How much math do we need to prove all simple numeric identities?

Question

Consider real numeric expressions build only from integers, operators $+,-,\times,/$ and taking a positive expression to a power (no variables involved), e.g. $$\frac{2}{7},\ 2^{1/2},\ \sqrt[5]{2+\sqrt{11}},\ 2^{\sqrt{3}},\ ...$$ Now we can write identities between such expressions that are either true, e.g. $\sqrt{3+\sqrt{8}}=1+\sqrt{2}$, or false, e.g. $\frac{3}{5}=\frac{2}{7}$.

Is it possible to prove all such true identities and disprove all false identities using only usual "high-school" algebra rules?

Answer 1

As far as I know, this is an open problem. It might be the case that even $ZFC$ axioms are not sufficient to settle all such questions, or that the problem is even undecidable (although it seems unlikely). The main issue that we know very little about behavior of repeated exponentiation. For example, it is unknown if the following number is an integer: $$2^{2^{2^{2^{2^{2^{1/2}}}}}}$$ Personally, I expect that there are no surprises to be discovered in the repeated exponentiation, that this number will eventually be proved to be transcendental, and that there are no unexpected identities in your class of expressions that could not be explained using only "high-school" algebra rules. But these are only conjectures.

Answer 2

I'll answer no.

Suppose you could prove all identities using such rules. Then there could get written some infinite list of all such identities (you could enumerate all such identities by a characteristic function which returns "true" for each true identity and "false" for each false identity). But, that would imply you could have a bijection between the set of all natural numbers and all of those identities. So, you could encode all of those identities via the natural numbers. Included in this list identities is all identities involving real numbers including pi=pi, e=e, sqrt(2)=sqrt(2). That set of identities clearly has the same cardinality as the set of all real numbers. But, there does not exist any bijection between the set of real numbers and the set of natural numbers. So there does not exist any bijection between the set of all natural numbers and that of all such "high-school" algebra identities. Therefore, you cannot prove "all such true identities and disprove all false identities using only usual "high-school" algebra rules".