Yup, "undefinable" would be a good way to think about it.
Mathematicians say "undefined" because they mean to say that
is undefined within
elementary algebra. However, they see
as definable since they're not constricted to elementary algebra.
If we were to take a more
abstract algebra approach, we could just state
.
And I know what you're thinking: how's that help, right? Well, simple: the
function doesn't return a normal number, but rather some new mathematical entity. This new mathematical entity obeys
,
so it doesn't suffer from the automatic inconsistency that all elementary entities do in the post that I made above.
What other properties does this new mathematical entity have? I dunno, I'm too lazy to make them up right now. And, heck, it's named after you, so
you make them up. You're allowed to make up whatever you want so long as you rigorously avoid contradicting yourself.
To sum this up, yeah, you're right that "undefinable" would make more sense to most folks since we usually think about math in the context of elementary algebra. However, since mathematicians don't see things that way, to them it's moreso "undefined" than "undefinable".
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Separately, I forgot to address your question about how undefined and indeterminate are somewhat the same. Well, in a way, they're actually exact opposites:
- undefined: No elementary description fits.
- indeterminate: The set of elementary descriptions which fit includes non-equivalent members.
However, these two terms are similar in that they refer to cases in which there isn't exactly one known set of equivalent elementary representations.
By which I mean:
- Most elementary entities are defined by only one set of equivalent representations. Examples of representations for :
- Undefined entities have no elementary representations. Examples of representations for :
- [None. can't even represent itself because it leads to a contradiction; you're literally not even allowed to write it! It's meaningless gibberish.].
- Indeterminate entities have more than one possible set of equivalent representations. Examples of possible sets of equivalent representations for :
- Set of representations in which , including:
- Set of representations in which , including:
- Set of representations in which , including:
- [Etc., since can be just about anything in this case.]
To sum this up, "undefined" and "indeterminate" are similar in that they both indicate that there's not exactly one set of equivalent elementary representations. However, "undefined" indicates zero sets while "indeterminate" indicates more than one set.