Comment: I'm not saying the definition would change

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I'm not saying the definition would change

I'm simply saying that they would be fundamentally different (due to the absence of a right to forcibly tax their citizens and to forcibly include non-consenting persons in their citizenry) and so would instead be thought of as voluntary agencies that would perform the agreed-upon functions.

Of course, I don't think it is likely that people would decide to create one, centralised agency that would dispense justice, defense, etc, all in one. It may, and indeed could, happen. But it is likely that any body of people who had moved into Anarcho-capitalism in the first place, would be wanting to separate those functions into agencies that would specialise in them. There also may be a concern that there is competition in each field.

All of this amounts to a re-thinking of what governance is, and how it would take place.

One thing, too, is that the definition above states:
"the group of people..."
"a particular system..."
"the process or manner..."
"the governing body..."

If this is the definition of government, then in some (or many) cases under Ancap'ism, the definition may indeed change. As there would not necessarily be only one agency dispensing justice within a given geographical area.

But my answer remains the same, as people within a given geographical area could indeed all voluntarily form or join one organisation that performs these roles, and thus, it would be defined as a government.

"Is life so dear, or peace so sweet, as to be purchased at the price of chains and slavery? Forbid it, Almighty God! I know not what course others may take; but as for me, give me liberty or give me death!" - Patrick Henry