VIEWS: 11 PAGES: 6 CATEGORY: Education POSTED ON: 9/3/2009 Public Domain
Re: Godels theorem is meaningless Re: Godels theorem is meaningless Source: http://sci.tech−archive.net/Archive/sci.math/2009−03/msg00685.html • From: Rupert <rupertmccallum@xxxxxxxxx> • Date: Wed, 4 Mar 2009 20:31:46 −0800 (PST) On Mar 4, 10:39 pm, byron <spermato...@xxxxxxxxx> wrote: On Mar 5, 12:28 am, Rupert <rupertmccal...@xxxxxxxxx> wrote: On Mar 4, 4:58 pm, byron <spermato...@xxxxxxxxx> wrote: On Mar 4, 6:17 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote: On Mar 4, 11:35 am, byron <spermato...@xxxxxxxxx> wrote: It has been pointed out that godels theorem is meaningless http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf as it turns out that godel had no idea what makes a Re: Godels theorem is meaningless 1 Re: Godels theorem is meaningless mathematical statement true as peter smith notes himself quote Gödel didn't rely on the notion of truth thus his incompletness theorem becomes meaningless quote http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir Gödel's first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that: For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. Re: Godels theorem is meaningless 2 Re: Godels theorem is meaningless And Peter smith notes godel is talking about true mathematical statements quotehttp://assets.cambridge.org/97805218...40_excerpt.pdf 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 The sentence GT itself is unprovable in theory T. So G T is true if and only if T cant 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, So with out knowing what makes a mathematical statement true the incompleteness theorem is meaningless There are semantic and syntactic versions of the incompleteness theorem. Your first quote gives a syntactic version, your second quote Re: Godels theorem is meaningless 3 Re: Godels theorem is meaningless gives a semantic version. Gödel gave both in his paper but made no claim to rigour for the semantic version because the notion of truth had not yet been defined. However, Tarski defined truth shortly afterwards and both versions are perfectly fine. you say Tarski defined truth shortly afterwards and both versions are perfectly fine. there are many theories of truth long before Tarski the point is any one of those theories could be claimed to account for godels "truth" it is not enough to say taraski or aristotle etc gave a theory of truth and hope that saves godel your tarsk is to SHOW just how tarski theory of truth or any other theory of truth out there fits godels claims as it stands godel had no theory of truth thus his theorem is meaningless −its your task seeing you make the claim to show just how taskis theory fits godel and to note taski theory requires metalanguge No−one before Tarski gave a *mathematically rigorous* definition of truth. Tarski's definition *does* save Gödel because it enables us to make the argument for the semantic incompleteness theorem mathematically rigorous. If you want to know the details you'll need to do some study; do you want me to recommend a textbook for you? and to note tarskis semantic theory of truth has problems Re: Godels theorem is meaningless 4 Re: Godels theorem is meaningless What problems? the problem with tarski is it requires a metalangauge What of it? and we get an ad infinitum Why? if a grammar of a language must be in its metalanguage, as Tarski. seems to require, than the grammar of this metalanguage must be in its metalanguage. thus we have a notion of truth in the object langauge on dependent on the notion of truth in the metalangauge.but the notion of truth in the metalague is itself dependent on the notion of truth in its meta−meta−language No, it's not. We simply *use* the metalanguage. We do not need to define truth in the metalanguage in a metametalanguage. you say No−one before Tarski gave a *mathematically rigorous* definition of truth. Tarski's definition *does* save Gödel because it enables us to make the argument for the semantic incompleteness theorem mathematically rigorous. it is your tarsk as i say to show just how tarskis theory fits godels theorem i can even say a coherence theory of truth fits tarski but i would Re: Godels theorem is meaningless 5 Re: Godels theorem is meaningless need to show that rather than state it tarskis theory is not rigourous as his theory leads to an ad infinitum in regard to "truth I am not talking about the philosophical issues raised by Tarski's work. Mathematically, as the definition of a concept, it is unimpeachable. It can be used to do mathematics. . Re: Godels theorem is meaningless 6