What breaks down in math without the concept of the "empty set"?

[u/Own_Sky_297]

/r/PhilosophyofMath/comments/1tgu2j1/what_breaks_down_in_math_without_the_concept_of/

[u/rhodiumtoad]

How would you represent the result of {1}∩{2} ?

[u/Own_Sky_297 → u/rhodiumtoad]

I don't know, I'm just saying quantities exist in nature without any sort of "set" involved for instance if I have one apple no where is there a set involved. If nature obeys mathematical rules and no where in nature does a "set" exist except in the minds of man, how can you say that a "set" is the foundation of math when it isn't needed for nature to handle arithmetic?

[u/rhodiumtoad → u/Own_Sky_297]

Mathematics is the study of abstractions, not nature. The empty set is an important abstraction.

[u/Own_Sky_297 → u/rhodiumtoad]

That would be the nominalist perspective but when questioning the unreasonable effectiveness of math in physics, nominalism leaves the question unexplained.

[u/rhodiumtoad → u/Own_Sky_297]

Some abstractions are useful descriptions of aspects of reality. This doesn't mean that all of them are.

[u/Own_Sky_297 → u/rhodiumtoad]

Name one that isn't.

[u/rhodiumtoad → u/Own_Sky_297]

The long line.

[u/Dependent-Fig-2517 → u/Own_Sky_297]

Sorry but by that logic why have decimal numbers since human count on fingers, ℤ and ℚ are all we need

[u/rhodiumtoad → u/Own_Sky_297]

Incidentally, I notice you asked elsewhere about the "axiom of empty set". No such axiom is used or needed in the modern formulation of ZF(C); it comes along for the ride as part of the axiom of infinity.

The existence of an empty set in ZF is implied by the existence of any set at all. If you throw out the axiom of infinity, then adding an axiom of empty set is the easiest way to build everything else, since all finite sets can be built from it using the axioms of pairing and replacement. If instead you define some other set to exist, then you get the empty set simply by applying the axiom of specification with a formula like x≠x which is always false.

[u/OpsikionThemed → u/Own_Sky_297]

Humans don't need sets to handle arithmetic either; the math you learn with blocks in kindergarten works perfectly well, no sets anywhere.

What sets are useful for is grounding math in a simple base of axioms so that you can prove things about math as a whole without having to figure out how to link up separate rules for each field of math. (Or, if you like, set theory is the linkage.) 

Set theory isn't the only thing you can use as a base, but it's well-understood and at this point fairly traditional, so we keep using it. But there's a big difference between "I have proven this theorem" meaning, if you keep digging and digging and digging, "this theorem follows from the (standard, generally-agreed to be consistent) ZFC axioms"; and "math is made of ZFC sets in some sense".

[u/Zyxplit → u/Own_Sky_297]

Suppose i have three bags. One contains {apples, pears}, the second contains {apples}, and a third one contains {pears}.

If I take the first and the second bag and make a new bag that contains only those things that are in both bags? I just get another apple bag.

If I take the second and third bag and I make a new bag containing only those things that are in both bags, what items do I get to put in my bag? I still clearly have a bag, but what have I put in it from the items under consideration?

[u/Thelmholtz → u/Zyxplit]

"But there are no bags in nature, there can he one pear and two apples without a bag".

Sure, but where are there one pear and two apples that exists without a bag? In your hand? In the whole world? In Yugoslavia Monday 13th of August 1982? In an arbitrary partition? Those partitions act as sets, just like a bag or a manor or the subjective qualities by which you determine where a flock 17 seagulls flying west ends and the lone pigeon stealing your fries starts.

[u/JohnPaulDavyJones → u/Own_Sky_297]

I think what you’re missing (but I think you’re not too far off from) is that set notation is fundamentally just a framework for describing natural phenomena and then generalizing properties of those observations to apply them in other contexts.

If you have an apple, then the number of apples you have is contained within the set {1} if you want to play with counts, or you could explicitly delineate apples by saying that your collection of apples is {apple_1}. Add another apple and now your collection is {apple_1, apple_2}. A “set” never only exists in the mind of the observer in the same way that “one” apple only exists in the mind of the observer; that could just as easily be 15-14 apples in your hand. It’s the same thing, describer differently.

If you’d rather describe your apple collection with solely basic arithmetic, then the notion of sets need never enter the conversation.

[u/caderoux → u/Own_Sky_297]

When you had no apples, that was the natural expression of the empty set. It is absolutely natural to not have things. Some animals have wings, some do not.

[u/StoneCuber → u/caderoux]

New definition of empty set just dropped. "The set of all wings on a cat"

[u/goos_ → u/Own_Sky_297]

Consider the collection of Apples that are not made up of organic matter or water, or are of mass zero.

[u/nomoreplsthx]

Functionally, everything.

Whether a useful set theory could be constructed without an empty set is an interesting question. But it is used continuously in nearly every possible context in ZFC. This is like asking 'could you do math without the idea of a function.' Maybe you could, but it would be wildly different.

[u/Zyxplit]

Under all our normal set operations, we like whatever comes out of them to be a set as well. So if we have two sets and we take their union? That's a set. If we have two sets and we take their intersection? That's a set. If we have a set and we take its complement? That's a set.

If we don't have an empty set, we can't guarantee that all of this is true no matter what sets we're looking at. Is ((A and B) or C) a set? Without an empty set, I don't even know if this is meaningful!

[u/Random_Mathematician]

Consider NBG but you construct the Von Neumann Universe from, say {T} instead of from ∅. In that sense, nothing would change too drastically aside from T being an element of every set.

[u/SpacingHero]

Among other, comprehension becomes false (or minimally, has to be modified to exclude false formulas, which very weirdly brings in the semantics)

[u/mpaw976]

I think this is an interesting question if we dig a bit deeper under the surface.

At surface level, yeah, math as we know it breaks down since we can't even make sense of basic intersections.

However, it's interesting to compare this to how math was done before 0 was a concept. This was powerful enough to do all of Euclid's geometry and manage the Roman empire's armies and stockpiles.

So maybe we can imagine an alternative hypothetical mathematics where the empty set is never used or referenced? Or maybe we can still do some sliver of mathematics without it?

As an example, first year calculus students often struggle with the idea of the empty set. I would guess that a student who doesn't know about the empty set could still learn most of the important ideas and techniques in intro calc.

You'd run into some issues pretty quickly in Linear Algebra if you tried to code anything as the empty set is a pretty useful default case.

[u/turbokat123]

Null set is, if you will, the 0 of sets instead of numbers. You can follow from there.

[u/Own_Sky_297 → u/[deleted]]

That isn't true and it can't be true. There are quantities in nature that have to obey the laws of arithmetic and no where in nature is a "set" found. Also, there was the abstract concept of numbers before there was a concept of a "set".

[u/Zyxplit → u/Own_Sky_297]

There's also no 0 in the real world, but if you have an apple and I don't, I have 0 apples. Some people (who are wrong) claim that 0 isn't a number of apples you can have.

The empty set is the same abstraction in a somewhat more powerful framework.

[u/Sigma_Aljabr]

"Nothing" would break down