Re: Formulating sentences in a possibly consistent ZF

Aatu Koskensilta wrote:
Nam Nguyen <namducnguyen@xxxxxxx> writes:

I think you've underestimated God's Wisdom. All He needs is *only 2 models*,
one in which GC is true and the other cGC is, and we'd *fail* to see
which one our beloved "standard model of arithmetic" is!

How does God present these two models [to] us?

He doesn't; He only presents *finite description* of *common part* of the
two, and the rest is an agony we've suffered for centuries, and could
still for an eternity. [Note by cGC, we mean the arithmetic formula
"There are infinite counter-examples of GC".]

Since we can't in any literal sense be presented with infinite
mathematical structures,

Right. If He literally did the structures in their *entirety*, we wouldn't
be able to "see" them.

the most natural assumption is that he does so in the form of some
mathematical description.

Some *finite* mathematical description, to be precise!

We may or may not then be able to tell which, if any, of these description is a description of the standard model.

Agree. That's also similar to what Torkel alluded to: "we can make
no similar observation about how a Goldbach-like statement can be
proved if it is true" (The Incompleteness theorem). So, there's
*a real chance* GC, a particular Goldbach-like sentence, can't be
known as true when it's actually true (however contradictory that
may sound)!

It seems the few techniques God would use in presenting an impossible
challenge for the mortal-knowledge like us to solve would be:

- Human Finite Knowledge/Describe-ability.
- The Known-Unknown Juxtaposition.
- One-to-Many Mapping.

Let me elaborate on these techniques a bit more.

Example 1: Suppose God has 2 infinite sequences of beads, all blue
in 1 sequence, and golden in the other. Now within a blink of an
human eye, He shuffles the 2 sequences into a new one. Now, suppose
as far as we could see (which is finite) all the golden beads we can
see happen to fall into the positions of the even number satisfying GC.
But if indeed all the golden beads happen to be that way, only God
can tell - human can't.

Example 2: Suppose God has 2 countably infinite sequences of ZFC sets,
all designated as "blue" in 1 sequence, and "golden" in the other.
Now within a blink of an human eye, He *chooses*, using Choice, a new
sequence from the 2 old ones. Now, suppose as far as we could see
(which is finite) all the "golden" sets we can see happen to fall into
the positions of the even number satisfying GC. But if indeed all the
"golden" sets happen to be that way, only God can tell - human can't.

Example 1 is the intuitive version of example 2, while 2 is the formal
version of 1. But in both cases the perceived difficulty (e.g. whether
or not "_all_ the golden beads happen to be that way") is a *juxtaposition*
of what we think/assume we know (e.g. the sequences' being countably
infinite [viz-a-viz Induction Principle]), and what we can't know (e.g.
the exact *Chosen* positions of all the golden beads [viz-a-viz Choice
Principle]).

***

Our case of the 2 models is simply another example of this juxtaposition,
but to see that we should examine 2 infinite sequences of the "naturals".
The 1st sequence (S1) would be related to GC and the other (S2) to cGC.
For each even natural satisfying GC, there corresponds the maximum prime
contributing to the satisfaction. E.g. 10 = 5 + 5 or 10 = 3 + 7; hence the
maximum prime for 10 viz-a-viz GC is 7. So if GC is true, we could define
S1 per this observation. The 2nd sequence, S2, would be the one defined by
cGC itself. I.e., for each even number e which is a counter-example of GC,
we'd take the nearest prime larger than e to be a defined prime in 2; hence
S2 would ultimately be formed accordingly.

Now, if we perceive N as the unique model of arithmetics of the naturals,
then N would contain all the n-ary predicates, each would correspond to
1 n-ary symbol (whether or not the symbol is a natural part of the language,
or a defined one). So we could treat both S1 and S2 as predicates that would
be represented by 2 function symbols 'S1' and 'S2', respectively. Of course
if N is non-empty then one of the predicates S1 and S2 must necessarily empty,
while the other is not. *But we don't which one is the case*.

Thus, in "the standard model" N we think we know, there's a juxtaposition
between the Induction Principle (used in the recursive definition of arithmetic
truths) and the the unknown-ability as to which one of S1 and S2 is empty.

Depending on the details of the descriptions kindly provided to
us by God, this observation may well amount to nothing more than
noting that we may or may not be able to decide whether Goldbach's
conjecture is true.

Are you sure about "nothing more"? If the "kindness" of God might not
grant us the arithmetic truth of GC, how certain could we be in *asserting*,
say, the truth of other Goldbach-like sentences typically seen involved
in Godel's work?

Do you think this indicates some problem in our on the face of it
entirely unproblematic understanding of the naturals?

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?

In fact, Godel's numbering doesn't insist which particular set of primes
to use, but does insist that the set must be infinite, to accommodate for
an arbitrary large number of (non-logical) symbols of a formal system
(language). So S1 or S2 could be used. But since *we don't know which* of
S1 and S2 is empty, the conclusion of G(T) is true is quite subjective,
without any of us ever knowing which is the case!

So in summary, the sense of "entirely unproblematic" is not correct here,
it'd seem to me.

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











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