On 09 Dec 2010, at 20:43, Brian Tenneson wrote:
Is there any first order formula true in only one of R and R*?
I would think that if the answer is NO then R < R*.
What I'm exploring is the connection of < to [=], with the statement
that < implies [=].
The elementary embeddings preserve the truth of all first order
formula. So it should be obvious that if A < B, then A [=] B.
In B there might be elements or objects or set of objects obeying
relations which are not consequences of the first order relations.
I think that all standard models of first order theories of finite
structures (like numbers, hereditarily finites sets, rational numbers,
etc.) are elementary equivalent with their non standard models. You
need second order logic to describe what happens in those models.
But I have not invest on model theory since some time.
Are there any other comparitive relations besides elementary embedding
that would fit with what I'm trying to do? What I'm trying to do is
one major "leg" of my paper: there is a "superstructure" to all
structures.
But sets and categories have been seen that way. This leads to
reductionism in math, in my opinion. Yet category theory provides
ubiquitous non trivial relations between many mathematical objects.
But Lawvere failed to found mathematics on the category of categories.
And categories with partial objects, like those which populate so much
computer science, are, well, quite close to abstract unintelligibility
(for me, but who knows). Category impresses me the most in knot
theory, and the buildings of models for weak logics (linear logic,
intuitionist logics, quantum linear logic).
What super means could be any comparitive relation. But
what relation is 'good'?
You ask a very difficult question. You might appreciate morphism of
categories (functor), or of morphism of bicategories, or n-categories,
if you want powerful abstractions.
But assuming mechanism, and the 'everything goal': I would insist on
the relations of 'dreaming', or partial emulation between numbers
relatively to universal numbers.
The infinite dynamical mirroring of the universal numbers. That just
exist if we assume the axiom of Robinson arithmetic, and we are
embedded or better: distributed, or multi-dreamed by or in it (with
our richer axioms!) and all, this with notions of neighborhoods and
accessibility between our consistent extensions (that you can extract
from studying what can and cannot prove sound löbian numbers about
themselves. See my papers for more on that, and good basic books are
Boolos 1979, 1993, Smullyan, Rogers, etc).
It depends on what you are searching for. If you want to include
psychology and theology, expect some universal mess diagonalizing
against all complete reductions.
Bruno
On Dec 9, 8:12 am, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 09 Dec 2010, at 05:12, Brian Tenneson wrote:
On Dec 5, 12:02 pm, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 04 Dec 2010, at 18:50, Brian Tenneson wrote:
That means that R (standard model of the first order theory of the
reals + archimedian axiom, without the term "natural number") is
not
elementary embeddable in R*, given that such an embedding has to
preserve all first order formula (purely first order formula, and
so
without notion like "natural number").
I'm a bit confused. Is R < R* or not? I thought there was a fairly
natural way to elementarily embed R in R*.
I would say that NOT(R < R*).
*You* gave me the counter example. The archimedian axiom. You are
confusing (like me when I read your draft the first time) an
algebraical injective morphism with an elementary embedding. But
elementary embedding conserves the truth of all first order formula,
and then the archimedian axiom (without natural numbers) is true in R
but not in R*.
Elementary embeddings are *terribly* conservator, quite unlike
algebraical monomorphism or categorical arrows, or Turing emulations.
Bruno
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