Re: Goedel's undecidable G

Kazimierz Kurz wrote:
>
> Jim Spriggs napisal:
> > LordBeotian wrote:
> >> I want to see the explicit expression of G in the language of PA, is there
> >> any web page where I can take a look to G?
> > No, is my guess, because G will be _very_ complicated when expressed in
> > the language of PA. Before saying more, I must admit to my ignorance
> > about what the language of PA is. Does it just have a constant for zero
> > and a function symbol for successor, or does it also have function
> > symbols for addition and multiplication?
> If formal theory has not multiplication there is no Goedel sentence
> construction for it.
>
> > To express G a long sequence of definitions is used and the G that
> > results is _not_ in the language of PA.
> You are wrong! Of course, and this is the clue, Goedel sentence
> is a senntence of PA annd it is just ordinary theorem about natural
> numbers! But simultaneously it may be interpreted as sentence about
> itself...

While looking for something else, I came across Peano's "Arithmetices
principia, nova methodo exposita" in English translation. It is quite
clear that the only non-logical constants are: number, one, successor,
is equal to. He introduces addition and multiplication by recursive
definitions though he fails to prove that "+" and "x" are eliminable
(and thereby breaks his own rule for definitions). The paper is
reproduced in van Heijenoort's Source Book.

--
I don't know who you are Sir, or where you come from,
but you've done me a power of good.
.

Relevant Pages

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