If the mathematical concept of 'x', in simplest definition means 'any number', what would the opposite of 'x' be? What I mean is, what is the symbol, if there is one, to represent 'ALL numbers'? If there isn't one, why not?

Also, I'm not talking about infinity, nor sigma, but the concept of a symbol representing all numbers as a whole, including negatives, fractions, complex/imaginary, etc.

  • Mathematicians have many different kinds of number systems. Beside the usual integers, reals, and complex numbers, and their higher dimensional analogs like the quaternions and octonions and beyond; there are the p-adics, the Gaussian integers, the perplex and dual numbers, etc. There are the transfinite ordinals and cardinals. The surreals and the hyperreals. There isn't any mathematical object that represents the collection of every type of number. There isn't even a standard definition of what a "number" is. A number is whatever seems numberlike to someone. – user4894 Jan 26 '17 at 5:49
  • ps -- That wouldn't in any way be the "opposite of x". x is a symbol, it doesn't really have an opposite. The literal question you asked doesn't have an answer or a meaning. Also of the sets you mentioned, the complex numbers are at the top of the food chain. They're symbolized as a fancy C, too bad this site doesn't support LaTeX. There's an example of it here. mathworld.wolfram.com/ComplexNumber.html – user4894 Jan 26 '17 at 5:53
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    x is a variable, i.e. a symbol, used to write formulae, i.e. expressions in formal (or semi-formal) languages: fullstop. – Mauro ALLEGRANZA Jan 26 '17 at 6:44
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    "Any" vs. "All" do exhibit an interesting linguistic difference, though both are usually thought of as universal quantifiers in logical circles. "Any", as opposed to "all", has the flavor of what we might call indeterminate or (following Ockham) "indifferent" quantification over the particulars of a domain in a distinctively "particularized" way that doesn't seem to assume a determinate class of entities in its range. "All", by contrast, seems to presuppose the entities in its range collectively, e.g., as a class constituting the domain of quantification.... – Dennis Jul 6 '17 at 3:30
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    ...cf. "Any real number can be named" vs. "All real numbers can be named". It's certainly true of any individual real in isolation, but false of the individual reals taken collectively (assuming the countability of language). Only the second seems to admit the false reading. Is this linguistic behavior relevant to your question? I ask because there's a heavy focus on "symbols" and similar elements of language. – Dennis Jul 6 '17 at 3:33

There is an interesting point behind your question, and it is broader than just the issue of numbers. The idea is that it is possible to negate a function or set, by complementation, but it is not possible to negate an individual. If x is a variable, then its job is to range over a domain of individuals, so it cannot be negated.

The logician Peter Geach used this fact to make an important point about the subject/predicate distinction in logic. If we see a sentence like "Alice is happy", how do we know which is the subject and which the predicate? In simple cases we might just observe that 'Alice' names a thing and 'happy' names a property, but in more complex cases, this explanation is not available. Geach's account of subjects and predicates is that a basic sentence can be thought of as a function satisfied by an individual. So "Alice is happy" is Happy(alice) in the same way we write a function as f(x). How do we know it is Happy(alice) and not Alice(happy)? Because if we negate the whole sentence to "it is not the case that Alice is happy" this means "Alice is not-happy"; it doesn't mean "not-Alice is happy". So we can negate (complement) 'happy' but not 'Alice' identifying it as the predicate.

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  • how do we know that "it is not the case that Alice is happy" does not mean that someone other than Alice is happy? – user6917 Feb 9 '17 at 5:19
  • That doesn't really answer to what we understand by the negation of "Alice is happy". Suppose I say "Alice is happy" and you wish to contradict me. You might say, "no she isn't" but you would hardly say, "somebody else is happy". If you say "no, but somebody else is" then you are saying two things: Alice is not happy and somebody other than Alice is happy. – Bumble Feb 9 '17 at 12:00
  • well it depends what we mean by "Alice is happy": we can be saying e.g. "Alice is happy". i'm not saying that Geach's analysis isn't helpful, but maybe you should add something more about sentences, about sentences only saying one thing, if that makes sense – user6917 Feb 9 '17 at 12:04
  • don't be offended but i'm going to make the smallest edit, and if it's a an issue at all just roll back – user6917 Feb 9 '17 at 12:10
  • -1, Because this is not an answer to the question. It is just an answer to the title; but, sadly, you didn't read the question really. The term "opposite" was meant by Sean in a naive way to designate something completely else than "opposite", and quite surely not "negation"! Please read the question and delete your answer. – user26880 Jul 5 '17 at 4:52

What you ask for is a bit tricky because you are using an informal concept to describe variables, rather than the very exacting formal definition of what 'x' really means. In general, informal concepts tend to run into trouble when you try to take the opposite of them, because they weren't sufficiently rigorously defined to survive such a treatment.

That being said, there are many cases where we wish to talk about a domain of "everything." Such a thing is often called the "Universe," and it encapsulates every "thing" which one may wish to talk about using the language of mathematics. It is often given the symbol "U," but it is typically referred to using words first, to make its meaning perfectly clear.

This "Universe" does not behave like a normal number. It behaves more like a collection of things. In fact, the mathematical term for it is a "class" (which is in contrast to calling something a "set," which is a easier to understand collection).

From there, you can look at the "for-all" notation: . I can write something like ∀x∈S, which is a series of symbols which means "for all possible values of x which are elements of the set S, the following expression is true." This for-all operator has its own behaviors in First Order Logic. You can also write something like ∀x, without the "element of S" specification. If this is done, it is assumed that x is any value within the domain of discourse (i.e. the Universe). This is probably as close to an "opposite of x" as you will get.

Of course, we also have several other meanings of the word "opposite." All of these are other concepts which are associated with the word "opposite," but have very different meanings than the one you appear to be referring to:

  • -x - additive negation of x
  • 1/x - multiplicative negation of x
  • ¬x - logical negation of x (strong)
  • not x - modal negation of x (weak)
  • x* - complex conjugate of x
  • xᶜ - set theory complement of x

If, in fact, what you were actually after was a symbol for "all numbers" and the whole discussion of "opposites" and "x" was a false path, we do have symbols for that. Most of them are written in the so called "blackboard font":

  • ℤ - The set of all natural numbers
  • ℝ - The set of all real numbers
  • ℚ - The set of all rational numbers

You rarely see a symbol for "numbers" in general because mathematicians don't have much use for such a symbol. The different sets of numbers have different enough properties that mathematicians typically want to be specific as to which set they are referring to.

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    -1 The questioner, Sean, does not really mean the opposite, he means counterpart in the sense of entirety. It is not a question of negation! "Opposite" (in the title) was just a naïve misleading paraphrase of what he means. And didn't you see that it is a question of numbers and not of propositional logic? — I already made an edit to correct this in the question. Could you please peer review this edit? Concerning this question everybody sleeps since months. – user26880 Jul 5 '17 at 23:32
  • @Zeus What evidence do you have to suggest that you understand the one possible meaning the OP is asking for? Myself, I assumed there was a deeper question the OP was going for. Otherwise, the answer is easy: the question should be migrated to Mathematics.SE – Cort Ammon Jul 5 '17 at 23:37
  • Did you already look into my edit of the question (in the queue for peer review)? I explain there in a note and in the justification for the edit that Sean is speaking explicitly about numbers in his detailed explanations what he wants. My evidence is the same as when a Freemason writes inadvertently: “I am against other races”, when he means “the organization of further auto races”, but then someone (e.g. you) thinks that he must be hauled to court because of racism. … – user26880 Jul 6 '17 at 0:08
  • … My evidence against this false interpretation is coherence, context. Not just taking for granted the isolated eye-catching suggestion of the (naively written) title of Seal’s question, which achieves absolutely no support in the further text. What I do is: abduction, reconstruction of something, which is (for any reason) obviously mutilated. – user26880 Jul 6 '17 at 0:08
  • @Zeus I have looked at your edit in peer review. As a recommendation in the future, I highly recommend not downvoting people's answers and insisting their answers are false before asking them to approve an edit whose primary purpose is to change the question such that it is more in line with one's own answer. Best of luck in your future philosophical endeavors! – Cort Ammon Jul 6 '17 at 0:23

"x" is a place-holder. It has no meaning or effect in any way. It's what replaces "x" that has meaning. By itself, is is just an abstract idea.

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  • -1, Because this is not an answer to the question. You didn't read the question really, probably because the title of the question is a blunder since it was difficult for the questioner to designate the term "cardinality" by paraphrase. – user26880 Jul 5 '17 at 5:05

no, because we cannot define "all numbers". Symbols like N for natural numbers work because we know (or can define) what natural numbers are.

A symbol like 'x' by itself does not stand for anything. In particular it does not mean "any number". It must be contextualized to something definite, e.g. "let x=3" or "let x be a member of N". Sometimes the context is not explicitly stated, but there always is one.

Suppose we were to propose that "A" is the symbol for all numbers. Then saying that x is a member of A would be vacuous. It would be a definition with no consequences, we would have no way if testing it.

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  • i don't get this answer, why would we need to be able to test every definition? and what if every x is a zebra, wouldn't it then fail the test of being a number? – user6917 Feb 9 '17 at 5:01

If x is our concept, then !x would be it's opposite. Logically, the intersection of x and !x x ^ !x = {} should result in nothing (or an empty set). In your simple case of x = any number, then !x can't be all numbers, because then the intersection of x and !x would still be x, I'd say the opposite of any number would be no number, and the opposite of all numbers would be none of the numbers.

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By x, you mean a single, arbitrary number. You are asking for a symbol to represent the whole class of numbers.

To represent multiple numbers, one possibility is to turn to set notation, typically represented by uppercase letters. For example, given S = {1,2}, S is a set that contains 2 numbers. By adding more elements, we can arrive at the concept of infinite sets.

You ask for "negatives, fractions, complex/imaginary, etc". There are standard mathematical symbols that cover the three classes specified, being the integers (ℤ), rationals (ℚ) and complex numbers (ℂ). (Fonts from Zah Lee.) For "etc", use standard set notation, e.g. S.

Your question appears to be asking for a convention for referring to all numbers. Note that ℂ includes the elements of the earlier two as subsets, but it's at least philosophically possible that there are even broader concepts of number. The convention I'm suggesting is set notation (which in turn has its own set of conventions).

set notation Sets are fundamental objects in mathematics. Intuitively, a set is merely a collection of elements or members. There are various conventions for textually denoting sets. In any particular situation, an author typically chooses from among these conventions depending on which properties of the set are most relevant to the immediate context or on which perspective is most useful. - wikipedia

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