Re: Godel cant tell us what makes a mathematical statement true

herbzet wrote:

Nam Nguyen wrote:
Baudouin Le Charlier wrote:

A statement is true just because it is true not because somebody
'relies on the notion of truth'.
Could you think of any circumstance in which 0=1 is true?

Sure. Let "0" be my dad. Let "1" be me. Let "=" be the
relation "father of".

Can't you think a bit more mathematical, like:

A1: AxEy[Sx=y]
A2: Ax[0=x]

Then (as a theorem) 0=S0, which is 0=1. Right?


Not sure where you're going with this.

Now T df= {A1 + A2} is consistent, so 0=1 is *true* in T. So,
unlike what Baudouin said, whether 0=1, or 0=/=1, is true does
rely on some notion of truth! There's no such thing as a formula
being true "just because it is true", and not relying on a notion
of truth.


To think that something is true only because
it has been proven is the wrongest idea you can conceive.
Suppose for a given formal system T, we define true sentences
as the following:

- T(F) = true iff T |- F.
- T(F) = false otherwise.

What would you think as "wrong" with this definition?

Well, if T |/- F and T |/- ~F then both F and ~F are false.
Is that problematic for you?

Of course not. It's *not a perfect definition* but:

a) If a formula is not defined to be true w.r.t. T, then there's
nothing wrong to equate it with being false. In this way then:

- A consistent system is one in which one of {F, ~F} is true and
the other false.
- An inconsistent system is one in which both F and ~F are true.
- F is undecidable in T if both F and ~F are false (which only means
F and ~F aren't provable in T!)

Nothing would seem "wrong" at all!

b) The standard definition of truth is *not perfect either*: there
are some T in which if T is consistent, there would be some formula
F which you can't tell whether or not F is true in any model
of T!

But the point here is a formula can't be just simply true. You have to
*choose* *some* selected/defined notion of truth!

--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.

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