Then what’s the difference between saying “5x - 5x = 5 has no solutions” and saying “0 = 5 has no solutions” instead?
Context matters. When someone asks, “What are the solutions to x^2 + x = 6?”, the usually-unstated context is that x is an unknown real number and we’re being asked to find all values of x that make that equation true. If we’re asked, “What are the solutions to x+y = 4?,” we’re being asked to find all pairs of real numbers (x,y) making that equation true. But what if the context for the first question was really to find all pairs of numbers (x,y) making the equation true? Then (2,0) and (2,-10) would both be solutions, in fact there would be infinitely many solutions, since once you have an x that works, any value of y can be paired with it.
So, when you asked for solutions to 5x-5x = 5, you’re implicitly giving us the context that you’re asking for value of x that makes that equation true. Of course that’s the same as asking for solutions to 0=5, but writing it that way hides your original context.
One way of making sense of the question, “What are the solutions to 0=5?” is then to say that the question is asking for values of a variable x that would make that equation true. But, as I think we all believe, there are no values of x that would do that, so there are no solutions.
What are the solutions to 2+2=4? If your context is to find values of a variable x making that true, then all values of x are solutions.
Again, context matters.
Let me spell out the terminology:
An equation is a statement that the value of one expression (the left-hand side) is equal to the value of another expression (the right-hand side).
A solution to an equation is a value of the variable(s) that makes the equation true (that is, makes the two sides have the same value).
The solution set of an equation is the set of all solutions. Depending on the equation, the solution set could be the empty set, or it could have any number of solutions.
To solve an equation is to find its solution set.
Equations can be classified as identities, conditional equations, and contradictions.
An identity is an equation that is always true (for every value of the variable(s) under consideration), such as x + x = 2x, or 5x – 5x = 0. The solution set of an identity would be the set of all real numbers (or all numbers of whatever set we are working with as possible values for the variable).
1 + 1 = 2 would be an identity, although, since it doesn’t involve a variable, it would be a bit odd to talk about its solutions. However, if it arose in a context where there was originally a variable, like 1 + 1 = 2 + 0x, or 1 + 1 + 5x – 5x = 2, it would be reasonable to think of it as being true no matter what value that variable had, and thus, all real numbers would be solutions.
But there is no frickin’ x in “0 = 5” Likewise, there is no x (or any other variable) in “2 + 2 = 4”.
Context matters.
I’m not sure a factual answer can be provided that will be accepted.
Of course, if you interpret “What is the solution to 0 = 5” as asking for the value of any other variable (or ordered n-tuple of variables, or any other sort of mathematical object), it doesn’t really matter, since the empty set is the empty set, regardless of what sorts of objects it’s specifically empty of.
In other words, the set containing no values of x and nothing else is the same set as the set containing no values of y and nothing else, and also the same set as the set containing no bananas and nothing else.
Isn’t “there is only one empty set” (or the equivalent) often used as an axiom?
Context matters.
If you have a set of sets, there can only be one empty set in it.
If you are using set theory to construct integers based upon the cardinality of sets and Peano’s axioms, the null set is the zero. (One is the set containing one element, the null set. Two contains zero and one, and three contains zero, one and two.
0 = \emptyset
1 = \{\emptyset\}
2 = \{\emptyset, \{\emptyset\}\}
3 = \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}
So lots of instances of the null set, but never more than one instance in any set. )
If you have some general collection of things you can have lots of identical things, including identical sets. The null set included.
Has any mathematics been developed in which “ \{\emptyset\} \neq \{\emptyset, \emptyset\} ” is meaningful?
\{\emptyset,\emptyset\} as a set is not really meaningful. It is identical to \{\emptyset\} simply because
\emptyset \cup \emptyset = \emptyset
Sets only contain one of any given thing. By definition.
You can define anything else you like, with whatever rules you like. Collections of things are fine. Lots of ways you can usefully build up collections. But people will frown on you if you try to call them sets, or wilfully reuse set notation.
The point about composing sets together is that we can build up numbers inside set theory. This is a big win in building a formal underpinning. Doing so in some less formal or rigorous way has a habit of leaving loose ends. The last century has see a lot of enthusiasm for the set theory approach.
The above glosses over the difference between Zermelo Frankel set theory and naive set theory, which is in and of itself a fabulous subject. Mostly we use ZF or ZFC set theory.
The theory of multisets is the study of collections that allow for multiple instances of each of its elements. However, I don’t know whether it has been applied to multiple instances of the empty set.
Yes, but I’m explicitly asking about mathematics that use (or not) the axiom “there is only one empty set” (or equivalents).
Just as “+” is different in real numbers vs integers vs modulo vs matrixes, etc. Notation is widely reused.
Thanks, @Petek.
note that
BUT, as you can imagine, [this is going beyond simple linear equations in one variable…] in algebraic geometry one can (algebraically) make sense of, for example, things like double points which are “fatter” than geometric points, also singularities, etc. by keeping track of the original polynomials, not just what the graph looks like (so for instance consider the equation x^2=0 contrasted with x=0.)
Also from the algebraic point of view you can overcome some of the technical limitations of sets by considering more general “groupoids”; that is not really what we are talking about though— that comes up when trying to for instance classify elliptic curves, it turns out that if you want them to form a nice geometric space you have to “remember” the symmetries of each curve rather than each one simply corresponding to a point.
I think the point the OP can take away from all this is that there is not a simple “explain like I’m 5” answer that is also rigorous to anything.
It can appear simple once you assume in that you’re working in a particular circumscribed realm, and also don’t pull out a magnifying glass to dig into the assumptions. Even if that realm is plain vanilla everyday arithmetic / algebra / logic over the reals.
But “simple” and “start only from first principles and be fully rigorous along the way” are mutually exclusive.
All I was ever looking for in the OP was a simple explanation of Proof by Contradiction along with a simple explanation of why it is a legitimate process. Something that could be used with people whose math skills are on the fifth-grade level. I work with several of those people. For reasons that are ENTIRELY MY FAULT, the thread quickly went into the weeds about the validity of the Law of the Excluded Middle. However, I can easily imagine a fifth-grader responding to “Mathematical statements can only be true or false.” with “Why?” That “why” question was eventually answered in the thread, more or less.
The short answer is “That’s part of the system of mathematics that we use. It’s possible to construct other systems of mathematics, but it’s really hard to do that, because most of them turn out to be really boring.”.
If you’ve got highly curious and motivated fifth-graders, then you can add “But if you really want to, you can try to construct your own system of mathematics, and see if it turns out to be something interesting.”. I did once see a class of seventh-graders develop a different system of mathematics, and do so correctly and interestingly (and it was also one that had already been invented by others, which was how I knew that it was correct and interesting, but I wasn’t about to spoil their fun).
I don’t have any fifth graders, just people on the fifth-grade level. They are neither curious nor motivated. At most, I might get, “Couldn’t I just Google it?”
I just noticed that I went from five-years old in the OP to fifth-graders in the antepenultimate (now preantepenultimate) post somehow. I don’t think it really matters.
I still stand by what I said:
If you say to me that you “do not dispute that”, then I’d explain it to you like you’re five years old — or like you’re in fifth grade — by simply repeating back to you what you just said regarding whether it has no solution.
I’d give a different explanation to someone else who says something else. But if a guy says to explain stuff to them like they’re five years old, and the very next thing they say is that they don’t dispute it, then I don’t see that there’s anything else to be said: there’s no additional info for me to relay; it’s a question of emphasis, about what’s already been said.