Re: Continuum hypothesis

On Aug 25, 2:10 am, herbzet <herb...@xxxxxxxxx> wrote:
Perhaps I can get used to the practice.

I agree that "structure" is an unintuitive term. If it were
up to me, "structures" would be "interpretations" (which people
say anyway when they're being slack) and "interpretations"
would be -- something else, maybe "valuations". We would
reserve "models" for sentences, sets of sentences, theories.

I stole this from somebody calling himself "honest rosewater" on a
physics forum.
It's just one example of somebody being comfortable with the
distinctions between
all these things in current usage.





The situation with structures is similar to the one with theories in
that we connect structures with a language in order to use them. An L-
structure is a structure that can be used to interpret all of the
symbols of a language L.

The interpretation and truth-value assignment are done with functions,
but you can use different definitions depending on your purposes, and
the form will depend on the form of the language that you're
interpreting. The most general form of an L-structure that I can think
of is an ordered pair (A, I), where A is your underlying set, or
domain, which contains the individuals of your structure, and I is the
set of functions that use A to interpret the symbols of your L-theory
and assign truth-values to your formulas. The variations on this (A,
I) pair would split I up into different functions or sets of
functions. For example, you might separate out the truth-assigning
functions (commonly called an L-valuation) or, if L has constant
symbols, you could specify that some function maps your constant
symbols to individuals in your domain. For simplicity, we'll keep
everything together under the umbrella of an L-structure.

If an L-structure assigns a truth-value of true, or whatever value we
have chosen to correspond to truth, to a formula, we say that the
structure models that formula or is a model of that formula.
Similarly, if a structure interprets every formula in a set of
formulas to be true, we say that it models that set of formulas.
Recall that a theory is a set of formulas. So, for example, a model of
an L-theory of sets is an L-structure that interprets every formula in
that L-theory as being true. If we turn our earlier example structure
of a set with the identity relation into a suitable L-structure, it is
a model of our L-theory of equivalence relations because the identity
relation does indeed satisfy the equivalence relation axioms, and due
to the properties of and relations among the entailment relations of
first-order logic and the axiomatization of our L-theory, any
structure that is a model of our axioms is also a model of our entire
theory; if it makes the axioms of our theory true, it must make all of
the other formulas of our theory true as well.


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