On Thursday, March 14, 2013 2:13:23 PM UTC+11, 1treePetrifiedForestLane wrote: > Godel's original proof is simple, > > with a lot of book-keeping. anyway, if > > you have never even proven the pythagorean theorem, > > how could you discuss "prrof?" > > > > Liebniz gave us the pretty-standard definition, > > "if and only if," meaning that you have to prove > > both "neccesity" and "sufficeincy," by using those words > > in a literate manner -- some how! > > > > but, it is still a good thing, to prove either one, > > > > > so tell us what goldel meant by the notion of truth
you say "Godel's original proof is simple, with a lot of book-keeping. anyway, if you have never even proven the pythagorean theorem, how could you discuss "prrof?"
Liebniz gave us the pretty-standard definition, "if and only if," meaning that you have to prove both "neccesity" and "sufficeincy," by using those words in a literate manner -- some how!
but, it is still a good thing, to prove either one,"
godel showed that provability is not a condition of truth
??from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system. The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system?
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Meaning_of_the_first_incompleteness_theorem ?Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250) For each consistent formal theory T having the required small amount of number theory ? provability-within-the-theory-T is not the same as truth; the theory T is incomplete.
so just tell us what goldel meant by the notion of truth if you cant then we dont know what makes ca maths statement true thus godels theorem with "the notion of truth ... introduced subsequently" is meaningless