# No number is equal to Zero, is this statement true or false?

While reading an article on logic, there is a sentence "No number is equal to zero" and we have to assign truth values to this sentence. I hope this is true and the article says it as false.

Can someone explain me why it is false? Or the author of the article is wrong?

• Why on earth do you hope that this is true? Dec 20, 2013 at 10:55
• @Gilles the model is set of natural numbers Dec 20, 2013 at 10:57
• @babou, it is not that this is non-obvious; it is that the question is not posed very clearly. For instance, the article is not written very clearly or precisely, so we have to make guesses about the intended meaning of that sentence. The fact that different people make different guesses about what precisely that sentence was intended to mean doesn't imply that it's a good question; if it anything, it suggests that maybe it's not such a great question. Just my opinion.
– D.W.
Dec 20, 2013 at 23:57
• @D.W. I beg to (dis)agree. What is not clear enough is the paper the OP is trying to understand, not his question. The paper is not self contained and you actually have to look at the hyperlinks, and understand them, though there are not all essential. The OP should probably have given some more details to explain his understanding, but it may have seemed obvious from his point of view: considering only the positive natural numbers. After all, it took a long time historically to consider zero as a number too. The role of the word "zero", and why it should mean anything is not well explained. Dec 21, 2013 at 16:49
• @TCSL What the article states is that statement (5) is false in all models. I understand that you object because you consider that positive natural numbers do not have a number zero. But the question is whether (together with the usual operation) they constitute a model. And then you have to address the central question: a model of what? It has to be a model of what the old man described. If the description included an element called zero, as hinted by the link referencing rings, then any model must include a zero. And if you call numbers the elements of your model, then zero is a number. Dec 21, 2013 at 17:25

It is false if and only if there is a number that is equal to zero.

"There is a number that is equal to zero" is true if and only if zero is a number. Without any context (there might e.g. be some very strange definitions given earlier in the exercise), we cannot say whether zero is considered a number in this case. If there is no other context given in the exercise, it should be clear that zero is a number.

For example, the domain of discourse might be only positive integers*, when zero is not a number, but then it would be very weird to use the word "zero" in the sentence in the first place.

*The set of natural numbers has two contradicting definitions, $\mathbb{N}=\{0,1,2,\dots\}$ (non-negative integers) and $\mathbb{N}=\{1,2,\dots\}$ (positive integers). This might be the source of confusion in your case.

• Don't you think the answer depends on the model we are using? Dec 20, 2013 at 10:57
• I agree that if zero is not a number, then no number is not equal to zero. On the other hand, the question mentions zero, so I thought it is assumed that zero is a number. Probably the textbook defines the set of natural numbers as $\{0,1,2,\dots\}$, while some define it as $\{1,2,\dots\}$, which may cause confusion in this case.
– JiK
Dec 20, 2013 at 11:05
• @TCSLearner A definition of “number” that excludes 0 would be really weird — ”number“ would be the wrong word in that case. Unless the definition is several centuries old, predating not only modern logic but also calculus and even algebra. Dec 20, 2013 at 11:22
• I edited the answer to include some of these points.
– JiK
Dec 20, 2013 at 11:33
• @TCSLearner I read the original article, but do not see your point. Statement (5) is false considering all models... for the old man's description. A model is something that fits some axiomatic description. There is no such thing as a model without a reference to a description the model must fit. So you must ask yourself what is that description. A statement is true for all models if it can be logically proved from the axioms of the description. Note however that, in some axiomatic theories, some statements may be true of all models without being provable (Gödel incompleteness theorem). Dec 22, 2013 at 1:22

What I gathered, form a cursory reading of the paper is that the old man described the algebraic properties of the rational numbers, by actually describing what a ring is, as the corresponding wikipedia article is explicitly referenced in the text (though the word ring is not given). Hence, he must have specified that the addition has a special element called zero wich is an additive identity.

Hence any model that fits the description given by the old man, i.e. that is actually a ring, must have an element called zero with the required properties. This is certainly true of the rationals, and it must be true of what you (as the character of the story) call the integers since you have no disagreement with the old man.

And, as stated in the article, it will actually be true of any model meeting the description of the old man. So, for any model you care to call "numbers", the statement "(5) No number is equal to zero" is clearly false.

However, I am bothered by statements (1), (3), and (4) that refer to the number 2, as the axioms for a ring do not define what 2 might be. It is not supposed to have been defined in the discussion, and it is therefore difficult to assign any meaning to a statement using it. They should have at least agreed that 2 is a notation for the result of "1+1".

Your confusion comes from trying to describe a mathematical statement in English -- but not very precisely. English often isn't very precise. Here it isn't clear exactly what the intended translation of the English into mathematics is. Let me outline a few possible translations:

• $\forall x \in \mathbb{Z} . x \ne 0$. This proposition is false. $\mathbb{Z}$ (the set of all integers) does include $0$.

• $\forall x \in \mathbb{N} . x \ne 0$. This proposition is false, using the usual definition of $\mathbb{N}$ as the set of all natural numbers (i.e., all non-negative integers, i.e., $\mathbb{N}=\{0,1,2,3,\dots\}$).

• $\forall x \in \mathbb{Q} . x \ne 0$. This proposition is false. $\mathbb{Q}$ (the set of all rationals) does include $0$.

• $\forall x \in \{1,2\} . x \ne 0$. This proposition is true, since the set $\{1,2\}$ does not include an element $0$.

• $\forall x . x \ne 0$. The truth value of this proposition is indeterminate, without knowing the universe over which $x$ is quantified. If we are letting $x$ range over all of $\mathbb{Z}$, this is true. If we are letting $x$ range over $\{1,2\}$, this is false.

From a practical perspective, I think we could say the following: in this particular example, I think nothing very interesting is happening. It's just a case of being imprecise with notation. So, from a practical perspective: take this as a lesson to try to be explicit about what you are quantifying over, to ensure there is no possibility of confusion.

From a theoretical perspective, here is what is going on. That article is talking about propositional logic. Propositional logic is, in some sense, a framework for reasoning about what is true regardless of what universe you are quantifying over. In other words, in some sense propositional logic wants to help us reason about propositions where we don't need to specify the set that $x$ ranges over, because the proposition will be true either way. This is where models come in. A model is basically a set that all quantified variables range over (it also specifies, for each predicate, which elements of that set make the predicate true). Now we can have a propositional formula that is true for all models; false for all models; or true for some models and false for others.

For more, see https://en.wikipedia.org/wiki/Interpretation_%28logic%29 and textbook introductions to propositional logic and first-order logic.

In the part of the article you're referring to, "number" means either "integer" or "rational" (specifically, the conceit is that you're talking to an old man and the old man thinks you're talking about the rationals, while you think you're talking about the integers). There is an integer equal to zero and a rational equal to zero so, in either case, the sentence is true: the number equal to zero is zero itself.

Quick Intro

The Theory Of General Polarity - fundamental logicaly based theory

{ 0 } = { +∞, 0, ∞- }

Testing stated to literate it trough fundamentaly - No number is equal to Zero

First, lets find out with what values we are going to deal and how are going to handle them:

• 0 / Zero - the most fascinating, uncountable, fully conservational, all conserved, all purpose/not usable, fundamentaly universal, unlimited, boundless, never starting, unreachable, unachivable, the master value - zero is only value that can divide infinity without any violation, making no any change to infinity's invalues, setting a zero division in middle. No need to ask - is there anything above absolute?
1. Do not exist - it is not a number it is absence of number and have no any value exposed, therefore, it is not interactive.

2. Always present - Natural number, always present and available and have a value exposed, value is empty so it contains nothing.

3. Non-polar - Not any positive and that much negative. Just zero.

4. Fully polar - Positive as much as negative, infinitely positive as negative.

5. Absolute - Absent out of a quantum mathematical possibilities, no factor ever been here. Cannot be reached in any sense ever.

• No number - Without number, or no any number here only, is going to be equal. Basicaly, I have a hard time finding out, why I wouldn't set a letter O there? In that logic, letter O is going to be equal to the zero: O = 0

There is also a values that lacks an exaction, meaning that same value is having a many exactions at once. I called those the universal values Example:

Statement.value = true

Statement.value = false

If you check this value as: Is statement value true? Ok, statement value is true.

If you check this value as: Is statement value false? Ok, statement value is false.

If you check this value as: Is statement value true and false together? Ok, statement value is true and false together.

If observed this deeper in "what a heck is this???" style, it is going to be obvious that maths handles (or even not cares at all, ignores) also non-exacting/many exactions at once - universal values.

To demonstrate this behaviour I'm going to divide exact value with zero (check above zero universal value definitions after)

100 /  = { 0 } - ready!
0. Operation of division can't be started and it won't budge any further before start, cause divisor is missing or not having any value exposed for operation to proceed. Untill I get value by what to divide it won't budge any.

100 / 0 = { +∞, 0, ∞- } - on air!
0. OK - Operation of division always starts but with both values. The divisor is fully having a value (empty value that contains nothing) exposed for operation to proceed.

*What is it? You are waiting something? Noooo, don't let me know you awaiting results of operation man, not right now xD*


Conclusion of these non-exactions: Zero = Infinity (positive+ negative-) Never started = never going to end Nothing = Everything True = False For sure = No dices Never = Always Body = Soul

Answer to the question: No number is equal to Zero - is true or false?