Why does the empty set or null set or void set (Φ) satisfies every property, every theorem, etc. in the set theory, topology?
The reason is actually quite fundamental. And make no mistake, this is a good question.
After all, might it be unsettling the concept of even implying something from nothingness?
Well, the reasoning is that it creates a whole lot more eloquent type of math. In mathematical logic, we try to formalize mathematical reasoning in terms of new constructs(that may use a very tiny amount of basic mathematics).
So, math is based on a whole lot more ‘absolute’ type reasoning. It will be clear what I mean by this as we go on.
In logic, there is a concept of ‘implication’. The idea that, some statement follows from another. We can consider whether the truthhood of whether it follows or not itself.
But, it is a bit unclear on what we mean ‘follows from’. So, we take a different approach to the concept of ‘implication’ in mathematics, in that we fix a property we certainly want.
We certainly don’t want false things to follow from true things. That will immediately get some things wrong, a rather serious crime.
So, in our mathematical logic, we will have an implication that is more ‘absolute’, called the ‘material conditional’ written p→q, where p and q represent sentences that can be evaluated as true or false, resepctively called the ‘antecedent’ and ‘consequent’, and, we will say p→q is false if p is true and q is false.
We also want to be able to make use of the fact it is an implication, which mean true things should follow from true things, in other words, we set p→q to true when both p and q are true.
Now, how about when p is false? Well, it is a lot more dependent on the statement p, then, in normal everyday logic. If I for instance say “If 2+2=52+2=5 then humans are living things”, clearly the ‘antecedent’ statement is false, and the ‘consequent’ one is true, however it is unclear how the consequent is deduced from the antecedent. Similar story when q is false. But, if I were to say “If a+b=a+b+1 then 0=1”, then it would be sensical to say this is true. The thing is, we clearly will encounter the second result more then the second, and, in that it does make, in some sense, to say in both of these cases, it is false. The reason for this is to create a simpler, more elegant system. Were it to be true, we would neglect these very important examples. Moreover, it would actually just reduce to the same logical idea as ‘and’, which we clearly don’t want.
We will now prove a very important result, the ‘principle of explosion’. It is in this sense, that we motivate our decision.
Suppose we derive a contradiction, p and ¬p. Then, it follows p and q is also true, for p is. However, since ¬p, then it follows q. So, from contradiction, anything follows. That is, whatever q is, it doesn’t matter, we have properly derived q from falsehood. Of course, falsehood isn’t necessarily contradiction, but certainly to be accurate, should include the case where the falsehood is due to contradiction, that the principle of explosion follows.
Hence, in this sense, the material conditional will be set true for false antecedents. This is called ‘vacuous truth’.
Another good way to think about this is that an ‘implication’ really serves as a promise for what will happen if something happens. So, if p is false, then regardless of what happens with respect to q, the requirements for my promise aren’t met and so my promise should still hold.
Why do we even want to introduce the empty set ∅? Same reasoning, it is to create a more complete system. Sets may be disjoint. They quite frequently are. If we restricted our set theoretic ideas to that of only objects to which there is at least one element, then what would result is we would constantly have to tip toe around the issue they may be disjoint. There is no harm for the existing nonempty sets in introducing operations with the empty set, and it creates a certain ‘closure’ to everything.
In topology, for instance, we may want to speak in general that the intersection of open sets is open. However, the intersection may actually be nonexistent. Certainly, we would lose a lot of elegance in having to say ‘if all sets with each other have existing intersection, then the intersection of such is another open set’. Gross. And while this doesn’t look bad, the complexities can exemplify when proving theorems.
Now, what actually results when we do this?
For instance, let us say A is a subset of B if for all x, x∈A→x∈B (most math papers just write this as ‘if … then… ‘ and understand that to mean a more mathematical form, but I will emphasize here the difference). Now, the problem is, I might have to actually specify “if A and B are nonempty, then…” to restrict it to cases where there exists value such that x∈A be true. However, if we allow empty sets, then we can account for all scenarios by saying the empty set is a subset.
So, the reason the empty set more or less satisfies every property and theorem in one form is it makes a more complete whole in simplicity of rules. Since the empty set is defined as that which the statement ‘x∈A' is false for all x, it follows additionally from the material conditional, the theorem follows on whatever implication since the antecedent is false.
Sure, some people might argue until they are blue in the face that you can’t actually deduce from that of falsehood, or that the empty set doesn’t actually exist, and interesting routes can be pushed in that direction, but it only hampers the ability to prove mathematical theorems into which the philosophical notice really doesn’t matter.
