Brief Explanation of Omega-inconsistency
I was thinking. . .
I parenthetically mentioned Godel's Theorem in my last post, expecting that to make clear how mathematical sciences being an invention of the mind would leave us knowing nothing. I now realize that is way too cursory an examination to make my point. But, since my explanatory power is inversely proportional to my interest in the subject, I will try to make this as easy to follow and interesting [ha] as possible. First, some background history . . .
Some of you my remember vaguely reading Dedekind. Roots of number theory and all of that, trying to separate arithmetic from Geometry. In the late 19th and early 20th centuries an attempt was proposed, that number theory could prove its own consistency. All that is saying that given a set of axioms, and simple "rules of inference" [symbolic logic] we could prove all theorems. Many attempts were made to have all of number theory derived from a very limited set of axioms.
On to the proof. Numbers and operations and so forth in this number theory [often called sentential calculus] are given letters and so forth with the standard symbolic logic "and" "or" "not" and so forth symbols. The "sentences" in sentential calculus represent "well-formed" strings. Analogically, the equation '8=' is not well-formed, because there isn't anything on the right of the equal sign. Godel showed that by giving words numbers and translating those numbers into "strings" that the strings were well-formed inside of the axiomatic schema. So, what does he do? He makes (roughly) the equivalent of "I cannot be proved in system A" (which we can call for simplicity, sentence G) a sentence written in system A. He then shows that it is "well-formed". Therefore, G is a theorem of the axiomatic schema, and a true theorem, as long as the schema is incomplete. But if we could prove G, then the axioms would contain a contradiction, because G itself says that it can't be proved. Thus he calls it "formally undecidable". But even if we added G as an axiom, to make a new, expanded schema, we could use the same method to create a new sentence "I cannot be proved in system A plus axiom G" and follow the same reasoning to get the same incompleteness.
To illustrate the point, Hofstadter, in "Godel, Escher, Bach" has a character with the best record player, who proclaims that it can play any record. So the other character make a record called "I cannot be played on so-and-so's record player". This record, when played, makes sounds that resonate in the natural frequency of the record player, causing it to vibrate and break. This is the result of axiomatic number theory - a broken record....
Back to last post. If mathematics is purely an invention, deductions following from inferences and axioms, then it is incomplete or inconsistent, for it follows Godel's proof. If that is the case then the human endeavor in mathematical physics is a waste of time. This is why I rejected choice number two.
