Re: What FOL Can't Prove

On Aug 28, 6:36 pm, RussellE <reaste...@xxxxxxxxx> wrote:
Which statement is true
depends on the domain of discourse.

Yes. What statements are true of a collection of things
depends on which things we are talking about. If we
say "all x have fur" and we are talking about mammals,
it is true, but if we are talking about prime numbers,
it's false.


FOL can prove
the domain of discourse is inconsistent,
but, FOL can't prove the domain is consistent.

Whoops! There's a problem here. Consistency isn't
a property of domains; it is a property of theories.
Asking whether a domain is consistent or not is like
asking what the daily calorie requirement for the set
of prime numbers is; you have misapprehended what
the term means.


Consider the statement:

Q = ExAy (x>=y)
~Q = AxEy (x<y)

Define the domain to be the set: {0,1,2}.
We see Q is true (x=2) and ~Q is false (x=2).

Let the domain be the set of non-negative
integers: {0,1,2,3,...}
Which statement, Q or ~Q, is true for this domain?

Obviously ~Q is true for this domain. The existence of
the successor function is a complete proof of this fact.
Actually it's even stronger than a proof, because it
tells you not only that for all x such a y exists, it tells
you how to find one.


If we assume PA is
consistent, we must say Q is false and
~Q is true. If we assume PA is inconsistent,
Q is true and ~Q is false.

What is the meaning of this brazen last-second mention
of PA? You asked a question about the non-negative
integers. PA? Bah. I don't need some johnny-come-lately
Italian theory to know how to calculate:

for all x, x<x+1


FOL can not prove the domain is consistent.

You need to review the meaning of "inconsistent."


Marshall

.

Relevant Pages

  • Re: What FOL Cant Prove
    ... Do you agree my domain of discourse is consistent with Ax x=x? ... Models and domains are not "consistent with" a theory or an axiom. ... of the domain does not have a successor. ...
    (sci.logic)
  • Re: What FOL Cant Prove
    ... Do you agree my domain of discourse is consistent with Ax x=x? ... But you can't prove PA inconsistent, ... Do you agree my theory is consistent with Ax x<S? ... In your model, whose set of naturals is supposedly, there is ...
    (sci.logic)
  • Re: What FOL Cant Prove
    ... I am assuming we can define the domain of discourse ... It may seem obvious to you that PA is consistent. ... You can prove PA is consistent using the successor function? ... If we assume PA is inconsistent, ...
    (sci.logic)
  • Re: What FOL Cant Prove
    ... I am assuming we can define the domain of discourse ... FOL can't prove the domain is consistent. ... If we assume PA is inconsistent, ... If x is a non-negative integer, then x+1 is not chopped liver. ...
    (sci.logic)
  • Re: What FOL Cant Prove
    ... I can derive Ax x<Sfrom the sixth axiom of PA. ... I see a finite set, for which your theorems are true, but any ... I am proving my theory is consistent with axiom one. ... Do you agree my domain of discourse is consistent with Ax x=x? ...
    (sci.logic)