Proving that it's impossible to prove irrationality of all real numbers.

Question

Proving that it&#039;s impossible to prove irrationality of all real numbers.

GrearomiaN3v · Accepted Answer

This claim is fallacious. Just replace the adjective “irrational” with “positive” and you would conclude that any set of reals that is provably positive must be countable. If you phrase it just right you could make this a true statement, but it clearly defies most reasonable interpretations of “proving positivity of all (positive) real numbers”.
It is not necessary for every individual element of

S

to have a distinct certificate of irrationality in order for the entire set to be provably irrational. The same proof can cover all elements of

S

without explicitly naming each and every one (which is clearly impossible).
For example, define

f : P (N) \to R

by

f (A) := e + \sum_{k \in A} \frac{1}{2^{k!}} .

This is irrational — in fact, transcendental — for every value of

A \subseteq N

, and there are uncountably many such values (

f

is injective so the range of

f

is indeed uncountable).

Proving that it's impossible to prove irrationality of all real numbers.

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