The un-referenceable may, at best, be inferred (although, of course, this statement is absurd in refererring to the un-referenceable).
Would you also say that a lot of mathematics is absurd in this sense? For example, almost all real numbers are un-nameable (because there uncountably many real numbers, but only countably many names you could give a number).
cousin_it is right about the undefinability of definability in ZFC.
Even the undefinable reals are taken to be in the “domain of discourse”. They remain “accessible” in some infinitary sense (e.g. corresponding to some infinite bit string expansion) even if they aren’t finitely definable.
Also, the predicate of being a real number in ZFC is definable. So the real numbers are definable in aggregate.
The physics case is about references like “the objective world” which are singular and aren’t any kind of infinitely long name.
In the case of the reals, a skeptical perspective will note that there exists a countable model of ZFC (by Löwenheim-Skolem), hence reals “could be” a countable set from the perspective of ZFC. Our formal systems may, thus, lack the capacity of referencing truly uncountable sets.
I think Jessica is right on this point. Within a system like ZFC, you can’t define the system’s own definability predicate, so the sentence “there are numbers undefinable in ZFC” can’t even be said, let alone proved. (Which is just as well, since ZFC has a countable model, and even a model whose every member is definable.) The same applies to the system of everything you believe about math, as long as it’s consistent and at least as strong as ZFC.
there are … only countably many names you could give a number.
If we take a name to be any pronounceable string pointing to a specific entity*, then in what way is that set limited? If you construct a list of syllables used for all names, and even limit your search to the items that start “the number”, you can always take an existing number name and append a syllable from that list to create a new name. That’s pretty much how set theory establishes integers, as I understand it.
I think there is a difference between “unnameable” and “unremarkable so far, so nobody’s bothered to name it”, which does describe nearly all numbers.
*This is an extremely narrow definition, but functional for this application. It could be extended to include any reproducible symbol including those that can be pronounced, scribed, or thought.
The existence of uncomputable/unnameable reals is usually asserted on the basis of some kind of continuity or absence of gaps.
Likewise in more concrete cases. If physics is correct in asserting that there is stuff beyond our cosmological horizon,then that is stuff we can never know about.
Would you also say that a lot of mathematics is absurd in this sense? For example, almost all real numbers are un-nameable (because there uncountably many real numbers, but only countably many names you could give a number).
There are a few things to say about this:
cousin_it is right about the undefinability of definability in ZFC.
Even the undefinable reals are taken to be in the “domain of discourse”. They remain “accessible” in some infinitary sense (e.g. corresponding to some infinite bit string expansion) even if they aren’t finitely definable.
Also, the predicate of being a real number in ZFC is definable. So the real numbers are definable in aggregate.
The physics case is about references like “the objective world” which are singular and aren’t any kind of infinitely long name.
In the case of the reals, a skeptical perspective will note that there exists a countable model of ZFC (by Löwenheim-Skolem), hence reals “could be” a countable set from the perspective of ZFC. Our formal systems may, thus, lack the capacity of referencing truly uncountable sets.
I think Jessica is right on this point. Within a system like ZFC, you can’t define the system’s own definability predicate, so the sentence “there are numbers undefinable in ZFC” can’t even be said, let alone proved. (Which is just as well, since ZFC has a countable model, and even a model whose every member is definable.) The same applies to the system of everything you believe about math, as long as it’s consistent and at least as strong as ZFC.
If we take a name to be any pronounceable string pointing to a specific entity*, then in what way is that set limited? If you construct a list of syllables used for all names, and even limit your search to the items that start “the number”, you can always take an existing number name and append a syllable from that list to create a new name. That’s pretty much how set theory establishes integers, as I understand it.
I think there is a difference between “unnameable” and “unremarkable so far, so nobody’s bothered to name it”, which does describe nearly all numbers.
*This is an extremely narrow definition, but functional for this application. It could be extended to include any reproducible symbol including those that can be pronounced, scribed, or thought.
Countable doesn’t mean finite. See https://en.wikipedia.org/wiki/Countable_set
The existence of uncomputable/unnameable reals is usually asserted on the basis of some kind of continuity or absence of gaps.
Likewise in more concrete cases. If physics is correct in asserting that there is stuff beyond our cosmological horizon,then that is stuff we can never know about.