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Bhartrhari's Paradox

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Re: Bhartrhari's Paradox

#31

Post by GPerson »

simonh wrote: post #28
pdonis wrote: post #17

Earlier quoted context omitted.

> 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…

Not a mathematician, so this question may be a bit thick. I see the problem with the set of reals > 0, but is it perhaps that in this case > 0 is the problem and for sets specified as >= 0 it's fine because 0 is a nameable real and the smallest element. Obviously you can't just exclude certain expressions arbitrarily though, so I don't know how you could justify that mathematically.

A well ordering on a set is a total order such that all non empty subsets have a minimum element with respect to this order. The standard ordering of the reals is not a well ordering, but the axiom of choice is equivalent to the statement that all sets possess a well-ordering. A well-order of the reals would probably look pretty chaotic though.

Re: Bhartrhari's Paradox

#33

Post by Xcelerate »

Pretty sure ZFC proves such things exist (and that it also can’t pinpoint any individual instances of course). Now, whether syntactical “∃” in the formal language of set theory corresponds to the platonic existence of some “thing”, who knows.

Re: Bhartrhari's Paradox

#34

Post by miksteyp »

b450 wrote: post #21

counterargument: 1) let x be a thing 2) I name x "Jeff" 3) all things are nameable (from 1 and 2) another way to put this is that it's natural to take the paradox as a reductio.

The problem with such sleight of hand counterargument is that you haven't even defined what "a thing" is nor "all things" are in this world. And such discussions will just come back to set theory, ZFC, axiom of choice and real numbers.

Re: Bhartrhari's Paradox

#35

Post by luc4 »

simonh wrote: post #28
pdonis wrote: post #17

Earlier quoted context omitted.

> 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…

Not a mathematician, so this question may be a bit thick. I see the problem with the set of reals > 0, but is it perhaps that in this case > 0 is the problem and for sets specified as >= 0 it's fine because 0 is a nameable real and the smallest element. Obviously you can't just exclude certain expressions arbitrarily though, so I don't know how you could justify that mathematically.

We just used the standard ordering < to define the set, it has nothing to do with the candidate well-ordering. If that's confusing, consider the set { 10^-x | x \in N } instead. It also has no minimum element in the standard ordering.

Re: Bhartrhari's Paradox

#36

Post by amavect »

bryanlarsen wrote: post #30
ngvrnd wrote: post #18

„Wovon man nicht sprechen kann, darüber muss man schweigen“ but if the thing can be interacted with, it can usually be mapped and defined and then named. But there will always be things which do not have names, at least in mathematics -- think of the real numbers.

"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 (constructively), and no unnameable real numbers exist.

Re: Bhartrhari's Paradox

#37

Post by lordnacho »

This reminds me of the 6 degrees of separation thing.

People tell you that you can connect more or less everyone by 6 degrees. But what struck me was, for anyone you try this with, you know their names, so you've already restricted yourself in how far out someone can be.

Re: Bhartrhari's Paradox

#38

Post by svat »

Incidentally, it is a matter of some debate whether Bhartṛhari the philosopher and Bhartṛhari the poet are the same person or two (or more, in the case of the anthology of verses). Oral tradition holds them to be the same person, scholars have debated back and forth. I have a collection of the poems here: https://shreevatsa.net/bhartrhari/web/ (will clean it up someday)

Re: Bhartrhari's Paradox

#39

Post by delecti »

miksteyp wrote: post #34
b450 wrote: post #21

counterargument: 1) let x be a thing 2) I name x "Jeff" 3) all things are nameable (from 1 and 2) another way to put this is that it's natural to take the paradox as a reductio.

The problem with such sleight of hand counterargument is that you haven't even defined what "a thing" is nor "all things" are in this world. And such discussions will just come back to set theory, ZFC, axiom of choice and real numbers.

That isn't a problem with the counterargument, because the "paradox" as-stated also uses the word "thing".

For that matter, the paradox is self-resolving. By labeling the entities it is concerned with as "unnameable things", it has named them. As a collection, entities not otherwise named can be simply referred to as "Bhartrhari's things".

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