Cybermorphic
03-17-03, 02:56 PM
Although I probably do not understand the math behind Godel any better than someone else who has read GEB, I feel that looking at the theorem even from a level more basic than TNT has led me to consider new implications from the Incompleteness Theorem on our basic system of reasoning; the Scientific Method. I will explore my considerations of both Godel's first Incompleteness Theorem, and Tarski's Undefinability of Truth Theorem using only English to express their important meanings.
When either Godel's theorem or Tarski's are expressed in English one is tempted to ignore them saying things like, "This is meaningless". If it is possible one should keep in mind that these theorems can both be expressed in mathematics so it is not a question of meaning but a question of what the implications of these theorems are. Tarski's theorem is not as well known as Godel's theorem and it is definitely not as understood. Tarski's theorem is constructed in the same fundamental way as Godel's theorem but with more startling consequences.
I will briefly go over Godel's First Incompleteness Theorem. Godel proved inside of the system Principa Mathematical which was designed not to allow self-reference which could lead to paradox that a self-referential statement could prove that Principa Mathmatica was incomplete, and also that all other formal systems complex enough to do simple arithmetic were incomplete. He did this by creating something that when translated from math looks like, "This is unprovable inside of Principa Mathmatica.". If you could prove that you would of course also prove that you can't prove it, and if you can't prove it then it is true but unprovable. Because of this we can say Principa Mathmatica is incomplete.
Tarski's Undefinability of Truth Theorem looks just the same as Godel's sentence but we replace 'unprovable' with 'not true'. It looks like; "This is not a true sentence of number theory". If that sentence were true it would be false, and if it were false we would see that it was true. To avoid this contradiction Tarski tells us we can not define what truth is inside of number theory.
Tarski's Theorem has implications which need to be considered. If we can not define what truth is in math I have to ask myself if we can define what truth is anywhere. Certainly we can not define truth in English because I have just explained his proof with English. Tarski spent the rest of his life after creating this theorem trying to find a way to define truth with other formal systems of his own imagination. What I consider concluding from studying this work is that any system of reasoning capable of referring to itself and defining truth can not use a definition of truth without contradictions. The scientific method is a system of reasoning that I am considering as meeting these conditions. The method, which is to first make a hypothesis and then test the hypothesis and finally make a conclusion based on the results of the test seems to fall appart when you test a Tarski sentence. Try this hypothesis, "This is not a true sentence when tested with the scientific method". To create the scientific test we only need to adopt a logical process of making an assumption about the hypothesis and following it through to see if we reach a contradiction. Sense we are using the scientific method the sentence contradictory when we have a definition of what truth is, or when we allow for testing the scientific method itself. So we might conclude that either the scientific method can make no conclusions or that the scientific method can not be used to test the scientific method without contradicting itself. Because we assume the scientific method does make conclusions, then we are left to consider that it may not be able to study itself without contradictions. I am considering that sense the scientific method may not be able to test itself without contradictions there is another method that may be able to test the scientific method and prove just as use full. A Cybermorphic Method.
I await your responses.
When either Godel's theorem or Tarski's are expressed in English one is tempted to ignore them saying things like, "This is meaningless". If it is possible one should keep in mind that these theorems can both be expressed in mathematics so it is not a question of meaning but a question of what the implications of these theorems are. Tarski's theorem is not as well known as Godel's theorem and it is definitely not as understood. Tarski's theorem is constructed in the same fundamental way as Godel's theorem but with more startling consequences.
I will briefly go over Godel's First Incompleteness Theorem. Godel proved inside of the system Principa Mathematical which was designed not to allow self-reference which could lead to paradox that a self-referential statement could prove that Principa Mathmatica was incomplete, and also that all other formal systems complex enough to do simple arithmetic were incomplete. He did this by creating something that when translated from math looks like, "This is unprovable inside of Principa Mathmatica.". If you could prove that you would of course also prove that you can't prove it, and if you can't prove it then it is true but unprovable. Because of this we can say Principa Mathmatica is incomplete.
Tarski's Undefinability of Truth Theorem looks just the same as Godel's sentence but we replace 'unprovable' with 'not true'. It looks like; "This is not a true sentence of number theory". If that sentence were true it would be false, and if it were false we would see that it was true. To avoid this contradiction Tarski tells us we can not define what truth is inside of number theory.
Tarski's Theorem has implications which need to be considered. If we can not define what truth is in math I have to ask myself if we can define what truth is anywhere. Certainly we can not define truth in English because I have just explained his proof with English. Tarski spent the rest of his life after creating this theorem trying to find a way to define truth with other formal systems of his own imagination. What I consider concluding from studying this work is that any system of reasoning capable of referring to itself and defining truth can not use a definition of truth without contradictions. The scientific method is a system of reasoning that I am considering as meeting these conditions. The method, which is to first make a hypothesis and then test the hypothesis and finally make a conclusion based on the results of the test seems to fall appart when you test a Tarski sentence. Try this hypothesis, "This is not a true sentence when tested with the scientific method". To create the scientific test we only need to adopt a logical process of making an assumption about the hypothesis and following it through to see if we reach a contradiction. Sense we are using the scientific method the sentence contradictory when we have a definition of what truth is, or when we allow for testing the scientific method itself. So we might conclude that either the scientific method can make no conclusions or that the scientific method can not be used to test the scientific method without contradicting itself. Because we assume the scientific method does make conclusions, then we are left to consider that it may not be able to study itself without contradictions. I am considering that sense the scientific method may not be able to test itself without contradictions there is another method that may be able to test the scientific method and prove just as use full. A Cybermorphic Method.
I await your responses.