"The Tao that can be told is not the eternal Tao"
Bhartrhari's Paradox
51–60 of 84 posts
Re: Bhartrhari's Paradox
#52Earlier quoted context omitted.
The reals can be ordered, just use x < y. I think you mean that if ZFC is true, we could enumerate unnameable reals (choose one with the axiom of choice, remove it, choose another one, etc.), but you could not enumerate them all. But it is true that you could get a "first" unnameable real.
> The reals can be ordered, just use x That ordering is not a well-ordering , which is what the GP specified. A well ordering requires that every non-empty subset has a smallest element. That's not true for the reals ordered by x 0 has no smallest element. No one has explicitly shown that the reals can be well ordered, but it's a consequence of the axiom of choice that every set can be well-ordered. So in ZFC there m…
> no one has found one
More than that: there is no way to build one (assuming the word "build" means some concrete construction), because it's consistent with ZF that the reals admit no well-ordering. Indeed, you can use forcing to construct a model of R in which there is an infinite but Dedekind-finite subset of R; and you can't well-order such a set, because a well-ordering would turn it into an ordinal, and any Dedekind-finite ordinal is finite. You must use some sort of choice principle to construct a well-ordering. (Of course, it's consistent that they can be well-ordered, too, as you say; or e.g. under the hypothesis V=L, where there's even a canonical well-ordering given by the lexicographic well-ordering L comes with.)
Re: Bhartrhari's Paradox
#53https://3quarksdaily.com/3quarksdaily/2014/03/boundaries-and... This is a beautiful article on the subject.
To prove that there are unnamable concepts, he uses cantor's diagonal argument. There are countably infinite names. Any subset of these names is a concept, which is same as the powerset of the set of names, and through cantor's diagonal argument, there are uncountably infinite concepts, most which are not namable.
Yes you can take a specific concept, and name it, but there are uncountably infinitely many, so even with infinite time, you cannot name them all.
Re: Bhartrhari's Paradox
#54How is it a paradox? Isn't this just a proof that there are many unnamed things, but no unnameable ones?
Many ancient paradoxes are not really paradoxes. Zeno's ones are resolved today with infinite series.
But this is a real one. Is it possible to describe an arbitrary real number? Almost all reals are not describable. But you cannot find a single such number.
The 'paradox' is that the search itself is self-failing - a broken strategy. Of course, now we have the language of sets and functions between them and cardinalities and we resolve this for us in a way that is meaningful. But still now you know the 'existence' of this thing? Can't be described.
It's interesting because of the property of creating with finite words universes of infiniteness.
Re: Bhartrhari's Paradox
#55Can't this also be used to justify things that are obviously nonsensical. Like for example: "I possess an immense undetectable sphere. How can I prove this? Well, any proof I offer you would by definition violate the undetectability of the sphere. So there's no way for me to prove it, I guess you'll just have to trust me bro."
Re: Bhartrhari's Paradox
#56Can't this also be used to justify things that are obviously nonsensical. Like for example: "I possess an immense undetectable sphere. How can I prove this? Well, any proof I offer you would by definition violate the undetectability of the sphere. So there's no way for me to prove it, I guess you'll just have to trust me bro."
I love philosophy Calvinball, so I would counter by asserting that undetectable implies no possession, an immediate contradiction. Or go further and assert that undetectable implies nonexistence. We all possess an immense undetectable nonexistent sphere. No bounds on assumptions means I can make up anything to annoy the interlocutor.
So, you're right. This shows why we should use formal math, so we can agree on the result yet bicker about the interpretation. Some folks point to Cantor's diagonalization theorem to show that some unnameable things exist, when the theorem doesn't say that at all.
Re: Bhartrhari's Paradox
#57Earlier quoted context omitted.
> No surjective function exists from definitions to real numbers. I'm not really up on maths so this is possibly a stupid question, but can't any real number be written as an ASCII string, which is basically an integer number, so there is a direct mapping there? Or is it because the ASCII number wouldn't be in order that makes the difference? Or is it that you can't write that mapping as a mathematical function perha…
Actually, and perhaps sadly, I asked an LLM and I understand now. But perhaps that's not such a bad thing that I can get answers to my foolish questions!
It’s not a foolish question, you got to learn about Cantor’s diagonal argument!
Re: Bhartrhari's Paradox
#58If 'it' is unnameable, there is no way to circumscribe or even describe what 'it' is. Even to show that what it refers to is an empty set, we need its description. If we use concepts like intention and extension, we can sketch out four scenarios:
extension, no intension (yes, we can point out things, which we can't describe)
extension, intension (we point out, and we describe)
no extension, intension (Yes, we can imagine and describe things vividly, but no referent in the world. Here, one can say these things exist in a Platonic world, but not the world we live in; this is where numbers, sets, ideas can exist. Neo-Platonism in Philosophy of Mathematics)
no extension, no intension (this paradox falls in this area).
Re: Bhartrhari's Paradox
#59Earlier quoted context omitted.
"Unnameable" and "unnamed" are two different things, in my opinion. Are there real numbers that are unnameable or are those just unnamed? There are some that unnameable with my mathematical understanding, but that's not saying much.
No surjective function exists from names to real numbers (diagonalization). With any naming scheme, some unnamed real numbers always remain. On the other hand, given any real number, I can name it. I'll run out of unique names, since no injective function exists from real numbers to names. So, some unnamed real numbers will always remain (non-constructively), I can make real numbers that escape a naming scheme (const…
What does "naming" mean?
Assigning a symbol? But who said that the set of symbols must be countable?
Re: Bhartrhari's Paradox
#60Earlier quoted context omitted.
You proved that definable implies nameable, and also unnameable implies undefinable. Obviously true. However, the idea of undefinable real numbers closely resembles a modern version of the paradox. No surjective function exists from definitions to real numbers. Really, the blurb about "seems impossible to verify this by giving positive instances" contains the tension between constructive math and non-constructive mat…
> No surjective function exists from definitions to real numbers. I'm not really up on maths so this is possibly a stupid question, but can't any real number be written as an ASCII string, which is basically an integer number, so there is a direct mapping there? Or is it because the ASCII number wouldn't be in order that makes the difference? Or is it that you can't write that mapping as a mathematical function perha…
It's because most real numbers are uncomputable. That means, most of the time, the only way to check that two numbers (i.e. names) are the same is to spend infinite time looking at all their infinite digits.
An unknowable name isn't a very good name, IMHO.