Re: Rational numbers, irrational numbers: each dense in real numbers

On Oct 5, 10:41 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
It's interesting that you mentioned the Pigeonhole Principle here,
since in the end, what TO (not sure about RF) ultimately wanted
to prove using "infinite case induction" was that the Pigeonhole
Principle holds for infinite sets as well as finite sets. These
so-called "cranks"

Why "so-called"? They're cranks.

I dislike the use of the word "crank" to describe the opponents of
ZFC, but since there's no simple word to describe collectively
RF, TO, WM, and the others, I use the phrase "so-called."

Yesterday, I posted for the first time in the thread of yet another
opponent of mainstream mathematics, Archimedes Plutonium, and
he was very upset at my use of the word "crank" to describe him,
even with the tag "so-called." Therefore nothing is accomplished by
repeated name-calling (both sides are guilty of this).

Of course, I already knew that this attempt is doomed.

Then why post all the rigmarole to prove it? We already knew it and
have proven it too.

You ask what I'm trying to accomplish by posting all of this.

As I've said before, much of what the opponents of ZFC wish to show
can be made quite rigorous using Nonstandard Analysis. But then
the mainstream mathematicians point out that Nonstandard Analysis
is derived from Standard Analysis and is rooted in ZFC.

What the opponents of ZFC wish to do is come up with a new set
theory that somehow isn't based in ZFC at all. They do not wish to
derive infinitesimals by first coming up with the standard set of real
numbers and using the Transfer Principle. They'd rather show that
infinitesimals must a priori exist without going through the standard
real numbers at all. (As one such person -- not a sci.math poster --
once put it, they want infinitesimals to be "naturally derived," not
"forcibly derived" from the real numbers.)

Indeed, I posted in the Archimedes Plutonium thread about how his
p-adics resembled hyperreals, AP argued that he didn't want them to
be derived from hyperreals at all. He considered the hyperreals to be
part of an algebra-based theory (and he considers all mainstream
mathematics to be algebra-based), while he wants his p-adics to be
more geometry-based. Thus AP completely rejects standard theory
and its reliance on algebra, just as many others reject mathematics
that's based on set theory (especially ZFC).

So obviously a new set of axioms is needed. These attempts to
come up with "infinite-case induction" fail. Yet still, I can't help
but
wonder whether there is a rigorous set theory in which:

-- no set, not even an infinite set, can be bijected with a proper
subset of itself
-- given infinity, one can find infinity-1, infinity-2, infinity/2,
sqrt(infinity), log(infinity) (mentioned by TO and others)
-- infinitesimals exist (TO, RF, etc.)
-- "cardinality" and "measure" are more intimately related (perhaps
by simply multiplying the former by an infinitesimal to derive the
latter, hinted at by several people)

but every attempt to come up with such a theory has failed.

Not to exclude WM, but some of his theories appear to be
even easier to axiomatize. WM doesn't believe in the existence
of uncountable sets. Obviously there are some theories, such
as ZF-Infinity and ZFC-Powerset, in which it is impossible to
prove the existence of uncountable sets. And in others, such
as Pocket Set Theory (which I mentioned in previous threads),
in which every set is countable -- uncountable classes turn out
to be proper classes.

Good call. Though, it would be interesting to me any progress you make
toward a systemm that upholds "infinite-case pigeonhole" and is
adequate for proving an alternative real analysis

That's still my goal, but of course it won't be easy (otherwise TO,
RF,
etc., would already have accomplished it).

.

Relevant Pages

  • Re: Calculus XOR Probability
    ... definitions in Cartesian geometry which even hint at infinitesimal ... Isn't the idea of limit to "sweep the infinitesimals under the rug", ... Possibly true.What do you mean by "universe"? ... to ZFC so I am unsure why you mention it. ...
    (sci.math)
  • Re: Rational numbers, irrational numbers: each dense in real numbers
    ... And I don't use the word 'crank' to mean 'opponent ... particular mathematics, including ZFC. ... derive infinitesimals by first coming up with the standard set of real ... reject UNCOUNTABLE sets. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... definitions in Cartesian geometry which even hint at infinitesimal ... Isn't the idea of limit to "sweep the infinitesimals under the rug", ... Possibly true.What do you mean by "universe"? ... to ZFC so I am unsure why you mention it. ...
    (sci.math)
  • Re: crank crank crank
    ... > Robinson's infinitesimals), as opposed to the ... But non-standard analysis is standard mathematics;-). ... of what they are meaning. ...
    (sci.math)
  • Re: crank crank crank
    ...  > Robinson's infinitesimals), as opposed to the ... This was pointed out by several posters. ... NSA is still "standard mathematics." ...
    (sci.math)