Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter

Bifurcations in Number Theory, and Metamathematicians So bankers, cloud-counters, and most of the rest of us need not worry abou ...

down. But this faith may be wrong. Perhaps every consistent formalization of number theory which humans invent will imply the ex ...

Now for the simplification of G. It has been shown that whenever you have a sufficiently powerful formal number theory, and a Go ...

Birthday Cantatatata ... One fine May day, the Tortoise and Achilles meet, wandering in the woods. The latter, all decked out ha ...

And it is only for that reason that I would ask you, "Do the foregoing sentences present enough evidence that I should conclude ...

Tortoise: Not at all. In fact, Achilles, I am looking forward to treating you to a fine birthday dinner, provided merely that I ...

are yet further Answer Schemas, such as w^3 , w^4 , w^5 , and so on. It goes on quite a ways, you know. Achilles: I can imagine. ...

CHAPTER XV Jumping out of the System A More Powerful Formal System ONE OF THE things which a thoughtful critic of Godel's proof ...

too. The trick will be to find a string which expresses the statement "I Cannot Be Proven in Formal System TNT +G." Actually, it ...

Its interpretation is "There is no number a that forms a TNT +G-proof-pair with the arithmoquinification of u'." More concisely, ...

Naturally, after a while, the whole process begins to seem utterly predictable and routine. Why, all the "holes" are made by one ...

In the Contracrostipunctus, one of the essential steps in the Tortoise's making an "unplayable record" was to get a hold of a ma ...

pleteness here is part and parcel of TNT; it is an essential part of the nature of TNT and cannot be eradicated in any way, whet ...

Once this ability for self-reference is attained, the system has a hole which is tailor-made for itself; the hole takes the feat ...

Among other things, it has to be able to do primitive recursive arithmetic. But for this very reason ... It is vulnerable to the ...

out-Codel the new machine, Codelizing operator and all. This has, in fact, proved to be the case. Even if we adjoin to a formal ...

FIGURE 76. Dragon, by M. C. Escher (wood-engraving, 1952). know it as we do. We could, in fact, carry the Escher picture any num ...

Now with this wonderful Escherian metaphor, let us return to the program versus the human. We were talking about trying to encap ...

There Is No Recursive Rule for Naming Ordinals Now offband you might think that these irregularities in the progression from ord ...

by C. H. Whitely, when he proposed the sentence "Lucas cannot consis- tently assert this sentence." If you think about it, you w ...