Godel had no idea what truth is so incompleteness theorem is meaningless
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From: byron on 8 Aug 2010 08:46 The australian philosopher colin lesie dean points out Godel had no idea what truth is so incompleteness theorem is meaningless http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE Now truth in mathematics was considered to be if a statement can be proven then it is true Ie truth was s equated with provability http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics ââ¦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, [b]true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.[/b] The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements [b]that are true but cannot be proven within the system[/b]â Now the syntactic version of Godels first completeness theorem reads Proposition VI: To every Ï-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r). But when this is put into plain words we get http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Meaning_of_the_first_incompleteness_theorem âAny 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, [b]there is an arithmetical statement that is true,[1] but not provable in the theory [/b](Kleene 1967, p. 250) For each consistent formal theory T having the required small amount of number theory ⦠[b]provability-within-the-theory-T is not the same as truth;[/b] the theory T is incomplete.â In other words t[b]here are true mathematical statements which cant be proven[/b] But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless at the time godel wrote his theorem he had no idea of what truth was as peter smith the Cambridge expert on Godel admitts http://groups.google.com/group/sci.logic/browse_thread/thread/ebde70bc932fc0a7/de566912ee69f0a8?lnk=gst&q=G%C3%B6del+didn%27t+rely+on+the+notion+PETER+smith#de566912ee69f0a8 Quote: Gödel didn't rely on the notion of truth but truth is central to his theorem as peter smith kindly tellls us http://assets.cambridge.org/97805218...40_excerpt.pdf Quote: Godel did is find a general method that enabled him to take any theory T strong enough to capture a modest amount of basic arithmetic and construct a corresponding arithmetical sentence GT which encodes the claim âThe sentence GT itself is unprovable in theory Tâ. So G T is true if and only if T canât prove it If we can locate GT , a Godel sentence for our favourite nicely ax- iomatized theory of arithmetic T, and can argue that G T is true-but-unprovable, but Gödel didn't rely on the notion of truth now because Gödel didn't rely on the notion of truth he cant tell us what true statements are thus his theorem is meaningless Ie if Godels theorem said there were gibbly statements that cant be proven But if godel cant tell us what a gibbly statement was then we would say his theorem was meaningless
From: Frederick Williams on 8 Aug 2010 13:20
byron wrote: > > The australian philosopher colin lesie dean points out Godel had no > idea what truth is so incompleteness theorem is meaningless As has been remarked before, one can state and prove G\"odel's incompleteness theorem without reference to any semantic notions. -- I can't go on, I'll go on. |