**Gödel, Escher, Bach/**

__Chapter 4/ Consistency, Completeness, and Geometry__**Insightful Summary**

Yet again, this book has given me much to think off, and much to ponder. Right at first it explains certain explicit and implicit meanings taken out of the dialogue read before about the Tortoise endeavor with a phonograph. I will not explain the phonograph situation in detail here, especially since it was discussed in the former chapter (chapter 3), but I will say it deals with isomorphism’s and a set of them instead of solely one. The dialogue in itself is a conveyor of various meanings, varying upon ones interpretations. Hofstadter suggests that there are two isomorphism’s in play, one out conveyed from the phonograph and one into it, at least that was what I supposed. Nonetheless, this partial phonograph explanation sets the ambient for further isomorphic discussion in regards to formal systems. For this to be more playable and properly understood we compare again with Bach’s fugues, and are given somewhat of an explanation of his last fugue, as it seems Bach lost his vision in the late stages of his life and somehow recovered it very soon before his actual death, point being that his work, yes was unfinished, but it gives us a wider scope of context into the factualness of events.

**Completeness and Consistency**

We had argue on and off about what is completeness and what is consistency, both in regards specifically to the definitions of the book as for the regular definitions we could come up with, this caused somewhat of a turmoil, discontent, and confusion. Roberto Blum in his introductory dialogue presented the words Completeness and robust, then we hear completeness and consistency, and also coherence and consistencies, such as it is these themes became recurrent or recursive in our dialogues (not the word recursive). To settle these matters, the book gives us in this chapter perhaps the clearest sense of their definitions, within the book of course.

__Consistency:__when every theorem, upon interpretation, comes out true (in some

imaginable world).

__Completeness:__when all statements which are true (in some imaginable world), and

which can be expressed as well-formed strings of the system, are theorems

Perhaps its meaning is not fully unraveled, and I am sure we will be looking more into these concepts further on our reading of the book. What is important here is this conceivability of an imaginary world, and how would our natural conceivable laws of now (the natural world as we know it) interfere with the imaginary world we are able to build. Roberto on our dialogue mentioned he was in fact capable of creating a round-square or squared-circle, he even gave proof and demonstrated. What is interesting is that inside Euclids system, that would be wrong, in fact it is wrong, but in another system it can be right, namely a three-dimensional one.

**Euclid**

Euclid once again takes the stage, for he is referred to by yet another author in another of our readings. This time he is quoted, paraphrased perhaps, in his 5 postulates, the fifth leaving mathematicians with unease. From here originated the Non-Euclidian Geometry, that began as the intention of proving Euclid’s fifth postulate. Instead of going into, as I say, the technicalities of the matter, I will simply speak of what I found significant. It seems Euclid thought himself to have had developed a complete and consistent system, which overtook all the component of our world, structured in a geometrical fashion (Geometry was God). Euclid was complete, but not consistent.

**Little Harmonic Labyrinth**

This dialogue is far too much to bear in one read, but the message comes out clearly, at least so I suppose. The dialogue is a recurrent integration of terms, words, realms, situations, ideas, etc. Even the dialogue is recurrent, the characters are recurrent, phrases are recurrent (such as Good Fortune), scenarios are recurrent, and the subject which is about recurrent subjects is recurrent in itself. The whole thing seems redundant and pointless, but is anything but pointless. The paradox to me seems incessant, and it appears on all situations…Bach, Escher, Gödel, Hofstadter, and even myself. Stories within stories within stories tend to confuse and disentangle the audience from identifying specific ideas, characters, or whatever the recurrent theme intending to elude is. The paradox as I have stated before, I believe, is another fascination of mine, one that I have looked into deeply but ironically only found myself back where I began.