**Gödel, Escher, Bach/**

__Chapter 8/Typographical Number Theory.__**Typographical number Theory**

I believe I mentioned chapter 7 as being a confusing chapter where I could not make sense out of the various symbols and systematic approaches Hofstadter gave to convey his particular meaning. If that was the case for chapter 7, chapter 8 got a lot more serious; perhaps stating I understood nothing is too much of an exaggeration, but in regards to the fullness of the proposed system, in this case TNT (Typographical number theory), I know very little. However, Hofstadter does not intend the reader to fully understand these systems, though it would be preferable that the reader could, what he really means us to understand is beyond this and beyond any of his proposed systems (MU, pq, etc). The reason he does use TNT is to analyze it and see if such a system could possible get to be complete and consistent, and rather than telling it to us straightforward, he only poses the questions. In itself, or at least partially, the TNT system has symbols and sequences that function mechanically in order to convey certain axioms, meaningful axioms, as more simplistic and general kinds (that of course carry the same interpretation). Problem is how far can we go with such a system, and not construct a inconsistent fallacy that negates itself.

**IS and IS NOT**

One of the general rules of consistency indicate that that which Is cannot at the same time not be, for instance a turtles shell cannot be “green” and “not green” at the same time. This kind of inconsistency tend to reveal themselves within the typographical number theory, and intending to adjust the inconsistency, then the system ceases to be complete; here lies the heart of the problem, or at least so I presume. If we wish to classify the TNT as a complete and consistent theory we would have to manage to find a way in which the above contradiction does not occur, and do this without the addition of another axiom; this seems impossible.

**Gödel**

“Circularity is inevitable”. That is the last sentence of this chapter; apparently, it is a concluded statement of Gödel, in which he proves that the TNT cannot be in itself complete and consistent at the same time. It appears that similarly to Euclid, the TNT has far too many rules that lead one to think some of them may be off, and applying the thought that the simplest explanation tends to be the accurate one, then the TNT fails. Hofstadter clearly states that the completeness of TNT is impossible, and goes on further to tell that this fact is a cause of either joy or mourn for people. Some came with the hope that they could generate a system that is simple enough, with simple axioms, that will fix any problems within the TNT system, however Gödel states its impossibility. Creating “thin ropes” which are believed to contain the “High ropes” already existent in the TNT are doomed to be at least as strong as the very “High ropes” one is intending to diminish, and hence it does not work properly.

**What now?**

It seems Hofstadter has done a very thorough and well planted job in taking systems and demonstrating to us how its consistency and completeness fail, and how it is either complete or consistent, but not both at the same time (Gödel’s theorem). What confuses me is whether Hofstadter himself believes such to be true, or does he actually think there exists a complete an consistent system? This will probably become clearer as we get further on to the end of the book, but for now it is unconceivable to acknowledge a system as both consistent and complete, though much more is left to be known about the matter…