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

Nam Nguyen wrote:
Nam Nguyen wrote:
herbzet wrote:

Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:
MoeBlee wrote:
On Sep 25, 6:03 pm, herbzet <herb...@xxxxxxxxx> wrote:
Don't you think the 2-tuple say (e1, e2) already
has a subjective semantic in that e1 could be *arbitrarily* interpreted
as the first - or last - element (and ditto for e2)?
The 2-tuple (a, b) is standardly defined as {{a} {a,b}}, which of
course is equivalent to {{b,a} {a}}, etc.
"Standardly defined", {{a} {a,b}} is *identical* to {{b,a} {a}}!
Nice catch. Also {{a} {a,b}} is equivalent to {{b,a} {a}}.
Minor complaint from me here but usually one would define the notion
*first* before using it!

What's the formal definition of 2 "equivalent" sets?
A ~ B iff A and B have the same cardinality: |A| = |B|.

Which of a, b, is "first"
or "last" is irrelevant; what is relevant is that (a, b) = (c, d) iff
a = c and b = d.
That doesn't prevent (b,a) = (d,c) iff a = c and b = d, does it?
Nope.

Are you saying (a,b) =/= (b,a) isn't significant?
No I'm not saying or implying that.

(a,b) = (b,a) iff a = b.
You've implicitly said that all right! According to you, (a,b) and (b,a)
are "equivalent",

They are equivalent.

which would make the fact they're different irrelevant!

The fact that they are equivalent makes the fact that they are
(in some cases) not identical irrelevant?

I don't know other cases but this is the case that no matter what
the formal (i.e. syntactical) definition of a 2-tuple is, such
definition will connote 2 opposite semantics for the 2-tuple. As such
any formal definition of equivalence of two 2-tuples will eternally
*not* pin down precisely which semantic it *must* be. Consequently,
model truth, which depends on the n-tuple semantic, must be necessarily
be subjective.

I've done demonstration on model subjectivity using a model with 2
elements before but MoeBlee and you didn't and still don't seem to
comprehend and appreciate it. So let me re-do the demonstration with
only 1 element; hopefully its being simpler would the whole situation
easier to understand.

But let me first go back to the 2 element example to clarify the
general notation used in model definition; then I'll proceed to
the 1 element example, with clarification having been explained.

Now let L2(a,b,<) be a language and let's consider the formula
F2 = a < b. Let's consider the following set M2:

M2 = {{e1,e2}, (a,e1), (b,e2), (<,{(e1,e2)})}.

Note that here "set" is a priori notion and not a formal ZF set
(though by practice we'd mean a ZF set by a "set"). Note also
the notation (x1, x2) means another priori notion of an finite
(2 elements in this case) ordered sequence. NOTE: in (x1,x2)
we have *not* yet stipulated/defined *which order* the sequence is!
Consequently, (x1,x2) would symbolize *either* one of the 2 ordered
sequences - in fact 2 2-tuples!

Now the truth of F2 in M2, denoted as M2(F2) is technically defined
as:

(M2(F2) is true) iff (P(a,b) is true).

But P(a,b) is true iff (a,b) is in the set I:

I(nterpretation) = {precise-2-tuple(e1,e2)}

where I is in this case a singleton, and where precise-2-tuple(e1,e2)
is the exact ordered sequence of 2 elements e1, e2 that *must* be
recognized by everyone; and we'd like the symbol (e1,e2) represent
this unique ordered sequence. But the uniqueness of the "sequence"
is not firstorderrizable, as we've alluded to above that we couldn't
stipulate it.

The good question is: could we demonstrate an example of this non-uniqueness
of an n-tuple in general? In the next example, we'll give such a
demonstration.

Let L1(theSky, isBlue) be a language where 'theSky' is an individual
constant symbol, and 'isBlue' an unary predicate symbol. Let's also
define the ZF set imageOfSky as:

imageOfSky = {(x,y),color} [where color is either 0.0 or 1.0, where
the set of all (x,y)'s is a rectangular, and the set
of (x,y,1.0) would form a shape of a cloud.]

Let M1 be the familiar-structured set

M2 = {{imageOfSky}, (theSky,imageOfSky), (<,{(imageOfSky)})}

I really hate typo but here it is again: instead of M2, we should
have had M1:

>> M1 = {{imageOfSky}, (theSky,imageOfSky), (isBlue,{(imageOfSky)})}


But what's exactly the *ordered* 1-tuple (imageOfSky)? (After all
there's only one element here!). According to Shoenfield, an 1-ary
predicate in the universe of individuals is a subset of the Universe;
and in this case with the universe being a singleton the predicate
is {{imageOfSky}}. So now the interpretation set I (for the formula
F1 = theSky isBlue) is:

I = {precise-1-tuple(e1)} = {{imageOfSky}}

which is *purely a set* now; and nothing is subject to interpretation
anymore, right? Well, what would happen to the "true" interpretation
set I' for F1, using the same singleton universe {imageOfSky} ?

Sorry for a typo: should have been "...interpretation I' for ~F1..."

Here's the final situation:

(1) F1: I = {{imageOfSky}}
(2) ~F1: I' = {{imageOfSky}}

If {{imageOfSky}} is the true-interpretation for both F1 and ~F1, that
would be so ridiculous that we should just stick with syntactical
provability, instead of introducing the truth notion to FOL. On the
other hand, if we insist only (1) be acceptable then we'd be at lost
as to why (2) be not acceptable, not to mention that we'd violate
all the intended usage for semantic within FOL.

The only solution we could take (and my little thesis here is the
founders of FOL have so taken), is relativize both (1) and (2)
in the following sense:

(a) We could choose either (1) or (2), but not both.
(b) Even if we individually choose either (1) or (2),
there's nothing wrong for others to claim we've chosen
the wrong choice, from their perspective.

Hopefully this demonstration has been technical enough to enable
you and MoeBlee understand what I have been talking about the
subjectivity or relativity of model truth.
.

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