Incompleteness Theorem 1 is about whether all mathematical truths can be proven.

The first incompleteness theorem says there will always be unprovable truths within every sufficiently powerful and non-contradictory mathematical system.

In order to prove these “unprovable” truths, you’d have to go outside that system into a bigger system, but that bigger system would then have its own unprovable truths as well, and to prove those, you’d need an even bigger system, and so on.

The exceptions are mathematical systems that you can’t use for arithmetic and mathematical systems that are self-contradictory (and therefore useless).

Why is this theorem true?

Step 1: Number all mathematical statements so that they can talk about each other

Gödel figured out how to assign a unique number to each statement in arithmetic. So, for instance, the statement “1+1=2” might be (just to make up a simple example) statement #1, and “4 < 5” might be statement #2 (realistically, these would be assigned huge numbers, but to keep things simple I’ll use small numbers).

Numbering all of the statements allows statements to make claims about each other in the language of arithmetic. For instance, maybe statement #3 says, “Statement #2 cannot be proven” (but it says it in the completely precise language of arithmetic, not in words – these statements are completely normal math, much like a typical mathematical statement you’d find in a math textbook, and so you’d expect each of these statements to either be true or false just like 1+1=2 is true and 1+1=3 is false).

Step 2: Create a self-referential statement

Gödel then considers a special statement, which he proved always will exist (say, for simplicity, that it’s statement #4). Statement #4 says, “Statement #4 *cannot* be proven in this system.” So, it refers to itself. In essence, in completely precise mathematics (rather than words), the statement says, referring to itself, “I cannot be proven.” That’s not the same as a statement that says “I am false” – being unprovable and being false are different, as we’ll see.

Step 3: Produce a contradiction

Now, we can show that statement #4 cannot be proven.

That’s because if the mathematical system *could* be used to prove that statement #4 is true, that would produce a contradiction. Statement #4 claims that statement #4 *cannot* be proved – so if you prove it’s true, then that implies you can’t prove it’s true!

Since we’ve assumed our mathematical system contains no contradictions, that means statement #4 can’t be proved in that system – which is precisely what statement #4 claims! So, statement #4 is true. Hence, it is a true statement that can’t be proved in that system.

Since Gödel showed that a statement like statement #4 can always be created in any mathematical system powerful enough to allow for arithmetic, any such system that has no contradictions has true statements that the system can’t be used to prove!

Incompleteness Theorem 2 is about whether you can prove that mathematical systems contain contradictions.

If a mathematical system contained a contradiction (e.g., that 1=2), that would make the system useless, so it would be really nice to be able to show that such a system contains no contradictions.

However, Incompleteness Theorem 2 says that a mathematical system with no contradictions that’s powerful enough to include arithmetic can’t prove that it itself has no contradictions. Or, more formally, that means that you can’t use the system to prove: “This system’s axioms do not lead to a contradiction.”

Yet another way to put it is, if such a mathematical system proves that it itself has no contradictions, then that system actually has contradictions!

Why? Well, let’s return to our statement #4 from before. Recall that this statement refers to itself, as it’s the statement that says, “I can’t be proven in this system” (though in completely precise mathematics, not in words).

Note that if statement #4 is false, that would produce a contradiction. Since statement #4 claims that statement #4 can’t be proven to be true, if that’s false, then statement #4 *can* be proven to be true, which is the same as saying it’s actually a true statement – so it being false implies it is true, which is contradictory!

What this means is that if we can prove that the system we’re working in has no contradictions, then that proof ALSO implies that statement #4 is true (i.e., that the system can prove statement #4).

But we already know from the first theorem that statement #4 can’t be proven to be true using the system!

Hence, the system must not be able to prove that it itself contains no contradictions (because if it did, it would violate theorem 1).

This tells us that any mathematical system powerful enough to include arithmetic can’t be used to prove that it itself has no contradictions (unless it actually contains contradictions).

A big thanks goes to Norman Perlmutter for his very helpful feedback on an earlier draft of this post!

This post was first written on November 17, 2023, and first appeared on my website on January 3, 2024.

But can mathematicians actually prove that math is true? If they can’t, does the fact that math is so useful in solving real-world problems provide evidence of its truth? And, if mathematics is not true, then does that imply that conclusions drawn from it are faulty or suspect? Let’s explore those questions.

The first attempt we might take to prove that math is true is to consider real-world situations where equations seem to appear. Some examples are:

• If I have three red balls in a bag and add two more, the bag will then contain five red balls (3 balls in a bag with 2 balls added to the bag gives 5 balls).

• If I am on a train traveling at three miles per hour and throw a ball at two miles per hour (measured with respect to the train), then the ball will be traveling at five miles per hour with respect to the ground (3 mph sped up by 2 mph gives 5 mph).

• If I had three dollars worth of goods yesterday and borrowed two dollars worth of goods from you today, then I have five dollars worth of goods in my possession (3 dollars of goods with an additional 2 dollars of goods borrowed yields 5 dollars of goods).

Each of these three situations seems to imply the equation 3+2=5. But do they actually PROVE that the equation 3+2=5 is true?

One problem with drawing conclusions about mathematics from these examples is that the number ‘3’ is not the same as ‘3 balls’ or ‘3 hours’ or ‘3 dollars’, and the operator ‘+’ is not the same as grouping balls or combining velocities or aggregating wealth.

While 3+2=5 is typically an excellent model for each of these situations, the equation is not precisely equivalent to these situations. Why not? Well, it’s true that when we group balls (by, in this case, placing them in a bag), the procedure generally behaves as though we are performing addition. But now suppose that the objects we are grouping together are made of packed sand. In this case, when we add new objects to our bag, they will sometimes fracture and split into multiple objects. Or if the balls are made of wet clay, they may fuse into a single object in the bag. The addition operator ‘+’ no longer models this situation well because when we place two new objects in the bag, it does not always increase the number of objects contained in the bag by two. So addition is not a perfect model for grouping physical objects in a confined space.

What about the other examples? Einstein’s theory of relativity tells us (in contradiction to the more intuitive but less accurate equations of Newtonian mechanics) that when a person on a train (which is moving three miles per hour with respect to the ground) throws a ball at two miles per hour (with respect to the train), then the speed of the ball with respect to the ground is actually very slightly less than 5 miles per hour, not equal to 5 miles per hour. So while addition is very accurate for modeling that situation, it’s known that it does not, in fact, give the exact correct answer.

What about the last example? If I had three dollars worth of goods yesterday and then borrowed two dollars worth of goods from you today, the total number of dollars worth of goods that I have possession of will not necessarily be five dollars if the value of my original goods changed between yesterday and today (as can happen in real economic markets).

What these examples show us is that the only reason to say that grouping balls or combining velocities or aggregating wealth encapsulates the idea of mathematical addition is that most of the time, the addition operator ‘+’ provides a good MODEL for these scenarios. We can no more conclude that 3+2=5 is a true statement simply because putting two balls into a bag that already has three balls usually produces a bag with five balls, then we can conclude that 3+2=5 is false, simply because if you’re using balls of packed sand, sometimes the balls will fracture into more balls when you place them in the bag. In other words, while real-world situations can motivate the equations of mathematics and provide justifications for applying them, they cannot prove that those equations are actually true.

We have stared at equations like 3+2=5 so many times in our lives that it can be difficult to consider them with fresh eyes in order to ask ourselves what it really is that they are saying. Clearly, ‘3’, ‘+’, ‘2’, ‘=’, and ‘5’ are not objects in the physical universe. You can go to the zoo and see three bears, or see the numeral ‘3’ printed on a sign, or perform arithmetic on paper using the symbol ‘3’, but nowhere in the universe can you find the actual (metaphysical) number ‘3’. This is hardly surprising, since ‘3’ is a concept or idea, not a physical thing. But this line of thought implies that 3+2=5 is a statement about the relationship among the concepts ‘3’, ‘2’, and ‘5’, and not a statement about physical entities that actually exist. The only time that 3+2=5 is a statement about physical things that actually exist is when we use it as a model for real-world properties that are sufficiently similar to the concepts for the model to be useful.

But how do we define the word “true” when it comes to relations among abstract concepts? One possible approach is to say that statements about abstract concepts are true if they follow as a logical consequence of the definitions of the concepts themselves.

This leads us to ask whether 3+2=5 and all other mathematical statements are simply true by definition as a consequence of our chosen definitions for ‘3’, ‘+’, ‘2’, ‘=’, ‘5’, and the other mathematical objects.

Unfortunately, this question cannot be answered without further qualification. How do we choose to define concepts such as ‘3’? Various authors have attempted to define mathematics by developing lists of axioms (which are simply assumed to be true) and then proving that the basic mathematical objects (e.g., integers) and theorems (e.g., a+b = b+a) follow from these axioms. There are a variety of different ways that math can be axiomatized (i.e., built up from basic axioms). Some approaches use sets as the most basic objects (as is done in what is probably the most popular axiomatization, Zermelo-Fraenkel set theory).

In contrast, others use Category Theory to provide the basic building blocks. Still, other theories attempt to axiomatize only small portions of math, such as Euclid’s Axioms of planar geometry, Hilbert’s axiomatization of Euclidean Geometry, and the Peano axioms for arithmetic.

What is even trickier (when it comes to deciding what is true) than having so many conflicting viewpoints for constructing math is that the axioms of these viewpoints are themselves not provably true. If you are willing to assume the axioms of math are “true”, then all of the resulting theorems that can be derived from those axioms are also true, but the axioms themselves must simply be accepted without proof in order for this process to work. If we could prove that the axioms were true, then they would be called “theorems” and not “axioms”!

Even those mathematicians who agree to rely on a single basic axiomatization (such as Zermelo-Fraenkel set theory) sometimes cannot agree on whether certain extra axioms (such as the continuum hypothesis, which concerns itself with the existence of sets of certain infinite sizes, or the axiom of choice which pertains to being able to select one element from each element of a set of sets) should be added or left out. And to top that off, mathematics (as defined by whichever axiomatization you like) has not even been proven to be consistent, meaning that no one has been able to mathematically demonstrate that the axioms of any single axiomatization do not contradict each other. In fact, Gödel’s 2nd incompleteness theorem shows that if mathematics is in fact consistent, then it will not be possible to use math to prove that no inconsistencies exist!

In conclusion, numbers and other mathematical objects are simply concepts, and not things that are actually observable in the universe, so we cannot say that statements like 3+2=5 are true in the same way that we can say that the statement “massive objects exert forces on other massive objects” is true. We might like to think that mathematical statements are true by definition. Still, this idea is complicated by the fact that there is more than one way to axiomatize mathematics, and therefore more than one definition that we might choose in order to define numbers, operators, and other mathematical objects. But even if there were truly only one way to axiomatize math, the axioms themselves would still not be provably true (they would only be assumed to be true), and hence it would hardly seem fair to then conclude that mathematical theorems are “true” in some objective and universal sense.

In the end, while it hardly seems fair to say that math is “false”, it also does not seem fair to conclude that math is “true” in the usual sense of the word. It’s true, conditional on the axiom that we choose to accept, and only insofar as it is talking about the concepts that it defines (rather than the physical world).

Of course, math undeniably provides extraordinarily useful models for making predictions about what will happen in our physical universe. This will perhaps seem less surprising if we remember that mathematics was not originally developed from the ground up using axioms, but rather piece by piece in order to find solutions to problems that appear in the real world (like those related to calculating the size of plots of land, counting money, measuring roads, tracking the movements of the stars, understanding heat flow in cannons, etc.). Humans chose mathematical definitions to model physical reality so that we could make useful predictions, not to encapsulate metaphysical truth, so should we have expected math to be “true” rather than merely (very) useful?

This piece was first written on January 18, 2009, and first appeared on my website on March 3, 2026.