What does it mean for a mathematical statement to be true?

As I understand it, mathematics is concerned with correct deductions using postulates and rules of inference. From what I have seen, statements are called true if they are correct deductions and false if they are incorrect deductions. If this is the case, then there is no need for the words true and false. I have read something along the lines that Godel's incompleteness theorems prove that there are true statements which are unprovable, but if you cannot prove a statement, how can you be certain that it is true? And if a statement is unprovable, what does it mean to say that it is true?

$\begingroup$ Philosophers argue ad infinitum about correspondence versus coherence theories of truth. $\endgroup$

$\begingroup$ For many mathematicians, the truth/untruth of a statement is not quite the same as whether it can be formally deduced from a given set of axioms! For instance, I might regard a given axiom as "false". $\endgroup$

$\begingroup$ As I understand it, writing is concerned with putting words together into grammatically correct sentences. $\endgroup$

$\begingroup$ @Qiaochu: Yes, I knew I had the idea from somewhere, but couldn't remember where. Information overload and all that. $\endgroup$

15 Answers 15

Tarski defined what it means to say that a first-order statement is true in a structure $M\models \varphi$ by a simple induction on formulas. This is a completely mathematical definition of truth.

Goedel defined what it means to say that a statement $\varphi$ is provable from a theory $T$, namely, there should be a finite sequence of statements constituting a proof, meaning that each statement is either an axiom or follows from earlier statements by certain logical rules. (There are numerous equivalent proof systems, useful for various purposes.)

The Completeness Theorem of first order logic, proved by Goedel, asserts that a statement $\varphi$ is true in all models of a theory $T$ if and only if there is a proof of $\varphi$ from $T$. Thus, for example, any statement in the language of group theory is true in all groups if and only if there is a proof of that statement from the basic group axioms.

The Incompleteness Theorem, also proved by Goedel, asserts that any consistent theory $T$ extending some a very weak theory of arithmetic admits statements $\varphi$ that are not provable from $T$, but which are true in the intended model of the natural numbers. That is, we prove in a stronger theory that is able to speak of this intended model that $\varphi$ is true there, and we also prove that $\varphi$ is not provable in $T$. This is the sense in which there are true-but-unprovable statements.

The situation can be confusing if you think of provable as a notion by itself, without thinking much about varying the collection of axioms. After all, as the background theory becomes stronger, we can of course prove more and more. The true-but-unprovable statement is really unprovable-in-$T$, but provable in a stronger theory.

Actually, although ZFC proves that every arithmetic statement is either true or false in the standard model of the natural numbers, nevertheless there are certain statements for which ZFC does not prove which of these situations occurs.

Much or almost all of mathematics can be viewed with the set-theoretical axioms ZFC as the background theory, and so for most of mathematics, the naive view equating true with provable in ZFC will not get you into trouble. But the independence phenomenon will eventually arrive, making such a view ultimately unsustainable. The fact is that there are numerous mathematical questions that cannot be settled on the basis of ZFC, such as the Continuum Hypothesis and many other examples. We have of course many strengthenings of ZFC to stronger theories, involving large cardinals and other set-theoretic principles, and these stronger theories settle many of those independent questions. Some set theorists have a view that these various stronger theories are approaching some kind of undescribable limit theory, and that it is that limit theory that is the true theory of sets. Others have a view that set-theoretic truth is inherently unsettled, and that we really have a multiverse of different concepts of set. On that view, the situation is that we seem to have no standard model of sets, in the way that we seem to have a standard model of arithmetic.

$\begingroup$ Can the phrase "these various stronger theories are approaching some kind of undescribable limit theory" be made formally precise in some sense? $\endgroup$

$\begingroup$ It is extremely difficult and is the subject of current philosophical work by Woodin, Koellner, Maddy, Hauser and others. But many large cardinal set theorists seem to espouse a view of this sort. $\endgroup$

$\begingroup$ Can you clarify what the "intended model of arithmetic" is? It seems to me that the Incompleteness Theorem is really a statement about this model, that no theory can prove every statement that is true in it. $\endgroup$

$\begingroup$ The intended model is $\langle N,+,\cdot,0,1,\lt\rangle$, also known as the standard model of arithmetic. And you are right that one way to understand the Incompleteness Theorem is that it says we cannot write down a complete axiomatization of the theory of this structure. $\endgroup$

Part of the reason for the confusion here is that the word "true" is sometimes used informally, and at other times it is used as a technical mathematical term.

Informally, asserting that "X is true" is usually just another way to assert X itself. When I say, "I believe that the Riemann hypothesis is true," I just mean that I believe that all the non-trivial zeros of the Riemann zeta-function lie on the critical line. (Note in particular that I'm not claiming to have a proof of the Riemann hypothesis!) This insight is due to Tarski. If you know what a mathematical statement X asserts, then "X is true" states no more and no less than what X itself asserts. Now, there is a slight caveat here: Mathematicians being cautious folk, some of them will refrain from asserting that X is true unless they know how to prove X or at least believe that X has been proved. So in some informal contexts, "X is true" actually means "X is proved." As we would expect of informal discourse, the usage of the word is not always consistent.

The word "true" can, however, be defined mathematically. Truth is a property of sentences. If you have defined a formal language $L$, such as the first-order language of arithmetic, then you can define a sentence $S$ in $L$ to be true if and only if $S$ holds of the natural numbers. So for example the sentence $\exists x: x > 0$ is true because there does indeed exist a natural number greater than 0. Here it is important to note that true is not the same as provable. The formal sentence corresponding to the twin prime conjecture (which I won't bother writing out here) is true if and only if there are infinitely many twin primes, and it doesn't matter that we have no idea how to prove or disprove the conjecture.

Now, perhaps this bothers you. Is it legitimate to define truth in this manner? Some people don't think so. However, note that there is really nothing different going on here from what we normally do in mathematics. When we were sitting in our number theory class, we all knew what it meant for there to be infinitely many twin primes. Why should we suddenly stop understanding what this means when we move to the mathematical logic classroom? If we understand what it means, then there should be no problem with defining some particular formal sentence to be true if and only if there are infinitely many twin primes. It is as legitimate a mathematical definition as any other mathematical definition.

Now, how can we have true but unprovable statements? And if we had one how would we know? Joel David Hamkins explained this well, but in brief, "unprovable" is always with respect to some set of axioms. Therefore it is possible for some statement to be true but unprovable from some particular set of axioms $A$. In order to know that it's true, of course, we still have to prove it, but that will be a proof from some other set of axioms besides $A$.

I say that $$ \int_<-\infty>^ <+\infty>e^ dx = \sqrt <\pi>$$ is true . and any talk about "correct deductions using postulates" is a rationalization added later. Mathematicians knew this fact (yes, fact) even before the field of mathematical logic began.

$\begingroup$ Do you have a similar view about the Continuum Hypothesis? This question was also asked before math logic began, first by Cantor, and then famously, as the first Hilbert problem. $\endgroup$

$\begingroup$ If it was not possible to prove this, would you still be confident in calling this true? $\endgroup$

$\begingroup$ If someone asks you why it is true, then you will deduce it from something more basic. And if they ask you why that is true, you will deduce it from something more basic still. What you won't say (unless you want to leave your questioner very dissatisfied) is that it's just a Platonic fact about the mathematical universe. Obviously these "why" questions have to come to an end eventually, and . bingo . there are your postulates. $\endgroup$

$\begingroup$ The "truth" of the above integral equality is only possible because "we know what you mean". But the movement to establish foundations for calculus, etc. only arose because people started to lose confidence that they really did know what was meant. --- How do we know what you mean? Because the integral is defined in a certain way; specifically, with sufficient detail that it becomes possible to demonstrate the truth of the equality above. Intuitions regarding interesting relationships do not become "true" or "false" until crystallized into axiomatizable definitions admitting proofs. $\endgroup$

Both the optimistic view that all true mathematical statements can be proven and its denial are respectable positions in the philosophy of mathematics, with the pessimistic view being more popular. The question is more philosophical than mathematical, hence, I guess, your question's downvotes.

Neil Tennant 's Taming of the True (1997) argues for the optimistic thesis, and covers a lot of ground on the way. I recommend it to you if you want to explore the issue.

$\begingroup$ How can it be philosophical? Godel's theorem is mathematics, so the terms have to be defined. $\endgroup$

$\begingroup$ @NR: it's worth noting that Gödel's incompleteness and completeness results deal with a vey specific kind of hypotheses and proofs; that is to say, first order. In general, a statement is true if whenever the hypotheses are true, the conclusion is. In the case of first order logic this can be restated as: for any model where the hypotheses are true, the conclusion is. However, whether or not you're in first order, what you really care about is whether there's an example where the hypotheses are true but the conclusion is wrong. $\endgroup$

$\begingroup$ It occurred to me that there's another point to make here. Even if you care about groups, fields, etc., your theorem can be stated in the language of sets. You may, if it's first order in the language of groups, etc., treat it as such. But the more general notion is being first order in sets. $\endgroup$

$\begingroup$ @Negative What if the terms can't be defined? en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem $\endgroup$

$\begingroup$ @H Hasson: First sentence is true, but misses the depth of the result. Because the Halting Problem is undecidable and individual cases can be encoded as 1st-order sentences of number theory, the set Th(N) of true 1st-order sentences of number theory is not computable, and therefore is not computably enumerable. Now for any proof system worthy of the name, the set of provable [1st-order number theory] sentences is c.e. It therefore cannot coincide with Th(N) — either the system proves false sentences, or (preferably) fails to prove true ones. One only need encompass 1st-order theory. $\endgroup$

The Stanford Encyclopedia of Philosophy has several articles on theories of truth, which may be helpful for getting acquainted with what is known in the area. Their top-level article is

There are several more specialized articles in the table of contents.

Let me offer an explanation of the difference between truth and provability from postulates which is (I think) slightly different from those already presented. (Although perhaps close in spirit to that of Gerald Edgars's.)

First of all, if we are talking about results of the form "for all groups, . " or "for all topological spaces, . " then in this case truth and provability are essentially the same: a result is true if it can be deduced from the axioms. (There is the caveat that the notion of group or topological space involves the underlying notion of set, and so the choice of ambient set theory plays a role. This role is usually tacit, but for certain questions becomes overt and important; nevertheless, I will ignore it here, possibly at my peril.)

But other results, e.g in number theory, reason not from axioms but from the natural numbers. Of course, along the way, you may use results from group theory, field theory, topology, . which will be applicable provided that you apply them to structures that satisfy the axioms of the relevant theory. But in the end, everything rests on the properties of the natural numbers, which (by Godel) we know can't be captured by the Peano axioms (or any other finitary axiom scheme). How do we agree on what is true then? Well, experience shows that humans have a common conception of the natural numbers, from which they can reason in a consistent fashion; and so there is agreement on truth.

If you like, this is not so different from the model theoretic description of truth, except that I want to add that we are given certain models (e.g. the standard model of the natural numbers) on which we agree and which form the basis for much of our mathematics. (Again, certain types of reasoning, e.g. about arbitrary subsets of the natural numbers, can lead to set-theoretic complications, and hence (at least potential) disagreement, but let me also ignore that here.)

In summary: certain areas of mathematics (e.g. number theory) are not about deductions from systems of axioms, but rather about studying properties of certain fundamental mathematical objects. Axiomatic reasoning then plays a role, but is not the fundamental point.

$\begingroup$ But ultimately, won't the independence phenomenon still arrive? After all, even for the strongest known axiomatizations of mathematics, such as ZFC+large cardinals, Goedel's theorem still provides arithmetic statements that are neither provable nor refutable. Although ZFC proves that there is a standard model of arithmetic, different models of ZFC can have non-isomorphic standard models (and even non-elementarily equivalent). So the question of what is true in the standard model depends on the set-theoretic background. $\endgroup$

$\begingroup$ Dear Joel, This is the question swept under the rug by my parenthetical decisions to ignore set-theoretic issues. My own position on this is unabashedly platonist, in that I believe that there is a true standard model for the natural numbers (the one in my imagination!), and perhaps a little naive. Thankfully, the comment box doesn't give me enough space to try to seriously defend my position! $\endgroup$

There are two answers to your question:

• A statement is true in absolute if it can be proven formally from the axioms.
• A statement is true in a model if, using the interpretation of the formulas inside the model, it is a valid statement about those interpretations.

Assuming your set of axioms is consistent (which is equivalent to the existence of a model), then
$\qquad$ truth in absolute $\Rightarrow$ truth in any model.

Conversely, if a statement is not true in absolute, then there exists a model in which it is false.

Let's take an example to illustrate all this. Let $P$ be a property of integer numbers, and let's assume that you want to know whether the formula $\exists n\in \mathbb Z : P(n)$ is true. Three situations can occur:
• You're able to find $n\in \mathbb Z$ such that $P(n)$.
• You're able to prove that $\not\exists n\in \mathbb Z : P(n)$
• Neither of the above.
In the latter case, there will exist a model $\tilde<\mathbb Z>$ of the integers (it's going to be some ring, probably much bigger than $\mathbb Z$, and that satisfies all the axioms that "characterize" $\mathbb Z$) that contains an element $n\in \tilde <\mathbb Z>$ satisgying $P$.

This is a question which I spent some time thinking about myself when first encountering Goedel's incompleteness theorems. I should add the disclaimer that I am no expert in logic and set theory, but I think I can answer this question sufficiently well to understand statements such as Goedel's incompleteness theorems (at least, sufficiently well to satisfy myself).

One one end of the scale, there are statements such as CH and AOC which are independent of ZF set theory, so it is not at all clear if they are really true and we could argue about such things forever. Even for statements which are true in the sense that it is possible to prove that they hold in all models of ZF, it is still possible that in an alternative theory they could fail. Even things like the intermediate value theorem, which I think we can agree is true, can fail with intuitionistic logic.

On the other end of the scale, there are statements which we should agree are true independently of any model of set theory or foundation of maths. For example, I know that 3+4=7. There are simple rules for addition of integers which we just have to follow to determine that such an identity holds. You might come up with some freaky model of integer addition following different rules where 3+4=6, but that is really a different statement involving a different operation from what is commonly understood by addition. Similarly, I know that there are positive integral solutions to $x^2+y^2=z^2$. To verify that such equations have a solution we just need to iterate through all possible triples $(x,y,z)\in\mathbb^3$ and test whether $x^2+y^2=z^2$, stopping when a solution is reached. In this case we are guaranteed to arrive at some solution, such as (3,4,5), proving that there is indeed a solution to the equation.

More generally, consider any statement which can be interpreted in terms of a deterministic, computable, algorithm. If we simply follow through that algorithm and find that, after some finite number of steps, the algorithm terminates in some state then the truth of that statement should hold regardless of the logic system we are founding our mathematical universe on.

So, there are statements of the following form: "A specified program (P) for some Turing machine and given initial state (S0) will eventually terminate in some specified final state (S1)". If such a statement is true, then we can prove it by simply running the program - step by step until it reaches the final state. Such statements, I would say, must be true in all reasonable foundations of logic & maths. Identities involving addition and multiplication of integers fall into this category, as there are standard rules of addition & multiplication which we can program. So does the existence of solutions to diophantine equations like $x^2+y^2=z^2$. Existence in any one reasonable logic system implies existence in any other.

At the next level, there are statements which are falsifiable by a computable algorithm, which are of the following form: "A specified program (P) for some Turing machine with initial state (S0) will never terminate". For example, "There are no positive integer solutions to $x^3+y^3=z^3$" fall into this category. You can write a program to iterate through all triples (x,y,z) checking whether $x^3+y^3=z^3$. Fermat's last theorem tells us that this will never terminate. We can never prove this by running such a program, as it would take forever. However, the negation of statement such as this is just of the previous form, whose truth I just argued, holds independently of the "reasonable" logic system used (this is basically $\omega$-consistency, used by Goedel). That is, if I can write an algorithm which I can prove is never going to terminate, then I wouldn't believe some alternative logic which claimed that it did. In the same way, if you came up with some alternative logical theory claiming that there there are positive integer solutions to $x^3+y^3=z^3$ (without providing any explicit solutions, of course), then I wouldn't hesitate in saying that the theory is wrong. Statements like $$ \int_<-\infty>^\infty e^\\,dx=\sqrt <\pi>$$ are also of this form. Assuming we agree on what integration, $e^$, $\pi$ and $\sqrt$ mean, then we can write a program which will evaluate both sides of this identity to ever increasing levels of accuracy, and terminates if the two sides disagree to this accuracy. The identity is then equivalent to the statement that this program never terminates.

Going through the proof of Goedels incompleteness theorem generates a statement of the above form. i.e., "Program P with initial state S0 never terminates" with two properties. (1) If the program P terminates it returns a proof that the program never terminates in the logic system. (2) If there exists a proof that P terminates in the logic system, then P never terminates.

So, if P terminated then it would generate a proof that the logic system is inconsistent and, similarly, if the program never terminates then it is not possible to prove this within the given logic system.

In fact, P can be constructed as a program which searches through all possible proof strings in the logic system until it finds a proof of "P never terminates", at which point it terminates. The assumptions required for the logic system are that is "effectively generated", basically meaning that it is possible to write a program checking all possible proofs of a statement.