HPX.23 A consistent ToE would be a model of some logic (logic system). As this system must incorporate number theory, we have immediately that this system would be either inconsistent or incomplete, thus contradicting the claim that we have a ToE.
Goedel, and others, extended Goedel's initial incompleteness result to show that a great many diophantine equations are undecidable in terms of axioms. These expressions are encoded forms of statements in the ToE system which are undecidable: they may be true, but not provably so.
So then, any supposed ToE contains a vast number of irresolvable problems.
Note that this result cannot be voided by appeal to some topology for the cosmos.
One may object that a countable infinity of trivial axioms should take care of this technicality to the satisfaction of most physicists. This may not be the case. If we call a system that has been extended by an additional axiom L', we have the potential that L' itself contains an infinity of irresolvable problems (that are encoded as diophantine equations). Hence we face the potential that the set of all Ls would require an uncountable infinity of axioms. Most physicists are unlikely to regard that eventuality as satisfactory.
If one is a mechanist, materialist or conventional physicalist, then the proofs of Goedel and Turing strongly support some form of Cartesian dualism. To reject Cartesian dualism means to reject the former philosophical approaches.
That is, the cosmos, and in particular, the human cognitive function (AKA "mind") cannot be reduced to Boolean logic circuits. This is shown by their proofs that mathematics cannot be reduced to logic alone -- unless one posits an infinite nested class of logics.
Thus, we draw the conclusion that not all mathematical insight and activity can be algorithmic; it cannot be reduced to a Boolean, or any other, logic system.
Goedel dealt with this fact by tending toward the divine. Turing, an atheist, tried to deal with it by mechanizing his logic system that was based on Oracles, which are not subject to the limits found by himself and Goedel. Yet, though Turing may have succeeded to some extent, one cannot void the fact that his Oracle remains unanalyzable, and hence as non-scientific as many a theological notion.
Goedel, and others, extended Goedel's initial incompleteness result to show that a great many diophantine equations are undecidable in terms of axioms. These expressions are encoded forms of statements in the ToE system which are undecidable: they may be true, but not provably so.
So then, any supposed ToE contains a vast number of irresolvable problems.
Note that this result cannot be voided by appeal to some topology for the cosmos.
One may object that a countable infinity of trivial axioms should take care of this technicality to the satisfaction of most physicists. This may not be the case. If we call a system that has been extended by an additional axiom L', we have the potential that L' itself contains an infinity of irresolvable problems (that are encoded as diophantine equations). Hence we face the potential that the set of all Ls would require an uncountable infinity of axioms. Most physicists are unlikely to regard that eventuality as satisfactory.
If one is a mechanist, materialist or conventional physicalist, then the proofs of Goedel and Turing strongly support some form of Cartesian dualism. To reject Cartesian dualism means to reject the former philosophical approaches.
That is, the cosmos, and in particular, the human cognitive function (AKA "mind") cannot be reduced to Boolean logic circuits. This is shown by their proofs that mathematics cannot be reduced to logic alone -- unless one posits an infinite nested class of logics.
Thus, we draw the conclusion that not all mathematical insight and activity can be algorithmic; it cannot be reduced to a Boolean, or any other, logic system.
Goedel dealt with this fact by tending toward the divine. Turing, an atheist, tried to deal with it by mechanizing his logic system that was based on Oracles, which are not subject to the limits found by himself and Goedel. Yet, though Turing may have succeeded to some extent, one cannot void the fact that his Oracle remains unanalyzable, and hence as non-scientific as many a theological notion.
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