On 09 Jun 2015, at 18:59, John Clark wrote:
On Mon, Jun 8, 2015 Bruce Kellett <[email protected]> wrote:
> What axioms led to arithmetic?
The Peano axioms.
Or the Robinson axiom, or many other systems. but they don't disagree
on any formula. Even the theories having weird axioms like "PA is
inconsistent" will not disagree on what they say for the standard
natural numbers. They disagree only on religion, somehow.
They were chosen because they are very simple and self evident. You
need to be very conservative when picking axioms, for example we
could just add the Goldbach Conjecture as an axiom, but then if a
computer found a even number that was NOT the sum of 2 primes it
would render all mathematical work done after the addition of the
Goldbach axiom gibberish. Or take Zermelo–Fraenkel set theory
(ZFC)
Well, that is Zermelo–Fraenkel set theory + the axiom of choice.
and the Continuum Hypothesis which says that there is no infinite
number greater than the number of integers but less than the number
of Real Numbers; in 1940 Godel showed that ZFC cannot prove the
Continuum Hypothesis to be incorrect, and in 1963 Paul Cohen showed
that ZFC cannot prove the Continuum Hypothesis to be correct either.
So ZFC has nothing to say about the Continuum Hypothesis one way or
the other. You could just add an axiom to ZFC saying "the Continuum
Hypothesis is true" but you could just as easily add "the
Continuum Hypothesis is NOT true", so which one do you add? The
problem is that neither of these axioms are simple and neither are
self evident.
> Could one have chosen different axioms?
It's never a good idea to change axioms unless somebody finds a set
of axioms that are even simpler and even more self evident.
It depends what we need. RA is interesting because it is the simple
essentially undecidable theory. Take any of its axioms,
0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
and remove it. You get a theory which is undecidable, but not
*essentially* undecidable. It means you can extend those subtheories
into decidable theories, like the theory of real numbers. But RA is
already essentially undecidable: all its consistent effective (RE)
extensions are undecidable and incomplete (with respect to
arithmetical truth).
But RA cannot prove many things. It is simple to see that 0 + x = x is
undecidable in RA. And RA is not Löbian. It is Turing universal, but
cannot prove it, unlike PA, ZF, ZFC, ZF+kappa, etc.
Once Löbian, they get the same theology, and the same testable "comp"
physics.
Note that ZF and ZFC proves the same formula of arithmetic.
Bruno
John K Clark
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