The argument against the above mentioned proof arises from philosophy: Actual infinities cannot exist. The infinite can only exist as a potentiality to increase/decrease. The actual is always finite. For example, 0.999…. mangoes cannot exist. But 0.99999999999 mangoes and 0.999 mangoes can.

Then you might say: "But Edwin, The repeating decimal does exist and comes from our representation of fractions using base ten arithmetic. Using Edwin's logic, a third of a mango couldn't exist (.333…) but I can certainly cut a mango into three equal pieces.".

Unfortunately, infinitely divisible mangoes do not exist. At some point of time we will end up trying to divide the fundamental building block of matter and we will fail unless a mango has a number of such building blocks which is divisible by 3. The same argument can be used to argue that no fraction of a mango can exist for all mangoes. This is because whether a fraction exists for mango will depend on its number of fundamental building blocks.

This is were we get to one of the core controversies in mathematics & philosophy: Can concepts in our mind like yellowness, oneness, one-third-ness exist apart from things that are actually yellow, one or one-third? Disciples of Plato & Kant, which includes most modern mathematicians, economists & theologians believe that concepts like yellowness, oneness do actually exist independent of our minds and apart from objects which are yellow and one.

As a disciple of Aristotle & Rand, I ask these Platonists & Kantians: Does kgrbd-ness exist? If you answered no, trying asking why not. You will end up reasoning that only words with connotation can exist. When do words have connotation? Only when they actually exist independent of our minds in reality.

So kgrbd-ness cannot exist because there are no kgrbd things as of now.

Thus we see the primacy of existence over consciousness yet again clarifying things for us. BTW do not fear that the last 2000 years of mathematics is wrong. A school of mathematicians called the constructivists have already cleaned up the almost all of the Platonist & Kantian scum that underlies mathematics and replaced it with more rational ideas. Thanks to them we Aristotelian and Randian "nuts" can sleep peacefully at night.