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Topic: What makes a mathematics statement true?
Replies: 16   Last Post: Mar 14, 2013 12:38 AM

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byron

Posts: 881
Registered: 3/3/09
Re: What makes a mathematics statement true?
Posted: Mar 14, 2013 12:38 AM
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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

Now truth in mathematics was considered to be if a statement can be proven then it is true
Ie truth is equated with provability
http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics

??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,[1] 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



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