Re: Why? [was Re: Cantor`s powerset theorem is false?]

david petry wrote:
MoeBlee wrote:
david petry wrote:
An axiom in a formal system is merely a string of symbols. In order to
say that it is either true or false, so I claim, it first needs an
interpretation. The theory under consideration here, Cantorian set
theory, does not include any rules for interpreting its axioms and
statements.

The rules are in the meta-theory.

You skipped responding to that.

And meaning is also that of consequence (whether or not an axiom is
true, certain statements that are conditionals are not just true, but
logically true, with an axiom as its antecedent). And meaning is also
structural. Whatever the truth of the axioms, the theorems of the
theory are related among themselves to structurally express at least
the ordinary mathematics we wish to express.

A key issue for this discussion is the meaning of the word "meaning".
I have given a definition: the meaning of a statement may be equated
with its observable implications. When I asked you before to define
your terms (specifically, the term "meaningless"), you gave a circular,
warm and fuzzy, touchy-feely nonsense definition. Until you can do
better than that, there will be no point to further discussion.

No, I gave a gamut of the senses in which one may understand a
mathematical activity to be meaningful. I started the list with the
formal sense of that of models, and I mentioned other senses that are
quite definite. Then I moved on to mention other senses that are more
subjective. And there was no circularity in any of it. That you
mischaracterize my response as including only the more subjective
senses is intellectually dishonest as well as it is the weak form of
strawman in which one attacks the least part of an argument as if one
has attacked the greatest part of it. And you use your
mischaracterization as a general pretext not to response to my
challenges. And by this time you're on the path to making a canard with
your repeated mischaracterization of my response.

Meanwhile, you have not answered all kinds of questions, including as
to your larger understanding of Feferman's philosophy, your source for
your claim that Cantor claimed mathematics to be only empty formalism,
your claim that there was a tacitly understood procedure for
determining the meaningfulness of mathematical statements prior to
Cantor (as well as how that jibes with your statement that the
statement that Goldbach's conjecture is true is meaningless and also
what meaning mathematicians took to be that of, for example, the twin
prime conjecture), and what axiomatization, if any, you propose for
proving theorems of ordinary mathematics.

And, my latest points still stand: another sense in which we may take
the meanings of theorems of set theory is that of conditional sentences
in which axioms are the antecedents. These sentences are not only true,
but they are logically true. And set theory does provide an
axiomatization of at least the ordinary mathematics we expect to have.
And I said more about that in another post, which you have not
responded to.

What standard interpretation? There is no standard interpretation of
set theory.

There seems to be widespread agreement among set theoreticians on how
to interpret set theory.

Interpretations are themselves mathematical. There is no standard
interpretation of set theory.

leads mathematicians to assert that there "exists" a world beyond the
world we observe (i.e. beyond the world of phenomena observable in the
world of computation).

To even assert that there are not just observations of phenomena but
that these phenomena form a world is itself an ontological commitment
that is not itself an observational statement.

Whatever. Maybe Ross could translate that last sentence of yours.

What I said is clear: You claim that there is a world of certain
phenomena. However, to claim that certain phenomena constitute a world
is itself an ontological claim. So a sentence such as "there exists a
world of phenomena" is itself a sentence that asserts an ontological
claim that is not observationally testable.

MoeBlee

.

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