Cardinality of the set of all mathematical statements

Question

Let A = {x|x is a mathematical statement}.

What is the cardinality of A?

This was a question my friend asked me yesterday.

At first, I thought A is a countable set because one can count the number of statements. Namely, I can establish a bijection between A and set of natural numbers, by assigning a unique natural number to each statement.

However, my friend then asked "How about having a math statement such that x = y, while x and y are arbitrary element of real number set? Then, the total number of mathematical statement would be equal to the cardinality of the power set of real number set?"

After listening to his argument, I became completely puzzled. Can anyone verify whether one of us have a correct reasoning? Or both of us are wrong?

Answer 1

There are countably many symbols used in stating a math statement.
Each math statement is a finite sequence of those symbols.
There are countably math statements of length n.
Add them all up for a countably many statements.

In practice there are finite many math statements.

Answer 2

Given the "set" A = {x: x is a mathematical statement}, there is no possible way to quantify the size of the "set". In axiomatic set theory- it actually would be considered a proper class and not a set and therefore not possible to quantify. Let me break this down more below:

Like your friend mentioned above, the set R = $\{x \in \mathbb{R}: x=x\}$ of all statements "x = x, where x is a real number" alone would have the cardinality $2^{\aleph_{0}}$, where $ \aleph_0$ is the cardinality of the integers (See Cantor).

How about $\mathcal{P}(R)$, the power set of R? It has a cardinality of $2^{2^{\aleph_0}}$. The union of R and $\mathcal{P}(R)$ would also have cardinality of $2^{2^{\aleph_0}}$.

$ \aleph_0$, $2^{2^{\aleph_0}}$, etc. are known as cardinal numbers. Take $\mathcal{P}^i(R)))$, the ith iteration of taking power sets of R. then we can define a set $A_i = \{x \in \mathcal{P}^i(R))): x=x\}$ of all statements "x=x, where x is an element of $\mathcal{P}^i(R)))$.

Now take the union of all such $A_i$ and call it U; U would be the collection of all mathematical statements x=x, where x is an element of some $\mathcal{P}^i$(R). Given the definition of A above, every statement in U would also be in A.

We see that each successive union of the elements of $A_i$ would have larger and larger sizes of infinite cardinalities and therefore when all of them are combined into U it cannot converge to any size. Since all statements in U would also be statements, it follows A cannot have a definitive size.