing anything with it. I am perfectly fine with just knowing that it exists.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 5:07 pm
by Skepdick
godelian wrote: ↑Thu Apr 10, 2025 5:00 pm
I am not interested in doing anything with it. I am perfectly fine with just knowing that it exists.
Everything you define into existence exists in Mathematics.
That's why Platonism is trivially stupid.
Re: Torn between classical and constructive mathematics
Posted: Thu Apr 10, 2025 11:10 pm
by godelian
Skepdick wrote: ↑Thu Apr 10, 2025 5:07 pm
godelian wrote: ↑Thu Apr 10, 2025 5:00 pm
I am not interested in doing anything with it. I am perfectly fine with just knowing that it exists.
Everything you define into existence exists in Mathematics.
That's why Platonism is trivially stupid.
ChatGPT: Most real numbers are undescribable and uncomputable
Yes, that's true—and it's a pretty wild fact when you think about it.
Here's the idea: the set of real numbers is uncountably infinite, while the set of describable or computable numbers (like those we can write down or define with a finite algorithm) is only countably infinite. That means the "majority" of real numbers can't be described or computed in any finite way.
According to constructivism, the "majority" of real numbers do not exist. That is a misleading take on the matter. Constructivism is therefore trivially misleading.
Re: Torn between classical and constructive mathematics
Posted: Fri Apr 11, 2025 5:47 am
by Skepdick
godelian wrote: ↑Thu Apr 10, 2025 11:10 pm
According to constructivism, the "majority" of real numbers do not exist. That is a misleading take on the matter. Constructivism is therefore trivially misleading.
Rinse. Repeat. Which real numbers?
According to some constructivists not only do the real numbers exist; they are also countable.
https://arxiv.org/abs/2404.01256
Of course, it's down to what you mean by "THE real numbers" again...
Re: Torn between classical and constructive mathematics
Posted: Fri Apr 11, 2025 5:51 am
by Skepdick
godelian wrote: ↑Thu Apr 10, 2025 11:10 pm
That means the "majority" of real numbers can't be described or computed in any finite way
Obviously! Because finiteness corresponds to termination (coming to an end!) in a computational sense. But who says a computation has to terminate?
We have infinitary processes. We have the process calculus. We have co-induction.
Never mind all that jazz! Who says that a computation even has to START? There's a Turing machine with a particular configuration. Don't turn it on.
Since every real number has an infinite decimal expansion; and every infinite decimal expansion can be encoded on the tape of a Turing machine that NEVER RUNS.
This is such a trivial conjecture (no, I am not going to call it a theorem - it's falsifiable!); that all you have to do to contradict it is to specify a Real number which can't be encoded as an infinite expansion of decimals.
It's basic empiricism.
Re: Torn between classical and constructive mathematics
Posted: Sat Apr 12, 2025 6:10 am
by godelian
I am perfectly fine with the work of Andrej Bauer, in which he builds a countable version of the reals inside the topos of realizability. However, I doubt that his construction is deductively closed. The set of all subsets of countable reals, is uncountable again. Most of these subsets are ineffable again.
The realizability topos looks very much like a chroot environment on Linux:
ChatGPT: Escaping a chroot environment
Escaping a chroot environment can be a serious security concern and typically comes up in contexts like penetration testing, system hardening, or debugging misconfigured environments. If you’re authorized to test or audit a system, here’s a high-level overview of how chroot escapes might happen—and how to prevent them.
How chroot escapes can happen:
1. Privilege Escalation + chroot:
If the user in the chroot has root privileges (UID 0), they can break out using chroot itself:
mkdir /tmp/newroot
chroot /tmp/newroot /bin/sh
If /tmp/newroot is actually a bind-mounted full filesystem or not properly isolated, this can escape.
2. Mount Points & Bind Mounts:
If /proc or /sys are mounted within the chroot, and tools like pivot_root or mount are available, users might be able to remount the real root filesystem.
3. Badly Configured Binaries or Scripts:
If there are suid binaries or interpreters inside the chroot (like bash, python, or perl) that allow command execution outside the jail.
4. Device Nodes:
If /dev exists and includes devices like /dev/mem or /dev/kmem, these can be used to interact with the system memory directly (very old-school and rare on modern hardened systems).
5. Exploiting Kernel Vulnerabilities:
With sufficient privileges, a user might exploit kernel bugs that allow arbitrary code execution outside the chroot.
---
Example (as root user):
If you're already root inside a chroot, this is a basic escape:
mkdir /tmp/real
mount --bind / /tmp/real
chroot /tmp/real /bin/sh
---
How to prevent chroot escapes:
Never run untrusted code as root inside a chroot.
Use containers (LXC, Docker, etc.) or VMs instead of chroot for real isolation.
Do not include dangerous binaries like bash, mount, python, or gcc inside the chroot unless needed.
Use seccomp, namespaces, and AppArmor/SELinux for stronger isolation.
If you're working on a specific setup, I can help review your configuration or show how to properly isolate it. Let me know.
I strongly suspect that it is not too difficult to escape from the realizability topos by using a relatively simple program. Adding a finite number of axioms will never manage to close all the loopholes.
Skepdick wrote: ↑Fri Apr 11, 2025 5:47 am
Of course, it's down to what you mean by "THE real numbers" again...
I am not as proficient in these matters as Andrej Bauer.
He is capable of constructing an alternative version of the reals. I am not. I just make do with the standard classical definitions by Dedekind and Cauchy.
Even though I often acknowledge the concerns voiced in constructive mathematics as valid, I still mostly tick to classical mainstream mathematics.
For example, Godel's and Tarski's work are staunchly classical while I find the constructive alternatives to their work too poor to be particularly enlightening.
Re: Torn between classical and constructive mathematics
Posted: Sat Apr 12, 2025 7:54 am
by Skepdick
godelian wrote: ↑Sat Apr 12, 2025 6:10 am
I am perfectly fine with the work of Andrej Bauer, in which he builds a countable version of the reals inside the topos of realizability. However, I doubt that his construction is deductively closed.
It is deductively closed. Because the internal language of a topos is higher order intuitionistic logic.
But this is; of course; a trivial tautology. The internal logic is exactly that which determines what statements can be proven about objects in the topos.
When I say "it's deductively closed with respect to its internal logic" what I am saying is "you can prove exactly what the system allows you to prove".
The real question isn't whether it's deductively closed (it is by construction/definition), but what specific theorems can be proven within this particular system.
godelian wrote: ↑Sat Apr 12, 2025 6:10 am
The set of all subsets of countable reals, is uncountable again. Most of these subsets are ineffable again.
Yeah; and? You don't even need the set of all subsets. Not all subsets of countable sets are countable.
godelian wrote: ↑Sat Apr 12, 2025 6:10 am
He is capable of constructing an alternative version of the reals. I am not. I just make do with the standard classical definitions by Dedekind and Cauchy.
Did you even read the paper? It's precisely about the countable Dedekind reals.
The real numbers, whatever they are, ought to form a metrically complete
space. Thus we must disqualify the Cauchy reals, because without the axiom
of countable choice we cannot show that a Cauchy sequence of Cauchy reals
has a limit which is a Cauchy real.
If your construction method doesn't guarantee completeness, have you really constructed the real numbers at all?
godelian wrote: ↑Sat Apr 12, 2025 6:10 am
For example, Godel's and Tarski's work are staunchly classical while I find the constructive alternatives to their work too poor to be particularly enlightening.
This is incoherent. The construction of an abstract object tells you more about the object than a mere classical existence proof.
OK. The real numbers exist. And then? What is their relevant structure? Are they; or aren't they a complete metric space?
Don't just tell me something exists - show me what it is, how it works, and what properties it has.