I too would like to know the specifics of how this has been axiomatized.
Here is how I've viewed it in my own philosophy/epistemology
(http://www.bayside.net/NPO/BMI/autevol/ghw_epi.htm -- "Growth of
knowledge"): "Paradox" plays a central role in transitions to a new
paradigm. Any theory exists within a set of founding assumptions
(metaphysics, or worldview). Rigorous construction of all possibilities
within that framework will eventually exhaust the explanatory value of the
view itself. The better worldviews last longer and explain more. The poorer
ones don't go very far. But "exhaustion" of the worldview will appear as an
increasing number of paradoxes -- two "correct" statements within the
particular view that are also contradictory within that view. The classic
example is constancy of the speed of light and the principle of relativity.
The new paradigm (e.g., the theory of relativity) removes the conflict,
holding both seemingly contradictory statements to be true. I tried to
flowchart this, but get into axioms as such.
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John J. Kineman, Physical Scientist/Ecologist
National Geophysical Data Center
325 Broadway E/GC1 (3100 Marine St. Rm: A-152)
Boulder, Colorado 80303 USA
(303) 497-6900 (phone)
(303) 497-6513 (fax)
jjk@ngdc.noaa.gov (email)