Theorems relating to the limitation of mathematics

Question

At one point, mathematicians believed that they may be capable of expressing all of mathematics in one system of ideas, and that their abilities were unlimited. Unfortunately, things like the Godel's theorems and the Halting Problem show that mathematics is not unlimited, but has clear limitations. Of course, this does not detract from the work of the mathematicians who discovered these theorems, or any mathematician; no other field can so rigorously identify its own limitations.

In honor of these theorems, I would like to collect a list of the theorems relating to limitations of mathematics.

Some notes:

• A limitation is not the same as something not existing. The fact that we cannot find integers $a, b$ such that $\frac ab=\sqrt 2$ is not a limitation, because they simply do not exist. Things not existing does not qualify something as a limitation of mathematics. I would even consider Arrow's impossibility theorem to fall in this category.
• On the other hand, the halting problem would count. It talks about certain turing machines not existing, but this can be applied to mathematicians and logic systems. By showing that a there are tasks that turing machines cannot do, it also shows that mathematicians probably cannot do those tasks (and the task actually has an answer).
• I cannot give a rigorous definition of "limitation of mathematics" (that may actually be a theorem). As such, I have marked this soft-question. If you have any questions, ask in the comments.

Let us assemble a list of mathematics limitations. (Bonus to whomever shows that there are limitations which exist, but we are unable to find/know about.)

Answer 1

Comment

Gödel's Incompleteness Theorems applies under specific conditions :

First incompleteness theorem. Any consistent formal system $F$ within which a "certain amount" of elementary arithmetic can be carried out is incomplete.

Thus, in order to conclude that "it show that mathematics is not unlimited, but has clear limitations", we have to make some extra-assumptions, like :

mathematics is a formal system,

assumption that are not so trivial.

For a very good discussion of the theorem and its implications, see e.g. :

• Torkel Franzén, Gödel's Theorem : An Incomplete Guide to Its Use and Abuse (2005).

Answer 2

Like you said, it is difficult to define what is a gueninely limitating result and what is just the discovery of. Both the irrationality of $\sqrt{2}$ and the Halting problem can be stated in the form "there is no [...] such that [...]".

I think any interpreted formal language is limited, and one can see maths as a formal interpreted formal language. One important example of limitation is that which appears when we can formulate negative reflexive assertions such as $x \notin x$, $g(n) \neq f(n)(n)$ with which we can derive "diagonal arguments". However those limitations and negative results can also be seen as positive results depending on how you formulate them modulo ZF.

-There is no set containing every set. A positive "restatement" is that for every set $X$, there is a set $Y \notin X$ ($Y = \{x \in X \ | \ x \notin x\}$)

-If $A$ is a set, there is no surjective map $A \rightarrow \mathcal{P}(A)$. A positive restatement is that there are sets of cardinality as high as one wants.

Similar examples are Gödel's first incompleteness theorem, Tarsky's theorem of undefinability of truth, the uncountability of the set of real numbers...