Re: True = [ proven | provable ]

From: |-|erc (h_at_r.c)
Date: 01/19/05 

Date: Wed, 19 Jan 2005 13:53:22 +1000

"LordBeotian" <pokipsy76@CANCELLAMIyahoo.it> wrote in >
> "|-|erc" <h@r.c> ha scritto
>
> > > > Mathematicians don't need the word true.
> > > >
> > > > For "I think its true" say "I think its provable".
> > > >
> > > > For "G is true" say "G is proven"
> > >
> > > How do you prove 1+1=2 ?
> > >
> >
> >
> > Usually like this:
> >
> > Define the number 0
>
> Difined in which way?

E0, 0 e N
zero is a number! you can deny 0 is a number, but then 1+1 may not equal 2.
what we're actually proving is "1 + 1 = 2 for the usual interpretation of the terms".

if 1 = todays breakfast and 2 = a razor and + = a mirror and "=" = air ticket
then "breakfast mirror breakfast ticket razor" may not always be true.

>
> > Define all other numbers as the successor of some other number.
> >
> > 0 = 0
> > 1 = s(0)
> > 2 = s(s(0))
> > 3 = s(s(s(0)))
> > 4 = s(s(s(s(0))))
> > and so on
> >
> > Define addition as
> > 0 + 0 = 0 rule 0
> > s(a) + 0 = s(a) rule 1
> > s(a) + b = a + s(b) rule 2
>
> Uhm... are these true statements we do not prove?

rule 0 and rule 1 are given by the *meaning* of +.
for the usual interpretation of plus, 0 + x = x
again, you can deny 0+x=x but again 1+1 wont equal 2.

rule 2 is the distributive law, it could be composed of these definitions
s(a) = a + 1. #1
s(a) + b = a + 1 + b = a + b + 1 = a + s(b) #2

#1 and #2 are a smaller grain than rule 2, but its just a "definition of the usual interpretion of plus"
rule 2 is not that hard to decipher as it stands.

> Are they true "by definition"?
> So not only is true what is proven but also is true what is stated as a
> definition or as an axiom?

bingo! there are few allowed truths as part of proof systems themselves,
like EXISTS X. but proofs are generally, A & A->B then B

A axiom (taken for granted, eg. the natural numbers appear in order)
A->B (given the axiom eg rule 1 and 2 and 0 defn, implies B! eg 1+1=2 )
B (1+2 = 2 is proven)

> What if I chose my axiom/definitions in different ways so that I "prove"
> 1+1=3? Would 1+1=3 become true?
>

Not for the usual interpretations of 1, 2, 3, +, =
The trick is your axioms must be equivalent to what the terms actually mean in real life.

Herc

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