Re: Cantorian pseudomathematics

MoeBlee wrote:
> Han.deBru...@xxxxxxxxxxxxxx wrote:
> > MoeBlee wrote:
> > > MoeBlee wrote:
> > > > Now, you've added to the page that
> > > > what your inconsistent theory does is show that set theory "blows up"
> > > > when you add your own axiom to it.
> > >
> > > Correcting that, as I recall, the 'blows up' commentary was there
> > > before I mentioned the inconsistency, so I should have said that what
> > > has been added is the commentary in reaction to having the
> > > inconsistency (which was commented upon by other posters too).
> > > pointed out.
> >
> > Yeah, I'm still dizzy and confused about this x = { x } stuff :-(
>
> You needn't be confused.
>
> > And though it seems ridiculous, I'm not going to remove that
> > page until I've found a satisfactory explanation why my physical
> > intuition is continously in conflict with the mathematics of sets,
> > even at the most elementary level.
>
> The explanation is simple: You're not allowing that abstractions such
> as those of set theory work structurally not literally. Your conception
> of meaningfulness is too shallow and too narrow. As long as you insist
> on keeping to the reduction of meaningfulness that you've arrived upon,
> then you will always be confused trying to make sense of thinking that
> doesn't fit that reduction. You have made your own procrustean bed.
>
> > I'm not even saying that set
> > theorists are wrong.
>
> You do SEEM to be saying that.
>
> > It would help, though, if I could get pointers
> > to a finitary version of set theory, iff such a thing exists.
>
> It does exist. And it is useful. But it is not clear, not at least at
> first, how it could provide an axiomatization of analysis. But there
> are other finitary systems that you can research too.
>
> > Is there
> > i.e. a subset of the ZFC axioms describing only _finite_ sets?
>
> You can just drop the axiom of infinity from ZF or ZFC, or (b) You can
> adopt the negation of the axiom of infinity, or (c) You can adopt a
> revised version of the axiom of infinity. With (a), it is left
> indeterminate whether there exist infinite sets. With (b), there are no
> infinite sets; thus you don't need the axiom of choice.
>
> About x = {x}:
>
> (1) You added Ax x = {x} to ZF. What makes that contradictory is the
> universal quantifier. You can add Ex x = {x} to ZF, but not Ax x = {x}.
> However, you can't add Ex x = {x} to ZFR ('R' for the axiom of
> regularity).
>
> (2) There are alternatives to ZFR in which Ex x = {x}. I.e., there are
> theories that contradict the axiom of regularity.
>
> (3) Quine in his 'Set Theory And Logic' discusses his own views,
> somewhat different from standard set theory, on x = {x}. You might find
> his reasoning to be interesting.
>
> (4) Brouwer has species. Does he stipulate that every species is a
> member of itself? And though you disdain intuitionism that you consider
> is too revisionist, there are different axiomatizations of
> intuitionistic set theory in Troelsta and van Dalen's 'Constructivism
> In Mathematics'.
>
> Aside:
>
> (5) On your web site you say that an inconsistent theory proves no
> formulas (or 'statements' or whatever word you use). What you should
> say is that an inconistent theory proves ALL formulas.
>
> /
>
> Why don't you just learn some mathematical logic and set theory? You
> don't have to think of the theorems as expressing truths if you don't
> want to. You can instead just enjoy the ingeniousness and austere
> beauty of the deductions and constructions, even if only as one would
> appreciate the ingeniousness and beauty of chess moves. I say this not
> to advocate that that is all set theory is or is worth, but rather to
> offer at least one way someone such as you might appreciate certain of
> its aspects. And it really is fun. If you allowed yourself to come out
> from behind that armor of yours, maybe to loosen up a bit, free of the
> constraints of that clanky warrior suit you wear, then you might enjoy
> yourself with a little set theory once in a while.
>
> I find it striking that there are mathematicians who have an
> appreciation of both constructivism and classical mathematics. These
> are mathematicians who can appreciate the differences and the relations
> between the approaches because these mathematicians UNDERSTAND both
> approaches. But on the Internet at least, there are people who just
> blindly attack classical mathematics while knowing virtually NOTHING
> about it. Instead of getting to a deeper level of understanding the
> differences, connections, and relatioins, these people just solidify
> their ignorance so that it is an even blunter weapon of their polemics
> in rounds and rounds of senseless controversy.
>
> MoeBlee

That's a nice post. It's not necessarily agreeable, but it's easy on
the eyes and the mind.

These reductionisms as you call them are not necessarily easy on the
mind. It's removal of a safety net of sorts, which can be dangerous
for where it is unnecessary, it is not so far beyond that where the
security of an abstract bedrock of foundation returns, but to some
extent there is involved a mental realization of basically the infinite
induction which is for some difficult in attainment, or detachment.

"There's little like a waking nightmare to make mathematicians mad."

The mathematical foundation, it is there. It's there already.
Whatever numbers you invent already were.

Basically I simply mental machinery that basically starts at a
conceptualization of the continuum that is inherent, yet to some extent
unobvious, i.e., easily arranged but difficultly grasped.

The universe is infinite. There's only one theory with no axioms.

Ross

.

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