Formal Contradiction implied by Godel's Theorem

A

It has long been disputed whether Godel’s incompleteness theorem supplies a counter-argument to the golem. The dispute has the appearance of an antinomy, but is not.

*golem*is in legend a clay figure brought to life by magic. In contemporary times the urge to make a golem finds expression in the claim of strong AI (in philosophy called*functionalism*) that a computer can be made that is indistinguishable as to behaviour from human life, and hence must*be alive*and conscious.It has long been disputed whether Godel’s incompleteness theorem supplies a counter-argument to the golem. The dispute has the appearance of an antinomy, but is not.

**The work of John Lucas asks us to look at the meaning of Godel's theorem, and to "see" that the theorem states something that computers cannot state. He is asking one to break out of the bounds of first-order logic.**

The dialectical opponents of Lucas refuse to do this. They insiste that Godel's theorem is a first-order theorem, that human beings also work (at the machine level of their brains) in first-order logic, so a human mind is limited in just the same way a computer is limited.

The breakout from this illusory antinomy is as follows. It is not disputed by first-order logicians that mathematical induction is a valid form of inference. By induction on all extensions to first-order theorems we deduce a universal Godel Theorem. Ifit is assumed that this universal Godel Theorem is first-order, then a formal contradiction arrises. Therefore, the universal theorem is true but not first-order. The resolution lies in realising that mathematical induction is a synthetic principle of human reasoning. The conclusion, a universal statement about a potentially infinite collection, contains more information than the two premises. (Here I assume the reader understands what mathematical induction is. It is not recherche, but all these technical pages are unforutnately aimed at technicians. It is equally unfortunate that these technicians are mainly mechanists with ideological commitment to materialism and atheism.)The dialectical opponents of Lucas refuse to do this. They insiste that Godel's theorem is a first-order theorem, that human beings also work (at the machine level of their brains) in first-order logic, so a human mind is limited in just the same way a computer is limited.

The breakout from this illusory antinomy is as follows. It is not disputed by first-order logicians that mathematical induction is a valid form of inference. By induction on all extensions to first-order theorems we deduce a universal Godel Theorem. If

Could John Lucas be right afterall?

So, Lucas is right after-all. Furthermore, he is quite right about the meaning of Godel's Theorem too - its "semantics". A second strand of explanation examines what it is about the models of Godel's theorem that make it true. It turns out that for a first-order logic to be essentially incomplete it has a model that contains the continnum. (The continuum is a model of Godel's theorem.) Within the continuum there is an inexhaustible boundary between the finite and co-finite subsets of the actual infinite (the set omega), and grasping this point is beyond anything that could be expressed in first-order language, and hence is non-computable. But this is particularly hard material to grasp. In essence the computer scientists assume that their clever manipulations of 1s and 0s can do everything, and the logicians have failed to examine what it means for a theorem to be true but not provable. Examination of that meaning takes one in the diectition of realising that Godel's theorem is modeled by the Cantor set as a representation of the continuum; from thence, into the theory of limits and to Cohen forcing. It can be shown that Godel's Theorem gives rise a forcing construction that generates a squences whose limit is a transcendental number. But the transcendental number itself exists is never constructed.

This problem about the non-constructiblity of the continuuum and its transcendental numbers has long been known even to the first-order logicians. They have sought to impose a limitation on mathematics, but limiting it entirely to the constructible, that is computable part. (For example, Garrett Bishop.) This is merely a fiat, and states, by fiat, that the human mind is a machine because we deem it so. Those of us who are not subscribers to that creed may see it for what it is: abitrary imposition of a false limitation on mathematical reasoning for the purpose of supporting a false ideology.

This problem about the non-constructiblity of the continuuum and its transcendental numbers has long been known even to the first-order logicians. They have sought to impose a limitation on mathematics, but limiting it entirely to the constructible, that is computable part. (For example, Garrett Bishop.) This is merely a fiat, and states, by fiat, that the human mind is a machine because we deem it so. Those of us who are not subscribers to that creed may see it for what it is: abitrary imposition of a false limitation on mathematical reasoning for the purpose of supporting a false ideology.