Re: Cantorian pseudomathematics

MoeBlee said:
> Tony Orlow wrote:
> > MoeBlee said:
> > > Tony Orlow wrote:
> > > > Randy Poe said:
> > > > > > There you say it ! But *I* don't think that "alternate decisions
> > > > > > could be made from the same axioms". And *I* dont think
> > > > > > that those theorems are "false", in the sense that they do
> > > > > > not follow from the axioms. What *I* think is that the axioms
> > > > > > themselves are false.
> > > > >
> > > > > What does it mean for an axiom to be false?
> > > > >
> > > >
> > > > It means that they contradict other axioms. Now, you have some small set of
> > > > axioms that don't contradict each other, but in the larger scope of mathemtics
> > > > at large, the conclusions drawn from those axioms contradict conclusions drawn
> > > > from other axioms of mathematics regarding sets such as Han is discussing. In
> > > > the context of those other areas of math and their axioms, the axioms of set
> > > > theory are flawed, as conflicts arise between them and the more solid
> > > > computational axioms of the rest of mathemtics, and reality in general. The
> > > > conclusions derived from the axioms are not correct.
> > >
> > > Please cite a theorem of set theory such that its negation is a theorem
> > > of some other axiomatized theory that you hold to be important. And
> > > what are these "other axioms of mathematics" that you have in mind?
> > >
> > > MoeBlee
> > >
> > > MoeBlee
> > >
> > >
> > No one axiom needs to be the exact negation of another for contradictions to
> > arise between them.
>
> I didn't ask for an example of an AXIOM and its negation. I asked for
> an example of a THEOREM and its negation.
>
> > Certainly, the notion of an infinite number of increments
> > never achieving an infinite value flies in the face of infinite series, where a
> > sum can only converge to a finite value if the terms have a limit of 0 at oo.
> > The constant increments of 1 have no such limit.
>
> That is gobbedlygook. I don't know of theorems of set theory that have
> terms 'infinite number of increments', 'acheiving a value', and
> 'constant increments'. I asked what THEOREM, that is, what provable
> FORMULA of set theory is the negation of another THEOREM of another
> theory that you hold to be important. Name the formula of set theory,
> and name YOUR theory that has a theorem that negates the theorem of set
> theory. Otherwise, your claim about "axioms" is just hot air.

All countably infinite sets have the same size. Is this not a theorem of set
theory with which you would say I disagree? According to my axioms of the
inverse function and N=S^L, different countably infinite sets can have very
different size, even with infinite ratios between sizes.

The infinitude of your omega as the size of the set of finite naturals is
derived, is it not? Given the definiton of the naurals as the set with an
identity relation between element count and value, this theorem is false, as
count cannot exceed value.

>
> > Wherever the axioms for
> > addition are stated, I am sure they are stated so as to avoid this issue, but
> > addition should always mean making greater, and subtraction should mean making
> > less.
>
> Name your axioms, name your theory. Otherwise, your claim about
> "axioms" is just hot air.

N=S^L, where N is the size of the set of strings of length L and S is the size
of the alphabet from which they are made.

The mapping function of a bijection describes the size of the infinite set
being mapped in terms of the size of the set it's mapped to.

Have you seen my axioms of finiteness? I've spewed a number of rules which
qualify as axioms, and are justified outside of the axiomatic system.

>
> > It may be by an infinitesimal fraction of the whole, but it should not be
> > nonexistent. That aleph_0-1=aleph_0 makes no sense to me.
>
> There is a bijection between Omega~{{0}} and Omega. In another post,
> you recognized that such bijections exist. That is all that is at
> stake. The rest is notation, which is NOT deductively creative (except
> that a cardinality operation does use either the axiom of choice or the
> axiom of regularity to formulate; but the 'equinumerosity' predicate
> does not), as I've explained ad nauseum.

That is all that is at stake for cardinality, but not for actual set size. If
the definition of set size as cardinality doesn't change anything, then drop it
and we'll be fine. Cling to it and insist that all other possible comparisons
for infinite sets are invalid due to your axioms, and we'll continue. You're
entitled to play with your system, but if you want to claim it's correct in any
objective sense, you're going to get complaints until the end of time.

>
> So, name your axioms, name your theory. Othewise your claim about
> "axioms" is just hot air.
>

see above

> > COncepts of set
> > density have been developed, but not gotten far enough to really remedy the
> > situation. Han seems to consider transifnite set theory to eb at odds with
> > probability theory, and has given some quotes and diatribes on the subject,
> > though I cannot say with great depth what foul conclusions set theory has
> > postulated for infinite samples of values. My guess is they suck and he's
> > right, even if he is a finitist. :)
>
> "Concepts of density". Name your axioms, name your theory. Othewise
> your claim about "axioms" is just hot air. And, by the way, set theory
> posits axioms; every conclusion is either itself an axiom or provable
> from the axioms in classical first order logic. Whatever "foul
> conclusions" you think are "postulated" are ones that you can only
> reject by rejecting either classical first order logic or an axiom of
> set theory. So far, over thousands and thousands and thousands of
> posts, you still haven't said what logic you do approve of and/or what
> axiom of set theory you reject.

I have, and claims to the contrary are bogus. Does none of what I said above
sound familiar? No, it just sounded "confused" to you. Well, it's not a perfect
seamless system yet, but I am not at all confused about it. Don't pretend that
I haven't responded to these questions, and then dismiss the responses I give
without reason. Try reading it again, why dontcha, and see if you can muddle
through it enough to formulate a question, how's about? I get the feeling that
mathemticians don't actually play with concepts, but only follow rules. I
really was hoping someone with more experience might pick up on some of the
ideas and formulate them to the satisfaction of their colleagues, but it seems
the goal is to stifle any new ideas in an ego contest of refutation. A sad
state of things, really, but I guess that's "natural selection" for you, in the
world of memes.

>
> MoeBlee
>
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: Infinity
    ... >> system and axioms, ... >> making your proofs in set theory. ... > infinite set, and I apologize, the original question had nothing to ... >> equivalent with that of the definition of 'Dedekind infinite'. ...
    (sci.math)
  • Re: Infinity
    ... > and axioms, ... > about 'Dedekind infinite' we're talking about set theory, ... since you don't know what set theory IS. ... While you're right that I misstated the Dedekind definition of an infinite set, ...
    (sci.math)
  • Re: Set Theory: Should You Believe
    ... Why, in your opinion, is the orthodoxy in set theory ... mathematics, ignoring the misgivings of geniuses like Poincare, Weyl, ... When NW said that "You don't need axioms", ... NAFL theories (but infinite proper classes, ...
    (sci.logic)
  • Re: Logarithm of transfinite numbers
    ... then your own axioms are subject to such scrutiny too. ... thatan infinitely long oline has an infinite length. ... criterion of fundamental truth you've challenged set theory with. ... qualitatively, as all primitives probably should for ease of comprehension, but ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... interpretation takes anything seriously at all. ... > believing the axioms are correct in some sense or something like that. ... Set theory was billed to me as the type-free be-all theory, ... It still doesn't mean that the reals ...
    (sci.math)