12

Is it possible for everything that exists to have a definition? I actually started out asking this in the linguistics - semantics stack and was directed here. By definition I mean at least in the dictionary sense, possibly including mathematical/logical statements. If so, would it be possible to "list" the definitions? Obviously you cannot read a infinite set in its entirety but I mean can you list them in the sense that you can list the natural numbers (I'm also interested in the mathematical aspect of this in terms of countable vs uncountable if someone is familiar with that). Would it be necessary that some thing have multiple definitions in order to have at least 1 definition?

I'd imagine something needs clarification here so please tell me if so, I am very interested in the answer.

13
  • 4
    Things don't have a definition, but words do. Do you mean "is it possible to name all things, with an unambiguous definition for each of those names?"
    – armand
    Aug 17 at 1:55
  • 1
    Yes although I'd also be interested in if it would also be possible to do so with ambiguous definitions (both cases) Aug 17 at 3:07
  • 2
    No, unless you count "defining" in a circle or infinite chains that go nowhere. To define something you need to use terms that are already defined, and tracing their definitions you'll have to stop at undefined terms at some point. If you fix some (finitely many) undefined terms and (finitely many) ways of combining them, as is done in formal languages, then yes, all definable terms can be enumerated by natural numbers and the total is countable. See definable real numbers as an example.
    – Conifold
    Aug 17 at 9:28
  • 7
    What do you mean by "definition"?
    – orome
    Aug 17 at 19:54
  • 4
    For that matter, what do you mean by “thing”?
    – Marq
    Aug 18 at 1:32
10

Here's some mathematical commentary which if nothing else illustrates some of the subtleties around making your question precise:

In the usual context of first-order logic and model theory, say that a structure 𝔄 is pointwise-definable iff for each element a of 𝔄 there is a first-order formula P without parameters uniquely identifying a in 𝔄. For example, the structure of the natural numbers with the usual ordering is pointwise-definable while the structure of the rational numbers with the usual ordering isn't.

Given that ZF proves that there are uncountably many reals, it's natural to assume that ZF can't have any pointwise-definable models. Hamkins/Linetsky/Reitz, however, showed that this is false - there are indeed pointwise-definable models of ZF (and many of its various extensions)! At first glance this may appear to contradict the obvious fact that ZF proves that the set of first-order formulas in set theory is countable, but this is not the case - the saving grace is the limitation imposed by Tarski's undefinability theorem.

In a bit more detail: ZF proves that, for example, there are uncountably many reals. Consequently if 𝔄 is a model of ZF, then in 𝔄 there is no bijection between what 𝔄 thinks is ℝ and what 𝔄 thinks is ℕ. This doesn't rule out the existence of "external bijections," though. Already the downward Lowenheim-Skolem theorem provides a strong positive result here: there are countable models of ZF, assuming of course that ZF is consistent in the first place (this was originally called "Skolem's paradox"). The existence of pointwise-definable models of ZF is just a further elaboration of the same internal/external nuance. Specifically, pointwise-definability of a model 𝔄 of ZF provides a surjection F from {formulas in set theory} to 𝔄 (send each definition to the thing it defines). However, as with Lowenheim-Skolem this is only a problem if this surjection F lies in 𝔄 itself - and Tarski's undefinability theorem shows that we can't inside 𝔄 figure out how to assign each definition to the thing it defines in an "all-at-once" way. This is a rather technical topic, and I think further questions about it would best be treated in another post - as a starting point there are already some questions on this at math.stackexchange and mathoverflow, and the linked Hamkins/Linetsky/Reitz paper gives a good summary as well.

Pointwise-definability isn't the only strong definability property around, though. For example, we could consider the weaker property "Every object is definable using only ordinal parameters." It turns out that, contra pointwise-definability itself, ordinal definability is internally definable - there is a sentence 𝜂 (usually written "𝖵=𝖧𝖮𝖣") in the language of set theory such that, for every 𝑀⊨𝖹𝖥, we have 𝑀⊨𝜂 iff every element in 𝑀 is definable in 𝑀 using only an ordinal in 𝑀 as a parameter. This is a sort of double surprise since Tarski blocks the obvious construction of such an 𝜂 - we need to use a further nontrivial result (the reflection principle). Yet another "internalizable" kind of definability is constructibility (due to Godel, and in my opinion not a great term), but this gets extremely technical.

9
  • This is a great answer! Pointwise-definability seems to point at a kind of Quinean idea of an "identified entity", and the idea of pointwise-definable models of ZF looks like a promising hint at the possibility of some kind of "universal substructure", even if we acknowledge that said structure itself ought not be definable simpliciter. Aug 17 at 10:35
  • I have some math background but I found this answer very confusing. Reals are uncountable, while the set of "definitions" is surely countable (such as the set of all strings). This proves that there are undefinable reals. So clearly I did not understand what it means for ZF to have a pointwise-definable model.
    – Pedro A
    Aug 17 at 21:42
  • 1
    @PedroA See the linked Hamkins/Linetsky/Reitz paper - essentially the point is that per Tarski we can't even say "every object is definable" in the object language, so there's a surprising lack of tension. (You might start with Skolem's paradox if you're unfamiliar with this sort of thing.) Aug 17 at 21:45
  • 1
    @PedroA Re: your second comment, the structure $(\mathbb{Q};<)$ consisting of the rationals with just the ordering - that is, no multiplication or addition - is not pointwise-definable. $(\mathbb{Q};+,\times)$ on the other hand is pointwise-definable. And it doesn't make sense to ask whether ZF as a whole is pointwise-definable - ZF is a theory, but pointwise-definability is a property of structures (some models of ZF are pointwise-definable while others aren't). Aug 17 at 21:46
  • 1
    @PedroA I've added a brief description of why this doesn't cause a problem. (Incidentally, the term "math tea argument" is a relevant one re: the intuition in your first comment.) Aug 17 at 21:56
19

This question is not answerable. First, we would need “everything” to be well-defined, and arguably, to be useful, the definition would not be self-referential. This being possible while decently approximately preserving our intuition of “everything” is questionable. Secondly, for canonical and the most useful approaches to logic, the idea requires quantification over everything, which is problematic even if “everything” is well-defined. In order for quantification to make sense canonically, it must be over a domain of discourse or your system must have restricted comprehension.

Not having a domain of discourse or having unrestricted comprehension leads to many paradoxes, such as Russell’s paradox and Curry’s paradox. The former is more famous, but I’d argue the latter is more relevant to your question. Once you specify a domain of discourse, there will be something outside of it, and restricted comprehension is precisely that – restricted, so not all-inclusive.

To see an example of what can go wrong, does your domain of discourse properly include itself? If it doesn’t, then it doesn’t include everything. If it does, then if we are speaking of sets, your system will fall prey to Russell’s paradox and will be inconsistent.

Does your system quantify over arbitrary formulas or can formulas be instantiated by variables? If your system does not, then you can’t give semantic definitions to all the formulas inside your system (they can still be well-formed, structurally speaking). If your system does, then your system likely falls prey to Curry’s paradox.

A specific, formal example of something that cannot be defined within the standard model of a system that can interpret arithmetic is “truth”. This is Tarski’s Undefinability Theorem. Informally, any consistent system that can interpret arithmetic cannot define its own truth predicate. There is an even more general form of the theorem with a wider scope.

What if we instead try to work with a natural language instead of a formal system? Not only will you still encounter Curry’s paradox, but you’ll have to deal with Richard’s Paradox as well…

Thus the resolution of Richard's paradox is that there is not any way to unambiguously determine exactly which English sentences are definitions of real numbers

Very informally speaking, to talk about everything, you want your system to be completely open (to include everything) while still being closed (in order to give rules, be manageable, and be consistent). This is not possible. The more inclusive your system is, the more you can speak about but also the more likely your system contains a contradiction. An intuitive analog to this is a bag which contains everything is impossible because everything must be inside the bag, but if there’s an inside then there must be an outside – there’s no such thing as “inside” without “outside” as well. Also, is such a bag properly inside itself?

So why isn’t the answer to your question “no”? Because these paradoxes emerged given the way we normally approach logic. There may be some other categorically distinct approaches we don’t know about. Because we don’t know them and because “everything” is not well-defined, your question is fundamentally unanswerable.

8
  • 1
    "If it [everything] does [contain itself], then if we are speaking of sets, your system will fall prey to Russell’s paradox and will be inconsistent." This is actually not true there are set theories believed to be consistent that do include universal sets or sets that contain themselves. For example Quine's set theory. Removing sets that contain themselves is the way ZFC solves Russell's paradox, but it is not the only way. Aug 17 at 15:16
  • 1
    @SriotchilismO'Zaic Given the previous paragraph, I am assuming unrestricted comprehension. Aug 17 at 18:58
  • 1
    "An intuitive analog to this is a bag which contains everything is impossible because everything must be inside the bag, but if there’s an inside then there must be an outside – there’s no such thing as “inside” without “outside” as well. Also, does such a bag contain itself?" Could we apply the same reasoning to the Universe instead of a bag? I'm thinking maybe we can, as long as we accept that the Universe is not an actual "thing".
    – D. Halsey
    Aug 17 at 22:36
  • 1
    IMHO, not being able to define "everything" answers the question perfectly. Aug 18 at 7:59
  • 1
    @TeroLahtinen I agree in spirit, but if the question is (essentially) “can everything be defined?” and I answer only with “‘everything’ cannot be defined”, it leaves an understandable dissatisfaction. Aug 18 at 8:05
6

In addition to the other mathematicians, I wanted to highlight some examples from Yanofsky 2003. This paper takes one single theorem (Lawvere's fixed-point theorem for Cartesian closed categories) and extends it to many examples:

  • Grelling words (p9): "Pentasyllabic" is pentasyllabic (at least in my accent). Such words are autologisms. "Autological" is autological. We can attempt to define its opposite, "heterological", but we will always obtain an inconsistency, even though "heterological" exists as a word.
  • Richard numbers (p11): Some real numbers are not just uncomputable, but indescribable. They exist by the sheer power of the Axiom of Choice, but no reasonable definition (no Richard sentence) can reach them.
  • Turing languages (p12): Similarly, some languages are uncomputable. In those languages, some utterances exist but it is not decidable whether they can interpreted.
  • Parikh sentences (p17): There are true sentences. They are provably true. The proofs are too long for the physical universe to contain.
  • Berry sentences (p21): There are definitions for numbers which require them to be constructed only after a certain number of steps, but the definition itself is shorter than the given number of steps.

Grelling words exist but cannot be defined without introducing inconsistency. Richard numbers don't exist; the Axiom of Choice is not generally true, because it would require the Law of Excluded Middle to be generally true (see nLab), and LEM is false in some topoi. (You don't have to agree with me on this point.) Turing languages exist. Parikh sentences exist, although I have not seen an explicit construction. Berry sentences exist and were how I was introduced to this topic as a child.

2
  • 3
    Parikh sentences exist, although I have not seen an explicit construction. Nice one, +1 for this sentence alone.
    – AnoE
    Aug 19 at 6:44
  • I should have been clearer, although I appreciate the humor of that sentence without context. I have not seen a Parikh sentence explicitly constructed where the the sentence is provably true to me but the shortest known proof exceeds physical limits. I have only seen the proof of the proof of the construction (!!), as well as small toy examples.
    – Corbin
    Aug 19 at 21:26
3

Many concepts lack formal definitions. This poses no problem for everyday use but has been taken up as a question in philosophy and linguistics.

Wittgenstein noted how all games have common features but no one feature is found in all of them, so he could not define necessary and sufficient conditions for something to be correctly called a 'game'. He proposed the family resemblance theory as an explanation.

Another answer is the prototype theory of Eleanor Rosch, which says there are prototypical examples of a concept and less prototypical examples of the same concept which we can recognize because we know the prototype.

The above are just two well known theories but there are others. A starting point for reading more would be Concepts by Margolis and Laurence, and there will probably be newer literature by now as well.

Finally, Mothy Python addressed this issue almost 50 years ago in their song 'Eric the Half a Bee':

Half a bee, philosophically
Must, ipso facto, half not be
But half the bee has got to be
A vis-a-vis its entity, d'you see?

But can a bee be said to be
Or not to be an entire bee
When half the bee is not a bee
Due to some ancient injury?
1
  • You can write formal definitions using prototypes, even for multi-centre concepts. They'll just be definitions that have the form of "something like X". Aug 19 at 0:36
3

Yes, but I don't think you will like the answer.

First, for every quanta of time list all the particles that exist at every time and give them unique names (so each particle is represented once for every Plank time of it's existance). Definition of each particle consists of their properties like mass, spin, position, time, direction, etc, as there is nothing else that can define it. Next list every possible grouping of those particles. You define all those groupings by listing particles that it contains. Some groupings will represent two particles at different ends of the universe. Some will represent a dog from the moment of being born to it's death. Some will represent that same dog, but on the day it will turn 10 years old, it will also include a part of a cloud for 4.8 seconds.

Most definitions will not be meaningfull for everyday life but I would argue there is no possible way to describe existing things accurately.

We as humans think in categories. "Cat" is a category and it's a usefull category for human communication and understanding of our world. When we see a cat, we know more or less what to expect. When we say "cat", the other person can easily imagine what we are talking about. "Every single cat on Earth, except that unremarkable, unnamed, homeless one in Cairo and 5 hadrons in the center of Andromed" is not a usefull category for human communication and it doesn't have a word associated with it. However not being usefull to us humans doesn't make this category less real. And now it has been defined.

Interesting thing happenens when we try to define "love" as something that exists. This is when we really get to the limits of human comprehention. What is "love"? Does it exist in this model? How would we define it? Is it a change over time in a person? Is it just specific organ or organs in a person? Is it just hormones in the brain? Which of the particle groups would contain all the "loves" in the universe? I don't know the answer to this question, so I leave it open as an excercise for a reader. And I feel if I had an answer, it wouldn't be the only correct one. Imagine that you could isolate only the selected group of particles and watch tham change over time like a movie. I'm sure you would see love if you watched all of them, but "love" would be much less accurate of a definition than what we have named it with our previous method.

5
  • 2
    This assumes that the universe is discrete and not continuous. The Plank constants (Plank time, Plank distance) are not meant to be discrete unites of the physical universe; i.e. we do not know that the universe is "pixelated" in Plank units. They are just so tiny that none of our theories can tell anything about smaller times or distances...
    – AnoE
    Aug 18 at 14:21
  • 1
    @AnoE I think it doesn't matter whether the time dimension is continuous or discrete (although I admit, this was my original assumption). If something is defined as existing in a certain period of time, Plank time being, to my knowledge, the smallest unit that can be observed, it would still be defined. Or in other words, If we define a "thing" as something with set of physical properties defined as a continuous function over specified time, it wouldn't change my argument. The difference is that instead of defining points in time, we define segments of time. Aug 18 at 14:51
  • 1
    And discrete vs continuous aside, even if we could have a description for some set of particles or region of space, we cannot have a full description of all particles or regions of space as this would require more particles and space than what we are describing. That is, a piece of paper of hard drive with all the information in the universe would be larger than the universe. Aug 18 at 18:31
  • 1
    @Shufflepants that is correct and this perspective could make for an interesting answer to this question. For my answer I assumed "the definer" is an external observer, like omniscient god-like consciousness. If I was trying to write an answer as an in-universe observer, the problems would start much earlier, with observing and registering everything in existence, including all of the past and future. The behavior of particles as observed and described in quantum physics makes it impossible to define even a single particle in matter I am suggesting. Aug 18 at 18:37
  • @ReverentLapwing, fair enough - to be totally honest I think the question is not well-defined (basically all the terms in it would need to be explained), so I don't really have a better idea than the different answers here. Just wanted to throw out the continuous thing because I found it fascinating when I first looked into the Plank units - they are more a trick of the physicists to get rid of some pesky notation than a fundamental property of the universe. If I were to start an answer, I would probably try to bring in Gödels incompleteness theorem ASAP, or something along those lines. ;)
    – AnoE
    Aug 19 at 6:40
1

The subject of a definition is implicitly a "class" of objects ("a something"), rather than a single specific object ("the something"). As the subject of definition approaches a specific object (i.e. the "range" of said definition approaches null), the definition itself would approach infinity.

All numbers exists, but how many can be defined? Integers, fractions, some real number constants like pi, e, and any number that you are willing to write down by brute-force, but not much more than those.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.