That capitalism is superior to socialism

Socialism (S) functions based on any other form of ownership: public, collective, cooperative.

Both approaches may be flawed in their own ways. However this does not rule out the possibility that one is superior to the other. If we can settle on a good system that is different from both C and S, then the superior one is whichever comes closest. 

These 2 theorems of logician Gödel proove the limitations of any axiomatic system. They show that such a system is either incomplete, or it is inconsistent as a whole. Of course, the fact only describes formal systems. Nonetheless, I extend its implications to indicate the problem I see with politics.

Politics is about governing large groups of people; it's about trying to settle what individual parts do and integrate them in a larger collective: a system with political values for axioms (i.e. general human rights). Therefore, as the theory suggests, it can't run perfectly because it is fundamentaly flawed; every struggle to fully control every aspect is bound to fail. Most of the solutions raise even more problems. Indeed, with more elaboration of some aspects of society, come more complicated quagmires. S consists in implementing an ideology  (i.e. trying to control the means of production) on a large society, whereas C allows private actions to operate freely. Consequently, we mark a theoretical point in favor of C. More often then not, a rigid theoretical system of management often fails to cope with the unpredictable social environment. In this sense, C has the upper hand as it leaves more room for trial and error. 

Superior means better, but in what way?

I guess that, in order to decide which one is superior, we first need to find a "common denominator". If they both aim at the same target, the superior one is whichever comes closest. I think we both agree on their ultimate goals, that is to make a better society. Now, how would a good society look like?

You suggested that the best model is something very similar to agorism. If we take agorism as an ideal for judging the topic, C has an advantage over S. I will be exposing the main points of Agorism:

1)The free market is a common factor between agorism and C, in contrast with rigurous economical planning which is fundamental in S.

2)Agorism's valuntaryism stems from a religious liberty of conscience which goes hand in hand with individual autonomy and ownership. (the leveller movement)

3)S is regarded as an atheistic ideology which forbids religion. This is relevant for my previous point and also to contrast with counter-economics ("the study or practice of all peaceful human action which is forbidden by the State")