WHY NOTHING IS TRUE AND WHY WE SHOULD BE HAPPY TO HEAR THIS.
In order for a theory to be consistently right it has to be set in the background of other theories that proved consistently wrong on the matter under question.
Furthermore consider this. We operate our thinking on such matters as products of a pliable cognitive drive as a consequence of an evolutionary process open to testing ideas that work and remain viable in a constantly changing habitat—this habitat now slowly being seen more and more as a massively expanding universe (approximately 2.1km/sec, even this is subject to constant change).
We distinguish between a real world and our knowledge (innate or acquired) and this knowledge is supposed to be true or at least partially correspondent to some part of the real world (for a period at least) in order for natural selection to take palce. Nevertheless “the real world is nothing but the world of our experience” (Gerhard Vollmer), our knowledge of it and all we can know about the world is what our cognitive system knows and allows. Moreover, our cognitive system is explained precisely as a product of evolutionary process. But so is the habitat we base our growing epistemology on also constantly changing and evolving.
We should as natural scientists by now at least realise that without change in both habitat and organism there can be no evolution...and no life.
So life (cognition) is a change of ideas in an evolving idea.
Set in such a background, now for the First Godelian Incompleteness Theorem:
If T is a consistent, strong, recursively axiomatizable theory, then there is a sentence “P” in the language of the arithmetic of the vast universe such that neither “P” nor “not: P” is provable in T.
As Earth based evolutionary products our perceptive means receive and interpret photons the way we do, this depending on your species and evolutionary niche achieved. Traveling faster than light is not possible... Merely thinking about this and the photons from the outer realms of the universe only now reaching us after 30 billion lightyears we have already travelled faster than light — so thoughts travel faster than light, already a good start to question an existing theory.
Second Incompleteness Theorem: If T is a consistent, sufficiently strong, recursively axiomatizable theory, then the theory:
T+ not: Con(T)
is consistent.
Here is the proof:
Let T1 be any consistent, sufficiently strong theory (e.g. Peano arithmetic). So, by Gödel’s second incompleteness theorem:
T2 = T1 + not: Con(T1) is a consistent theory.
We can deduct from this that “Con(T2)” is true. Now, consider the following theories:
(i) T2 + not: Con(T2)
(ii) T2 + Con(T2)
Since, as we have already seen, T2 is consistent, it follows, again, by the second incompleteness theorem, that the first theory:
T2 + not: Con(T2) is consistent.
But now consider the second theory (ii). This theory includes the claim that T2 does not prove a contradiction – that is, it contains “Con(T2)”. But it also contains every claim that T2 contains. And T2 contains the claim that T1 does prove a contradiction – that is, it contains “not: Con(T1)”. But if T1 proves a contradiction, then T2 proves a contradiction (since everything contained in T1 is also contained in T2). Further, any sufficiently strong theory is strong enough to show this, and hence, T2 proves “not: Con(T2)”. Thus, the second theory:
T2 + Con(T2) is inconsistent, since it proves both “Con(T2)” and “not: Con(T2)”. QED.
If we now place our pliable evolutionary cognition within its constantly changing epistemology in an equally pliable constantly changing habitat, we can perhaps sense the unlikelihood of absolute truths.
Lastly a very plausible thought if everything is know in a fixed environmet there can be no scope for natural selection or evolution (cognitive evolution) to occur. So we can conclude change and tempory ideas about ideas is the best we can try and achieve even in mathematics. This is good news in a world where limitations can be blinding and politicians overconfident in their believes and ideas.
T D Holtzhausen
July 2016
References
-Roy T Cook Consistency of inconsistency claims. Cited on the Internet http://blog.oup.com/2016/03/paradox-consistency-inconsistency/
• Kurt Gödel. Consistency of the Continuum Hypothesis. Princeton University Press
• Authors own unpublished personal work.
• Lewontin, R. C., "Organism and Environment", in H. C. Plotkin (ed.): Learning,
• Development, and Culture: Essays in Evolutionary Epistemology. Wiley, Chichester 1982, pp. 151-170.
• Lorenz, K., "Kants Lehre vom Apriorischen im Lichte gegenwärtiger Biologie", Blätter für Deutsche Philosophie 15 (1941): 94- 125. English translation reprinted in H. C. Plotkin, ibid.)
• Wuketits, F. M., Evolutionary Epistemology and Its Implications for Humankind. SUNY Press, Albany, N. Y. 1990.
In order for a theory to be consistently right it has to be set in the background of other theories that proved consistently wrong on the matter under question.
Furthermore consider this. We operate our thinking on such matters as products of a pliable cognitive drive as a consequence of an evolutionary process open to testing ideas that work and remain viable in a constantly changing habitat—this habitat now slowly being seen more and more as a massively expanding universe (approximately 2.1km/sec, even this is subject to constant change).
We distinguish between a real world and our knowledge (innate or acquired) and this knowledge is supposed to be true or at least partially correspondent to some part of the real world (for a period at least) in order for natural selection to take palce. Nevertheless “the real world is nothing but the world of our experience” (Gerhard Vollmer), our knowledge of it and all we can know about the world is what our cognitive system knows and allows. Moreover, our cognitive system is explained precisely as a product of evolutionary process. But so is the habitat we base our growing epistemology on also constantly changing and evolving.
We should as natural scientists by now at least realise that without change in both habitat and organism there can be no evolution...and no life.
So life (cognition) is a change of ideas in an evolving idea.
Set in such a background, now for the First Godelian Incompleteness Theorem:
If T is a consistent, strong, recursively axiomatizable theory, then there is a sentence “P” in the language of the arithmetic of the vast universe such that neither “P” nor “not: P” is provable in T.
As Earth based evolutionary products our perceptive means receive and interpret photons the way we do, this depending on your species and evolutionary niche achieved. Traveling faster than light is not possible... Merely thinking about this and the photons from the outer realms of the universe only now reaching us after 30 billion lightyears we have already travelled faster than light — so thoughts travel faster than light, already a good start to question an existing theory.
Second Incompleteness Theorem: If T is a consistent, sufficiently strong, recursively axiomatizable theory, then the theory:
T+ not: Con(T)
is consistent.
Here is the proof:
Let T1 be any consistent, sufficiently strong theory (e.g. Peano arithmetic). So, by Gödel’s second incompleteness theorem:
T2 = T1 + not: Con(T1) is a consistent theory.
We can deduct from this that “Con(T2)” is true. Now, consider the following theories:
(i) T2 + not: Con(T2)
(ii) T2 + Con(T2)
Since, as we have already seen, T2 is consistent, it follows, again, by the second incompleteness theorem, that the first theory:
T2 + not: Con(T2) is consistent.
But now consider the second theory (ii). This theory includes the claim that T2 does not prove a contradiction – that is, it contains “Con(T2)”. But it also contains every claim that T2 contains. And T2 contains the claim that T1 does prove a contradiction – that is, it contains “not: Con(T1)”. But if T1 proves a contradiction, then T2 proves a contradiction (since everything contained in T1 is also contained in T2). Further, any sufficiently strong theory is strong enough to show this, and hence, T2 proves “not: Con(T2)”. Thus, the second theory:
T2 + Con(T2) is inconsistent, since it proves both “Con(T2)” and “not: Con(T2)”. QED.
If we now place our pliable evolutionary cognition within its constantly changing epistemology in an equally pliable constantly changing habitat, we can perhaps sense the unlikelihood of absolute truths.
Lastly a very plausible thought if everything is know in a fixed environmet there can be no scope for natural selection or evolution (cognitive evolution) to occur. So we can conclude change and tempory ideas about ideas is the best we can try and achieve even in mathematics. This is good news in a world where limitations can be blinding and politicians overconfident in their believes and ideas.
T D Holtzhausen
July 2016
References
-Roy T Cook Consistency of inconsistency claims. Cited on the Internet http://blog.oup.com/2016/03/paradox-consistency-inconsistency/
• Kurt Gödel. Consistency of the Continuum Hypothesis. Princeton University Press
• Authors own unpublished personal work.
• Lewontin, R. C., "Organism and Environment", in H. C. Plotkin (ed.): Learning,
• Development, and Culture: Essays in Evolutionary Epistemology. Wiley, Chichester 1982, pp. 151-170.
• Lorenz, K., "Kants Lehre vom Apriorischen im Lichte gegenwärtiger Biologie", Blätter für Deutsche Philosophie 15 (1941): 94- 125. English translation reprinted in H. C. Plotkin, ibid.)
• Wuketits, F. M., Evolutionary Epistemology and Its Implications for Humankind. SUNY Press, Albany, N. Y. 1990.
