Is it true that the statement: " $\frac{1}{0}$=5 is false statement " is unprovable statement?

Question

I had a conversation on this site about some question, and a claim had been made by one of the users on this site that the truth value of this statement(" $\frac{1}{0}$=5 is false statement " ) is undetermined. And I had many objections on why I think that "$\frac{1}{0}$=5" is false statement. Like:

1-Assuming a=0 ,can i ask why you can't assign truth value for the consequent? If we assume that the implication is true we get a contradiction ($\frac{1}{a}$ is undefined and defined), so the negation must be true, so the whole implication is false.Therefore the primes is true and the conclusion is false.

2-But the statement x=5 means: x is defined and has the value 5. Since $\frac{1}{0}$ is undefined, then saying $\frac{1}{0}$=5 is like saying: $\frac{1}{0}$ is undefined and defined at the same time, so it's false statement.

3-Regardless of what is the precise meaning of undefined. Still the statement can be translated into a form of two mutually exclusive statements( defined and undefined ) that are true at the same time, which is a contradiction. If I can say: It is not the case that $\frac{1}{0}$=5 ( what ever the interpretation of that statement is ) is true statement, then it has to be false. There is no other alternative if we are talking in the context of propositional logic. Another example : If I say blue = cat, although this statement has no meaning, but since i can't say it is true ( "blue" is not an identity of "cat" ), then it is a false statement. If i'm wrong ( which is very possible ), I want to know what is wrong with saying: blue=cat is false statement.

But I'm not sure, I hope that someone could clarify what is going on. And if it is the case that this statement is unprovable, then it would be helpful if someone could give such a proof.

Answer 1

This is a case where we need to distinguish between whether we're doing formal logic or ordinary informal/semi-formal mathematical reasoning.

In formal logic, we define what it means to for a formula to have a truth value in a particular interpretation of the non-logical symbols in them. If we want to use the notation $\frac{\text{something}}{\text{something else}}$ in the formulas we deal with, the interpretation needs to provide some value for every pair of operands, including when the denominator is zero.

However, in formal logic it is also commonly the case that we don't specify our interpretations completely. Instead we write down axioms that we assume that our interpretation will follow -- and we're quite serenely comfortable with the fact that there may be many different interpretations that all satisfy our axioms. This is the case for division by zero, where the axioms we write down typically only demands something of the outcome of the division when the denominator happens to be nonzero. One interpretation could choose to make $\frac10=5$; another could make $\frac10=42$ -- as long as both of them satisfy all of the axioms whatever we derive from the axioms will be true of both of them.

The fact that the axioms don't constrain the value of $\frac10$ means that it is a pretty useless to write down, but in formal logic "useless" doesn't necessarily mean "forbidden".

In each particular interpretation, $\frac10=5$ is either true or false. Until we have selected an interpretation it doesn't make sense to ask about its truth value. Even then, however, we can still speak about "provable" and "disprovable" from the axioms. It might very well be that $\frac10=5$ is neither provable nor disprovable from our axioms.

In ordinary mathematical reasoning we're not doing formal logic, but natural language. We're not even trying to imitate formal logic (it is the other way around; formal logic is an attempt to imitate ordinary mathematical reasoning in a way that is convenient to reason about).

Your philosophical position that "every sentence that is not true must be false" is not really tenable for arbitrary natural language sentences. It breaks down for the liar paradox:

This sentence is false.

which is certainly not true because it denies itself, but it cannot be false either because then it would be true!

So at least for the liar paradox we need to come to terms with the fact that it is possible to say something that is apparently neither true nor false. And once we have resigned to work with such a category of sentences, we might as well declare that "one divided by zero makes five" is a sentence of the same kind. Doing so matches the way people usually speak about mathematics (an unmitigatedly good thing) and sweeps all the potential problems you point at off the table, without creatiung any new problems that we don't already need to deal with anyway.

Answer 2

The problem is that "$1/0 = 5$ is false" is not a well formed statement in the usual formal logics that we use to define what it means for a statement to be true or false. At the very least, if the quoted phrase is intended to be a well formed formal statement, it isn't clear what that statement it is supposed to be.

The most common systems for formal logic, such as first-order logic, have the property that "every term is defined". "Undefined" or "valueless" terms such as $1/0$, although used in informal natural-language math, are not used. So the issue in first-order logic is not so much that "$1/0 = 5$ is false" is unprovable - the issue is that "$1/0 = 5$" doesn't directly represent a sentence of first-order logic.

There are "free logics" in which some terms can be undefined. There are several different kinds of free logic that differ in how they handle equations with undefined terms. In most or all of these systems, though, "$1/0 = 5$" is disprovable, and so "$1/0 =5$ is false" is provable. This is because, if $1/0 = 5$ then (by the definition of "$1/0$") we have $5 \cdot 1 = 0$, which is not the case.