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


MoeBlee wrote:
david petry wrote:

Seriously? I have said repeatedly that truth requires meaningfulness,
and not all the axioms of Cantorian set theory are meaningful.

What about ZF? Do you think any of the axioms are meaningful? If so,
which ones?

And what axioms would you add to replace the ones you've deleted? Or,
if you make no replacements, how do you derive theorems such as those
used in basic calculus?

MoeBlee

In a sense the calculus, the integral and basically (but not totally)
equivalently differential calculus, where there are notions along the
lines of discrete and umbral and predicate and lambda and pi, type,
program, and concurrency, calculi, is defined in terms of points and
lines, or geometry, with basically some algebra, or in toto, algebraic,
analytic geometry.

Where that is so, there are notions along the lines of the circular
definitions of points and lines, from Euclid/Hilbert, which are defined
in terms of each other, and basically in this context what to consider
is points vis-a-vis the complex hyperspace, basically the NxN, or a
larger, orthonormal space.

So, to consider something along the lines of geometry or the
differential calculus, with its "infinitesimals", the nilpotent
"Cholera bacillus" differential, in terms of set theory and foundations
as generally practiced, then basically those primitive elements in the
set theory, or sets, become as well the geometric objects, or otherwise
the primary objects might be numbers or some such thing.

So, then, considerations of sets, and the empty set and universal "set"
and so forth, basically resolve to tautologies and as well various uses
of the law of excluded middle, or not.

There is no universe in ZF. That might seem surprising considering the
claim that there is a universal quantifier. Quantify over sets, in ZF
it's not a set, yet in ZF every thing is a set, so everything is a set,
else quantification s not universal, and that is quite the useful
notion, in terms of brevity.

Ross

.

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