BIG LIST: Statements that look obviously false but cannot be disproved

$$P = NP$$

is unlikely but still possible.

To explain it to a child, you could ask if it's easier to:

• check the definition of a word in a wordbook
• or find a word in the wordbook given its definition

It seems obvious that the former is easier than the latter, $P = NP$ would mean that both are equally easy.

My personal favorite :

The number $$2\uparrow 2\uparrow 2\uparrow 2+3\uparrow 3\uparrow 3\uparrow 3$$ is a prime number.

Since this number is very large (it has $3\ 638\ 334\ 640\ 025$ digits), it is very likely composite. However, according to Enzo Creti's calculation, there is no prime factor below $10^{12}$, so the number might be prime.

Outside mathematics I would vote for the statement : "God exists" and on the second place : "Our universe is not the only one"

To coin another mathematical statement :

Goldbach's conjecture is false

This is almost surely false , but as long Goldbach's conjecture is not proven, it cannot be ruled out.

The continuum is $\aleph_{37}$.