"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Wednesday, December 18, 2013

How empty is { }? Different orders of the Empty Set. Coursera Forum discussion.

Here is a raw cut-and-paste of the exchange on this topic from a Coursera forum.

I'd like to thank those who provided constructive criticism, or just picked at the idea with an unbiased razor of skepticism. Apparently most of the interlocutors were inspired, and some have even admitted to having been led to think in a new way --- to a philosopher, there's no greater reward :)

It's this sort of exchange that allows the idea to mature, and hopefully catch on :)

Mariusz Popieluch· 2 months ago
Let me introduce the discussion with a little humor.

The Poet, the Chemist, and the Physicist.
The trio met one day over some wine and discussed matters of rhyme, thyme and time. Soon the conversation turned to the notions of nothingness and emptiness - what is empty? The poet finished his glass, and pointed to it -- as far as I am concerned this glass is empty, as it is devoid of wine - the sparkling grape, the drink of gods. Hold on a minute! -- exclaimed the chemist -- surely it's not empty as it contains air - we'd have to pump all the air out of the glass, creating a vacuum in it, and then and only then would it be empty. It wouldn't make any difference to me -- replied the poet, shrugging his shoulders. Please let me interject at this juncture dear fellows -- interjected the physicist -- and let me put an end to your obvious confusion. Vacuum, as you describe it doesn't cut it at all, since we know that even empty space is a breeding ground for virtual particles whose immediate annihilation results in what we observe and call vacuum energy. The poet looked at the physicist with a frown of suspicion -- I know nothing when I see it good man, and I won't let anyone tell me otherwise - let the bartender settle this matter - haloo, good fellow! Another round please!


The above humorous scenario pertains to natural language and some theories of varying degree of conceptual refinement, but the idea extends nicely (I believe) to formal languages, theories and logics too, thus generalizing the idea of emptiness. In particular it intends to do to the notion of emptiness and it's various formalizations such as the set theoretic ∅, what Cantor has done with the notion of infinity - show that there is structure within it, and varying orders of it.
I know this is a radical idea, but rest assured that the idea of varying orders of infinity wasn't initially taken lightly either - even today many still think it's preposterous. But interestingly enough, there seems to be a demand for such a hierarchy - hinted at in the literature - but I'm not aware of a formal theory of various orders of the empty set being developed.
The general claim is that what counts as empty is dependent on the strength of a logic underlying the theory in which it is defined and the expressibility (richness) of the language of the theory. That is, in some theories there are more objects that fail to be members of the empty set (e.g. to the physicist there is more to the contents of the glass than wine and air), thus rendering the empty set somehow emptier (to put it simply).
Or to put it another way - we could just define the order of the empty set by the cardinality of all the impossible objects in a given language/logic/theory. Those impossible objects are not actually members of the empty set, but rather are assigned to it in some sense - by saying that some object (this includes linguistic objects) is an element of the empty set (in a given theory), is just another way of saying that this object is impossible (in that theory).
Consider classical logic CL and some paraconsistent logic, say LP. In LP contradictions are not considered as impossible. Now define a property G: of being green all over, and not being green all over. Now, in CL any object with that property is considered as an impossible object, take the object a, so in CL we obviously have Ga ∈∅, but in LP we have Ga ∉∅. Hence the suggestion to index ∅ by logics - this may become crucial when talking of various theories in some metalanguage/meta theory.

Philosophically speaking "{ }" is an object with a fixed extension across theories (it has no elements), yet varying intension. What the above aims to show is that the varying intension can be accompanied with a varying quasi-extension, i.e. rarity as opposed to cardinality.

Mariusz Popieluch· 2 months ago

Consider two urns in a marble shop. The shopkeeper labelled them in the following way - urn 1 holds only (and can only hold) white marbles, whereas urn 2 can hold both white and black marbles.

Now consider a new assistant that has just been offered an apprenticeship in the shop of his dreams (he loves playing marbles), and is not as yet familiar with the arcana of the labeling system, i.e. he hasn't as yet been told by the shopkeeper what the restrictions on the contents of the urns are.

Now consider the statements 'urn 1 contains no white marbles' and 'urn 2 contains no white marbles' - they entail different things, depending who is exposed to that information - the shopkeeper or the assistant. In particular, to the shopkeeper it means that 'urn 1 is empty'.

On a more technical side, we can substitute the terms 'has no white marbles' and 'is empty' interchangeably salva veritate in the context of urn 1 (charity permitted, for naturally one could generate oddly sounding sentences). This cannot however be done in the context of urn 2.

David M. Kaziska· 2 months ago
In your first post the set, {wine in the glass}, is empty after the wine is consumed (neglecting any residual wine).  For the poet's own purposes it is empty but we may his or her statement that it is empty as colloquial in light of the chemist's and physicist's later statements.  I'm not seeing mathematical consequences which would require redefining the notion empty sets.  {wine in the glass} may be an empty set but {air in the glass} may be nonempty.  

In your later post, the shopkeeper reasons as follows.  (P1)  All marbles in Urn 1 are white, (P2)  There are no white marbles in Urn 1, therefore (C)  Urn 1 is empty.  The assistant has (P2) but not (P1) and cannot draw the conclusion.  Urn 1 is still empty, though, it's not a matter which is relative to the observer.

Mariusz Popieluch· 2 months ago
Thank you for your reply David. I'll do my best to address your observations, and further clarify the idea. :)

The later post intends to illustrate the idea that in some theory T based on some logic L1 some formula F may express an impossibility (which we can express as that formula being "an element" of, or assigned to, the empty set), whereas the exactly same formula in an analogue to T based on a logic L2 which is stronger than L1 may not come out as impossible, and as such cannot be expressed as "belonging"/being assigned to the empty set.

Note: "element of" and "belonging" are terms I tentatively use to denote some kind of assignment of F to the empty set (or empty sets, since I'm arguing that they're needn't be unique across analogous theories based on distinct logics). Note also that for each single theory the empty set still remains a unique object/element, as theories are usually based on a single logic.

The epistemic states of the protagonists of the marble shop illustration (informal context) correspond to the assumptions/axioms/conditions/semantics of theories based on distinct logics in the formal context.

Some logics distinguish semantically the propositions "p & ~p" and "p & q". In particular they consider the first proposition to take a fixed value for all valuations (contradictions are always false, and only false in classical logic). Some other logics do not make that distinction.

Likewise the shopkeeper distinguishes the two propositions concerning the urn contents, whereas the assistant doesn't. Try to think of the urn labelling system and the protagonists' distinct epistemic states as an analogy for the distinct meta-linguistic propoerties of analogous theories based on distinct logics, and the propositions concerning the urn content as object language entities, i.e. formulae.

But if the informal analogies muddle, rather than clarify the idea, I'll be happy to talk more formally about it. :)

Louise Craven· 2 months ago
Every set is a subset of the relevant domain of discourse. Thus if we are only talking of natural numbers, the empty set is the set containing no natural numbers. The poet was thinking of drinks when he said his glass was empty, so it truly was. The chemist and physicist tried to confuse him by exploiting the fact that he had not explicitly stated the domain of discourse, and they were also right about the nature of the empty set within their individual domains of discourse.  In the marble example, the difference between the shopkeeper and the assistant is not in the 'size' of the empty set, but in the inferences they can draw from the fact of emptiness, which depends on other information.

I am, however, intrigued by your suggestion and it may be that in some applications of mathematics to real-world problems, the size of the empty set in the model may need to vary according to the significance of 'nothing' in the area modelled. For instance, complete extinction of a species is of much greater significance than the remaining existence of a number of breeding pairs much smaller than the population at a previous time.

I shall go now and give it more thought....

Mariusz Popieluch· 2 months ago
Hi Louise - thank you for taking the time to read the OP.

The informal anecdotes are mere illustrations of the general idea.

In the first scenario I intended to show, via the relevant domains of discourse illustration,that given the same language (in this case natural language), but distinct theories, identical propositions (formulae, i.e. linguistic objects) entail different things - in particular, where in one theory "no wine in glass" does entail "glass is empty", in other theories it doesn't. I stress again, that this is merely an illustration of meta theoretic properties, rather than a discussion concerning restricted quantification to relevant domains of discourse per se.

Think of the Poet, the Chemist and the Physicist as analogues of theories of "The stuff on the table, at the local restaurant.", albeit based on dictinct logics. In the poet's logic "there's no wine in the glass" can be associated with the empty set - this however isn't true in the chemist's or physicist's logics. Likewise in the chemist's logic "there is no liquid and no gas in the glass" can be associated with the empty set - but again, this isn't true on the physicist's logic.

As for the second scenario, please see my above reply to David. And please do not shy away from addressing the formal content of the OP.

Andrew Kelley· 2 months ago
First let me say that I enjoyed reading this post. It made me think in a way I hadn't thought before.

I think you may be on to something with recognizing different orders of the empty set. However I am as of yet unable to think of any examples where the knowledge of such different orders is relevant to solving a problem. Are you?

Mariusz Popieluch· 2 months ago
It may serve as the ontological foundation for a theory of comparative impossibility.

Francisco Vasconcelos· 2 months ago
I think that the analogy between empty set and infinite set is not the most correct one. The evident oppositions would be between empty and complete set, and infinite and infinitesimal amounts.

So on the empty/complete side, I'm not aware that there are different orders of completeness. Both these terms are usually used as binary categories applied to other sets, both finite and infinite. Yes, their meaning varies with language context, but so do everything else, such as the number "2", the operator "+", or the word "number".

On the other hand, I think it's easier to make the claim that there are different orders of infinitesimal quantities. For example, think about the probability of picking the number "5" at random from the set of all natural numbers. Since the set is infinite, the probability is 0. However, it seems that this "0" is still bigger than the "0" probability of random sampling the number "5" from the set of all real numbers, since the pooling set is infinite to a higher degree than the previous one.

Andrew Kelley· 2 months ago
I'm not buying that you can say that one 0 is bigger than another 0. Zero is zero. Can you prove it?

Mariusz Popieluch· 2 months ago
Francisco, thank you for your reply -- you're right that it's not the most correct analogy, as it's not intended to be a directanalogy, i.e. a dual of sorts. It's a weak kind of analogy, in the sense that in both cases of infinity and nothingness/emptiness we're dealing with concept which apparently don't admit to degrees. Whereas Cantor developed the idea of bijection as the criterion for equinumerosity, I'm using it in developing the notion of the rarity of some empty set, which is currently tentatively defined to be the cardinality of the set of all formulae which express an impossibility (aka impossibilia) in a given logic L, thus yielding an indexed empty set ∅L.

As I said before, in the above response to Louise, the contexts of relevant discourse analogy is again merely an illustrationof the more precise idea. See the above reply.

The observation you expressed in your last paragraph puts you in good company. In the context of probability theory the idea of different orders of “zero” has been hinted at by Andrey Kolmogorov and Bruno de Finetti as a possible candidate to solving some probability theory paradoxes – “Like Kolmogorov, de Finetti is occupied mostly with probabilities defined directly on arbitrary uncountable sets; but he views additivity differently, and is led to such anomalies as an unlimited sequence of layers, like an onion, or different orders of zero probabilities that add up to one, etc. ” (E.T. James: Probability Theory, the Logic of Science, 2011, p.656).

Francisco Vasconcelos· 2 months ago
Andrew, try not to read my "0" as the integer number 0, but as an infinitesimal quantity that is infinitely close to zero, something like the result of limx→∞1x. As infinity can have different sizes, the result of this limit quotient should be able to have different sizes too.

Mariusz, thanks for the info

" different orders of zero probabilities that add up to one"

this is definitely interesting and worth checking out.

Regarding your emptiness orders:

"I'm using it in developing the notion of the rarity of some empty set, which is currently tentatively defined to be the cardinality of the set of all formulae which express an impossibility (aka impossibilia) in a given logic L, thus yielding an indexed empty set ∅L."

I get it now. So I guess you're trying to say that in different theories the empty set can be connected to a set of propositions that can have a different finite or infinite size.
Another question: hypothetically speaking, do you think that within a single theory L (maybe with number references), it is possible to represent empty sets with different sizes in this sense?

Andrew Kelley· 2 months ago
We can mathematically prove that infinity can have different sizes by using bijections. But we cannot mathematically prove that limit of 1/x as x approaches infinity is not equal to the integer zero. In fact, we *can* prove that, for example, 0.999999 repeating is exactly equal to 1. Not a number which has an infinitesimally small difference than 1. 1.

Francisco Vasconcelos· 2 months ago
Andrew I'm still trying to figure out what different kind of "0" can mean, it seems that this concept raises some issues in probability theory, but lets forget that for a moment and define the following:

For any functions f(x) and g(y), assume that

Now suppose the following hypothesis:
For any arbitrary ϵ, there is NOT a bijection between all possible values of g(y) and f(x), for x,y≥ϵ

If this proposition is true, then the limit of g(y) and f(x) being "0" might have different meanings, since we know that for any given ϵ, one of the sets will always be larger than the other.

Mariusz Popieluch· 2 months ago
@Francisco: "Another question: hypothetically speaking, do you think that within a single theory L (maybe with number references), it is possible to represent empty sets with different sizes in this sense?"

Well, personally I'm not entirely on-board with that idea, for the reasons I stated in my reply to David above: "Note also that for each single theory, the empty set still remains a unique object/element, as theories are usually based on a single logic."

My position is such, due to what I mean by "different orders of emptiness". But this is not to say that I'm not open to the variation of this idea, which you and the company of famous thinkers suggest. :)

Andrew Kelley· 2 months ago
Francisco, thank you for this example. I think you may be right but I am finding it extremely hard to wrap my brain around it.

Hayden VanIderstine· 2 months ago
Francisco, in order for the hypothesis that there exists no bijection between all possible values of g(y) and f(x) when x,y>ϵ, then it is necessary that the cardinality of the set of all possible g(y) with y>ϵ be different from the set of all possible f(x) with x>ϵ. For this to be the case, then both sets cannot both be of any of the following cardinalities:

Countably infinite (set of naturals)
Uncountably infinite 1 (set of reals)
Uncountable infinite 2 (set of all functions defined from the set of reals to the set of reals)

What I am wondering, is what is the domain and range of f and g, because if they both have a range being a subset of the real numbers (or the entire set of real numbers), then there necessarily exists a bijection between them.

Francisco Vasconcelos· 2 months ago

I'm not sure I follow you. The way I see it, for there to be no bijection it is only required that the domain of f(x),x>ϵ has a different cardinality of g(y),y>ϵ. So, for example if the domain of f(x) is a subset of the rational numbers, and the domain of g(y) is an interval of the real numbers, then there is no bijection.

Hayden VanIderstine· 2 months ago
In my opinion, excellent post Mariusz. I am thinking in a new way now thanks to you.

What we consider as empty is dependent on what we acknowledge exists.

{n∈N | 1 < n < 2} =  ∅
Exactly  |N| elements fail to be in the set {n∈N | 1 < n < 2}

But exactly |R| elements fail to be in the set {x∈R | 1 < x < 2}.

According to your definition of the rarity of some empty set, would the rarity of the empty set in a logic which isn't strong enough to construct the reals, but is strong enough to construct the naturals, be |N|, because there are only a countably infinite number of sentences like 1 < n < 2 which could be made which are impossible?

(btw, I'm not well versed in formal logic, nor am I sure that there exists a logic which is strong enough for the naturals, but not enough for the reals).

Mariusz Popieluch· 2 months ago
Hayden - thank you for your insightful reply and interpretation of the ideas in this thread. Your observations, and question also made me think more carefully about what I'm proposing.

The natural numbers, or Peano arithmetic, is a first order theory, whereas the it is not possible to characterize the reals with first-order logic alone since the supremum axiom of the reals quantifies over subsets of the naturals, and is therefore a second-order logical statement. See the list of axioms here: https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach

The stronger logic in which the axioms of R are expressed is second order logic, which is stronger than first order. So to answer your question directly - first order logic is the logic "which is strong enough for the naturals, but not enough for the reals", for the reasons given above.

(Note on the terminology: Logic B is stronger than logic A iff all theorems of A are B therorems, and there exists some B theorem that's not an A theorem. Or equivalently B is a stronger logic than A iff the set of A theorems is a proper subset of B theorems.)

Mariusz Popieluch· 2 months ago
Also, to clarify - "What we consider as empty is dependent on what we acknowledge exists." is not entirely correct.

Rather "What we consider as empty is dependent on what we acknowledge can exist.".

Pedro Forquesato· 2 months ago
Hello Mariusz,

Nice post! It brought a good discussion, and that is the purpose of this forum. I think your stories are good examples of the importance of clear and rigorous (for example mathematical) sentences in philosophy (and other studies), and the danger of "spoke communication".

Let's try to solve it by translating to "mathematical language" what they are saying. In the case of the urns, for example, when the apprentice says "the urn is empty", he means (lets define U as urn and W as the set of white marbles):
U = { }
While when the master says "the urn is empty", rigorously he means:
U intersection W = { }

So while in English what they say is the same, actually they are saying different logical propositions, and thus it is not paradoxical that the second might be true while the first is not. It is not the empty sets that are different, it is the translation from English to logic that differs. (Naturally the same argument is valid for the poet's glass).

Mariusz Popieluch· 2 months ago
Thank you for your reply Pedro - if you're interested in a mathematical treatment of the idea, don't look at the intuitive illustrations of the marble shop, and the trio drinking wine. Instead focus on the content of the OP that follows the phrase "The general claim is...". :)


To be continued...

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