Re: Formulating sentences in a possibly consistent ZF

Nam Nguyen wrote:
herbzet wrote:

Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:
Aatu Koskensilta wrote:
[...]

Again, how sure are you about "entirely unproblematic"? For example,
if we can't know which one of GC and cGC is arithmetically true, then
we'd have to acknowledge that the notion about the naturals is
*subjective*, since you might take them to contain the truth of GC,
while I of cGC, right? And if arithmetic is subjective, then Godel's
assertions such as "G(Q) is true" is consequently subjective as well.
Right?
You might be interested to know that Goldbach's weak conjecture, that
every integer greater than 5 can be written as the sum of 3 primes,
is known to have, at most, a finite number of exceptions.

See http://En.wikipedia.org/wiki/Vinogradov%27s_theorem#A_consequence .
Of course, by "every integer greater than 5", it's meant to be
"every *odd* integer greater than 5".
You are correct: That every odd number greater than 5 is the
sum of three primes is the weak Goldbach conjecture, and
Vinogradov showed there are at most a finite number of
exceptions.

That every integer greater than 5 is the sum of three primes is
a stronger conjecture, equivalent to saying that every even number
greater than 2 is the sum of two primes, and is how Goldbach actually
phrased his conjecture in a letter to Euler (modulo 18th century conventions).

Found out today ( http://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture ):

In 2002, Liu Ming-Chit and Wang Tian-Ze lowered [the upper bound] to
approximately n < e^3100 =(approx) 2 * 10^1346. If every single odd
number less than 10^1346 is shown to be the sum of three odd primes, the
weak Goldbach conjecture will be effectively proved.

As such it seems make a huge
difference on the nature of difficulties between GC and wGC (the
weak GC mentioned above).
Yes, it's a weaker conjecture.

If we recall, an even number *could be purely defined* by a non-inductive
multiplicative way (as a product of primes one of which must be the smallest
prime). On the other hand, we can't define odd numbers in the same manner!
An odd number is a number that doesn't have 2 as a factor.
Sure. But there's still a difference: the even numbers (in an multiplication
only system) can be defined as direct function of what's known to exist:
the smallest prime and the other known primes. The way you define the
odd numbers wouldn't yield much information about them. For instance,
given a model of a system that has only multiplication, prime, and "<"
I could define the addition "+" that looks "a bit natural" as:

Ax(x+0=x)
Axy(x+y=y+x)
AxEp1,p2(x is even -> (p1, p2 are prime and x = p1 + p2))
...

Where (x is even) <-> Ey(x = 2*y) where 2 is the smallest prime.

How would you define addition so that it'd "partially look natural",
given the definition of odd above?

Don't know, don't care.

But I'll take this opportunity to give an alternate definition
of odd numbers -- a number is odd iff it is congruent to 1 (mod 2).
And how is "congruent" defined in term of multiplication?

a is congruent to b (mod m) iff m is a factor of a-b.

And how is 'a-b' defined without the addition '+'? As noted above, an
even number could be defined without addition.

I don't think you can define odd numbers with only '*', the way even
numbers can. And that, imho, seems to suggest on intuition level
*this difference* between even and odd numbers is the key reason why
if GC is true it'd be "absolute undecidable".

Let me elaborate a little more. Suppose we stipulate the existences of
the 2 infinite sets M1, M2, but where say M1's stipulation is based
solely on Induction Principle, while M2's stipulation is strictly based
on Dedekind's property of infinity. Suppose further that we now combine
both into one final set M = (M1 + M2).

Now there's a new thesis (a new Principle) related to M that I think we
should adopt in the foundation of mathematical logic or reasoning. And
that thesis would basically say the following. Suppose the statement F
is:

F = (H1 & H2 & ... Hn) -> C
and it's hypotheses H1 & H2 &... Hn are based strictly on Dedekind's
property of infinity, while the conclusion C could be expressed solely
by the Induction Principle, then:

if F isn't false, it's impossible to know F is true in M.

Sorry for a typo. Instead of "H1 & H2 &... Hn are based strictly on
Dedekind's property of infinity", it should have been "H1 & H2 &... Hn
can be strictly based on Dedekind's property of infinity".

Given this thesis then it's easier to see the relevancy of emphasizing
the fact that odd numbers can't be "positively" defined the same way
even numbers can, using only primes and multiplication. The reason being
is the part of arithmetic that involves only multiplication, primes, and
the binary predicate '<' is Dedekind's property of infinity which is
more generic property than that of Induction kind of infinity. And in
such case, which involves infinity, you can't not use what is general
(all the hypotheses H1, .. Hn) to conclude what is specific, namely C.

There doesn't seem to be a "proof" of this thesis or new principle.
But that's what it is: we can only reject or adopt it. I'm working
on an example simpler than GC to illustrate the thesis and would
present it here when it's done.

But the following is a "primeval" arithmetic system that has only
multiplication and < that I mentioned earlier as reflecting
Dedekind's infinity and not Induction infinity.

<Foot Note>

Consider the language L(0,1,*,<) and the following axioms:

A1 - Ax[(x*0=0) /\ (x*1=x)]
A2 - Axy[x*y=y*x]
A3 - ExAyz[~(x=0) /\ ~(x=1) /\((x = y*z) -> ((y=p /\ z=1) \/ (y=1 /\ z=p))]
(Prime number definition; x is called a prime and is denoted as P(x).)

A4.a - Axy[(x<y) \/ (x=y) \/ (y<x)]
A4.b - Axy[(x<y) -> ~(x=y)]
A4.c - Axyz[((x<y /\ (y<z)) -> (x<z)]
a4.d - (0 < 1) /\ (Ax[P(x) -> (1 < x)])
A4.e - Axy[(P(x) /\ P(y))-> (x < x*y)]

A5 - AxEy[P(x) -> (P(y) /\ (x < y))] [This means infinite number of primes].
A6 - ExAy[P(x) /\ (P(y) -> (x <= y))] [This is the minimal prime i.e "two".]


Note 1: An even number x can be defined as x = two*y, for some y.
In particular, 4 is defined as two*two.

Note 2: L actually could be L(0,S,*,<) such that wherever '1' occurs it could
be replaced by 'S0'.

</Foot Note>



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

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