In modern logic an axiom is a premise, it cannot be proven. It is a starting assumption against which other things may be proven. For example, in Euclidean geometry one cannot prove that parallel lines never meet. It simply defines the particular geometry.

Proving statements about Bitcoin requires reliance on axiomatic systems, specifically mathematics, probability and catallactics, and therefore the assumptions upon which they rely. However Bitcoin also relies on an axiom not found in these systems. Satoshi alludes to this in an early statement:

You will not find a solution to political problems in cryptography.

Yes, but we can win a major battle in the arms race and gain a new territory of freedom for several years.

Governments are good at cutting off the heads of a centrally controlled networks like Napster, but pure P2P networks like Gnutella and Tor seem to be holding their own.

Satoshi Thu Nov 6 15:15:40 EST 2008

In other words there is an assumption that it is possible for a system to resist state control. This is not accepted as a fact but deemed to be a reasonable assumption, due to the behavior of similar systems, on which to base the system.

One who does not accept the axiom of resistance is contemplating an entirely different system than Bitcoin. If one assumes it is not possible for a system to resist state controls, conclusions do not make sense in the context of Bitcoin - just as conclusions in spherical geometry contradict Euclidean. How can Bitcoin be permissionless or censorship-resistant without the axiom? The contradiction leads one to make obvious errors in an attempt to rationalize the conflict.

It is common for people to refer cynically to a Bitcoin-like system that omits the resistance axiom as just another "PayPal", a designation not without merit. Confinity originally attempted to create a system with a similar value proposition to Bitcoin. Having failed to do so it discarded the axiom, building the PayPal we know today.