axioms CREATE meaning

On May 6, 5:04 am, Phil <toob-head...@xxxxxxxxxxxxx> wrote:
You are mostly correct, except that everyone
claims that the ESSENCE of
an infinite set is that it has the same cardinality as a proper subset
of itself. Well, I am pointing out that that is true ONLY when that
infinite set is combined with the proper axioms.

ALL language is like that, DUMBASS!
"The natural numbers" does not MEAN the natural numbers
EXCEPT IN THE PRESENCE of the relevant DICTIONARY!
The dictionary plays, for natural languages, the role analogous
to that played by the axioms in formal languages.
N doesn't have to contain 0 until and unless you have an
axiom saying so. "Natural" doesn't have to mean
"finite, whole, and non-negative" until AFTER you have
a dictionary saying so; other meanings of "natural" might
tend to imply that the algebraic closure of the COMPLEX
rationals were the more "natural" numbers. ALL LANGUAGE
DEPENDS, for its meaning, for its effectiveness, on some
PRIOR AGREEMENT or conventions about meaning!
For symbols in formal languages, THE AXIOMS ARE that
prior agreement.

There is also an even deeper prior agreement about logical symbols
and first-order-language components like commas and parentheses.

.

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