So, I came to a very important realization today about what makes a word overly-ambiguous (meaning, has an infinite number of potential definitions) – that its definition is too broad. This reminded me of Russel’s paradox in set theory, which was the direct result of a lack of axioms, which enabled sets to be of unbounded size, so today, I will write about Russel’s paradox, it’s solution, and how this solution can be applied to a language.
Russel’s Paradox in Set Theory:
Before the times of the Zermelo-Frankel axioms of set theory, which changed the face of mathematics, sets, which are the simplest collections of objects were defined through properties. For example, here’s a property – “is a cat,” (well I like cats), so now we can make a set of all cats. Or, we can take the property “likes the color black,” and make a set of all objects that like the color black. Sounds simple, right?? Awesome, here comes the fun part:
Let’s define a property that a set can have like so: Let’s say that a set X has the property D if X is not an object in X (itself).
For example, the set {1,2,3,4} has the property D, because {1,2,3,4} is not an object in {1,2,3,4}. We can think about {1,2,3,4} as a box which contains 4 different numbers. Now, we want to know if this box (set) has the D property, which means that we want to know if the box contains itself. It doesn’t, and therefore, it has the D property. In general, nearly all sets that we can think of have the property D, so we can imagine already that the set of all sets with the D property is ridiculously large.
In a more visual manner, this treasure chest contains treasure, but it doesn’t contain a treasure chest identical to itself in every way, from size to color, to material, and therefore, this treasure chest has the D property.
We defined a property, D, correct? So, now what are we going to do?? Simple – we’ll define a set R={X: D(X)=1}={X such that X has the D property}={X, such that X is not an object in X}
Now arises the question, does R have the D property??? There are only two possible answers to this question – yes or no. Let’s see what happens in every case:
- If R has the D property – Then R must be an object in R, because every object with the D property must be in R. On the other hand, since R has the D property, according to the definition of the D property, R isn’t an object in R. Since we’ve arrived at a contradiction, this option is definitely incorrect, so the remaining option must be correct. Let’s evaluate it:
- If R doesn’t have the D property – Then R is not an object in R, because only objects with the D property are in R. But, since R is not an object in R, then R must have the D property, because this is the definition of the D property. We’ve once again arrived at a contradiction, and therefore, this option is also definitely incorrect.
Since all of the possible options lead to a very clear contradiction, there must be a problem somewhere along the way. There were several suggested solutions, and of those, one of them stood out and was very widely accepted in the mathematical community – the Zermelo-Frankel axioms.
The Solution to Russel’s Paradox – The Zermelo-Frankel Axioms
The primary idea behind the Zermelo-Frankel Axioms is that the set R from Russel’s contradiction is too large, and that in general, sets that are too large will cause contradictions. Therefore, the goal of the Zermelo-Frankel Axioms is to write a logical list of rules (axioms), which will enable us to create sets that are not ridiculously huge.
Here’s the list of the necessary and trivial Zermelo-Frankel Axioms:
- Axiom of Extensionality –
Two sets are equal if and only if they share the exact same objects.
\forall A,B, A=B if and only if, \forall x: x \in A if and only if x \in B - Axiom of Regularity (Foundation) –
Every set A contains a set B, which is disjoint to A, meaning that A and B have no common objects.
\forall A, \exists B \in A: x \in A if and only if x \notin B - Axiom of Restricted Comprehension (how to build the empty set) –
\exists \O the empty set, which can be built using a trivial contradiction, like:
\O={X: X!=X}={X: X isn’t equal to X} - Axiom of Pairing –
For any two sets in the world, A,B, there’s a set C, which contains A,B as objects.
\forall A,B, \exists C: (A \in C) and (B \in C) - Axiom of Union –
The union of two sets is a set, when union is defined using the logical OR.
\forall A,B, \exists C: C={x: x\in A OR x\in B} - Axiom of Intersection –
The intersection of two sets is a set, when intersection is defined using the logical AND.
\forall A,B, \exists C: C={x: x\in A AND x\in B} - Axiom of Power set –
For any set, A, the collection of A’s subsets, which is known as the power set, is a set.
\forall A, \exists C: C=P(A)={B: B\subseteq A} - Axiom of Minimal Set –
For any set A, the set containing the singleton A is a set.
\forall A, \exists B: B={A}
These axioms enabled mathematicians to define sets in a clear manner, without creating bizarre contradictions, such as Russel’s contradiction, by using the very basic idea that huge sets cause problems.
How exactly can we view huge sets??? We can view them as sets that contain too many objects, and now, let’s use this analogy in the world of spoken language:
Why We Should The Zermelo-Frankel Fix to Spoken Language:
We can use the huge set idea from Russel’s contradiction in spoken languages, by saying that a word is like a set, and it’s objects is the subjects/words which are a part the category which this word creates. For example, we can take a word like “Religion,” which creates a category, which contains other words, such as “Christianity,” “Paganism,” “Islam,” “Shinto,” etc. Another example is the word “comfortable,” which creates a category, containing words and phrases, like “satisfactory,” “within 1 standard deviation from the optimal softness,” etc.
In a similar manner to Russel’s contradiction, a word, which creates a category with too many words and phrases causes many problems and contradictions. How does this work?? Let’s see this in an example, one of the words that I hate very much, which is the word “natural.” The word “natural” creates a category, which is too broad, because it contains words from “nature,” to “habit,” to “God’s will,” to “social norms,” to “what an individual feels comfortable with.”
Did you catch those contradictions there??? Nearly every word here contradicts another. Let’s examine this:
- “God’s will” vs. “Habit” – I will begin laughing like crazy here, because God’s will differs among Gods and religions, and habit depends on so many factors, from religion, to education, to intelligence type, to… I can go on forever here. Also, if we were to assume that there is a God (let’s assume the Jewish god, because I’m familiar with this from school, unfortunately), then, he’d want me to stay away from electricity on Saturdays…. Like I’d ever do that. I need my internet on Saturdays, and that’s my habit, so obviously, “God’s will” contradicts my habits, and I’m obviously not the only atheist in the world who’s bothered by “God’s will.”
- “God’s will” vs. “What an individual feels comfortable with” – Let’s see, I feel comfortable baking my amazing chocolate chip vegan cookies with wheat flour during Passover, and “God’s will” is for me to stay away from wheat. So again, a clear contradiction, which is definitely not unique to me only.
- “Social Norms” vs. “What an individual feels comfortable with” – This one is way too ridiculous, because exists something called “not mainstream,” which is already enough evidence to support the claim that social norms contradict what some individuals feel comfortable with. For example, one of the most important social norms in Israel is mandatory army service for all people from age 18 until age 21-24. Yet, there are lots of individuals who do not feel comfortable with the army service during these years, due to an understanding that the mandatory draft is characteristic of a fascist regime, pacifism, hating to wear the same thing everyday, and a bunch more reasons. These individuals feel comfortable not joining the army, which is clearly a contradiction to the social norm.
- “Nature” vs. “What an individual feels comfortable with” – Most individuals feel comfortable with using a cell phone, a laptop, a television, and a bunch more technologies, which obviously do not exist in the Africans safaris or the tropical rain forests, which represent nature. So again, another contradiction.
- “Social Norms” vs. “Nature” – One of the most obvious social norms in the modern world is owning a car, a weapon which causes the death of many animals (in nature), exhumes greenhouse gases, which harm the atmosphere, which is crucial to nature, and harms natural resources in many other ways, so again, social norms contradict nature, not that I have any problem with owning a car.
- “God’s will” vs. “Nature” – Let’s be precise here, and say that God is the creation on mankind, and not the other way around. According to God’s (some person’s) word in the bible, the earth is some 4,000 years old. On the other hand, the earth, meaning soil, which is literally nature, has shown scientists through fossils, carbon sediments, and other tools, that the earth is precisely 4-5 billion years old. Tell me that’s not a contradiction (hahaha).
So, in short, we can see that words which create a category that’s too broad create contradictions, and therefore, we need to find some axiomatization of words, in order to get rid of broad categories. We are facing the exact same problem that set theory was facing with Russel’s contradiction, and therefore, we can try to use the same ideas behind the Zermelo-Frankel axioms in order to fix our current problem. The only question at the moment is how. I’ll be honest and say that I don’t have an idea at the moment, and therefore, I’ll think about it, and give an attempt (which may not succeed) in a future post. So until next time.
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