On Friday, June 15, 2018 at 9:39:50 AM UTC-7, Ross A. Finlayson wrote:
> On Thursday, June 14, 2018 at 1:52:26 PM UTC-7, burs...@gmail.com wrote:
> > Anyway fruit cake, reposts extremely old nonsense,
> > he has already posted in the past. Somehow something
> > is new in his post. It seems that
> >
> > he really believes that its time to overcome some
> > existing foundation, and replace it by something
> > new. As if the choice of a foundation so
> >
> > far was a transient phaenomenom, which is now
> > to vanish. Ha Ha. Sounds almost the same as
> > this question here:
> >
> > Is the multiverse a merely transient phenomenon
> > or can it legitimately claim to represent the
> > ultimate set-theoretic ontology?
> > http://logika.ff.cuni.cz/radek/papers/MCST_Synthese_Complete.pdf
> >
> > As if prime numbers, Gödels incompletness etc..
> > were some super nova stars from physics and not
> > some everlasting results from math,
> >
> > that might thus disappear in the future.
> >
> > Am Donnerstag, 14. Juni 2018 22:45:48 UTC+2 schrieb burs...@gmail.com:
> > > You sure? What do you mean exactly by that?
> > >
> > > Some mathematicians were very exited by the
> > > reductionist power a foundation offers, for
> > > example in real analysis. (*)
> > >
> > > Some foundations are so versatile, you can
> > > even do all kind of stuff with it. Almost
> > > everything you can imagine. (**)
> > >
> > > (*)(**) I owe references for all that
> > >
> > > Am Donnerstag, 14. Juni 2018 00:39:43 UTC+2 schrieb FredJeffries:
> > > > Mathematicians have never (consciously) "built" on any (formal) foundation.
> I kind of approach the "set theory multiverse"
> the same way as the "Many Worlds Interpretation
> (MWI) physics universe", there's just the one, it's a universe.
>
> Copenhagen -> Shelah (it's either)
> Everett -> Hamkins (it's both)
> Boehm -> Zermelo (it's one)
>
> "We show now that, with a little care, all reasonable
> properties of V formulated with reference to outer
> models are actually first-order ...."
>
> In the analogy to physics, much of the foundation
> in physics is ofcourse based on the stochastic, and
> and about Pauli exclusion. But, these days there
> are known the parastatistics, about that physics
> observes what should be the stochastic but that
> there are different observations and thus different
> probability theories about bosons and fermions.
>
> So, it is upon mathematics to discover the
> fundamental probability theories so relevant
> to automatically equip the relevant physical
> theories with a corresponding mathematical
> foundation.
>
> Then this corresponding notion of paraconsistency
> as about symmetry flex addresses many the same
> notions as "nonstandard models of arithmetic",
> "infinitely diverse and large integers", and about
> systems for cardinals with and without CH, though
> that some would have that forcing is disordered
> and that relevant independence results of "conclusions"
> in set theory have alternative means.
>
> So, my idea is basically about ubiquitous ordinals
> for the unbounded in both the horizontal and
> vertical following Zermelo, one universe with
> everything in it (including itself).
>
> Physics is always on the search for a mathematical
> "theory of everything" (ToE), for mathematics
> it's the same, a foundation. (A formalist's general
> mathematical logic.)

"I think [Z a.J.] asks: what is the ultraclass of ultraclasses? I coin that
the "group noun game", that either having sets is enough or sets,
classes, ultraclasses, ..., is never enough, there are not infinitely
many group nouns, were there a maximal element, might as well call it
a set. What is the collection of all (proper) ultraclasses?

You might look at one of Russell's various theories of types, in terms
of those kinds of encapsulations. (Classes are to sets as models are
to theories.)

I call classes and so on, in a set theory, non-sets, if the set is a
fundamental collection, defined by its elements, it should be
sufficient to fulfill its own definition.

Then, there are obviously perceived paradoxes about infinite integers
and infinite integers, or for example the consideration of less axioms
than ZF's deciding that ZF's universe is the Russell set, the sputnik
of quantification of infinity in infinity, (infinite) sets are
irregular, N E N. (ZF's universe is the Russell set.)"

Feb.9, 2014:

"Also:

Cantor's nested intervals theorem <-> Finlayson's EF as counterexample

Cantor's antidiagonal argument <-> Finlayson's EF as counterexample

continued fractions <-> Finlayson's EF as counterexample

Cantor's indicator function theorem <-> Finlayson's symmetrical mapping
as counterexample

Zuhair's binary tree theorem <-> Finlayson's BT = EF as counterexample

Cantor's powerset theorem <-> Finlayson's powerset as order type as
successor construction, and a dialetheic ur-element

Russell's negated correlates <-> Finlayson's note on statement of
structurally true languages

irrationals uncountable <-> Finlayson's "A function surjects the
rationals onto the irrationals"

I give myself a lot of credit.

It's not always so simple. Reason might arrive at that
there are features of the infinite numbers that can't
be taken up and put down like a hat or gloves or on
a whim (besides that we can neatly conjecture).

Surely axiomatics offer a neat way to categorize
modes of thought and given kinds of objects of
reason (including the absurd) but the reasoner
doesn't have to accept "axioms" except conscientiously
hold them as so for a theory. Then here as above
the numbers as so rich and ubiquitous objects of
theory, may have conclusions about them that can
be arrived at quite despite claims they don't
exist.

Basically this is pointing that having only an
"ordinary" infinity is as _wrong_ as having
none. That is _wrong_ in the sense that
ignorance is wrong, that axiomatics reflect
proper theories, but there are others.

This is not just that "the numbers exist",
they're the same to everyone, all of them.

Retro-finitist crankety trolls as computational
or numerical methods with bounded resources
and no internal rational agency to posit others
are simply enough closed systems, and physics is not.

Thanks, though, it's a gentle rejection.

"In 1893 Vivanti, the Italian mathematician, wrote
to Cantor suggesting that his rejection of infinitesimals
was unjustified. As du Bois-Reymond had shown, "the
orders of infinity of functions constitute a class of
one-dimensional magnitudes, which include infinitely
small and infinitely large elements. Thus there is no
doubt that your [Cantor's] assertions cannot apply to
the most general concept of number."

In defending his point of view, Cantor wrote back
to Vivanti with the vitriolic remark that to the best
of his knowledge, Johannes Thomae was the first
to "infect mathematics with the Cholera-Bacillus of
infinitesimals." Paul du Bois-Reymond, however, was
soon to follow. Cantor claimed that in systematically
extending Thomae's ideas, du Bois Reymond found
"excellent nourishment for the satisfaction of his own
burning ambition and conceit." Cantor went on to
discredit du Bois-Reymond's infinitesimals because
they were self-contradictory, since he rejected without
compromise the existence of linear numbers which were
non-zero yet smaller than any arbitrarily small real number.
Infinitesimals, Cantor replied, were complete nonsense.
He reserved his strongest words for du Bois-Reymond's
Infinitäre Pantarchie and his orders of infinitesimals:
"Can one still call such things numbers? You see, therefore,
that the `Infinitäre Pantarchie,' of du Bois-Reymond, belongs
in the wastebasket as nothing but paper numbers!" Cantor
placed the theory of actual infinitesimals on a par with
attempts to square the circle, as impossible, sheer folly,
belonging in the scrap heap rather than in print. Ironically,
much of Cantor's criticism could have been turned as
effectively against the transfinite numbers as against
infinitesimals. And his rejection of infinitesimals was
certainly fallacious in its reliance upon a petitio principii.
Having assumed that all numbers must be linear, this was
equivalent to the Archimedean property, and thus it was
no wonder that Cantor could "prove" the axiom. The infinitesimals
were excluded by his original assumptions, and his proof
of their impossibility was consequently flawed by its own
circularity." -- Dauben

On Saturday, August 26, 2023 at 12:30:21 PM UTC-7, Ross Finlayson wrote:
> On Friday, June 15, 2018 at 9:39:50 AM UTC-7, Ross A. Finlayson wrote:
> > On Thursday, June 14, 2018 at 1:52:26 PM UTC-7, burs...@gmail.com wrote:
> > > Anyway fruit cake, reposts extremely old nonsense,
> > > he has already posted in the past. Somehow something
> > > is new in his post. It seems that
> > >
> > > he really believes that its time to overcome some
> > > existing foundation, and replace it by something
> > > new. As if the choice of a foundation so
> > >
> > > far was a transient phaenomenom, which is now
> > > to vanish. Ha Ha. Sounds almost the same as
> > > this question here:
> > >
> > > Is the multiverse a merely transient phenomenon
> > > or can it legitimately claim to represent the
> > > ultimate set-theoretic ontology?
> > > http://logika.ff.cuni.cz/radek/papers/MCST_Synthese_Complete.pdf
> > >
> > > As if prime numbers, Gödels incompletness etc..
> > > were some super nova stars from physics and not
> > > some everlasting results from math,
> > >
> > > that might thus disappear in the future.
> > >
> > > Am Donnerstag, 14. Juni 2018 22:45:48 UTC+2 schrieb burs...@gmail.com:
> > > > You sure? What do you mean exactly by that?
> > > >
> > > > Some mathematicians were very exited by the
> > > > reductionist power a foundation offers, for
> > > > example in real analysis. (*)
> > > >
> > > > Some foundations are so versatile, you can
> > > > even do all kind of stuff with it. Almost
> > > > everything you can imagine. (**)
> > > >
> > > > (*)(**) I owe references for all that
> > > >
> > > > Am Donnerstag, 14. Juni 2018 00:39:43 UTC+2 schrieb FredJeffries:
> > > > > Mathematicians have never (consciously) "built" on any (formal) foundation.
> > I kind of approach the "set theory multiverse"
> > the same way as the "Many Worlds Interpretation
> > (MWI) physics universe", there's just the one, it's a universe.
> >
> > Copenhagen -> Shelah (it's either)
> > Everett -> Hamkins (it's both)
> > Boehm -> Zermelo (it's one)
> >
> > "We show now that, with a little care, all reasonable
> > properties of V formulated with reference to outer
> > models are actually first-order ...."
> >
> > In the analogy to physics, much of the foundation
> > in physics is ofcourse based on the stochastic, and
> > and about Pauli exclusion. But, these days there
> > are known the parastatistics, about that physics
> > observes what should be the stochastic but that
> > there are different observations and thus different
> > probability theories about bosons and fermions.
> >
> > So, it is upon mathematics to discover the
> > fundamental probability theories so relevant
> > to automatically equip the relevant physical
> > theories with a corresponding mathematical
> > foundation.
> >
> > Then this corresponding notion of paraconsistency
> > as about symmetry flex addresses many the same
> > notions as "nonstandard models of arithmetic",
> > "infinitely diverse and large integers", and about
> > systems for cardinals with and without CH, though
> > that some would have that forcing is disordered
> > and that relevant independence results of "conclusions"
> > in set theory have alternative means.
> >
> > So, my idea is basically about ubiquitous ordinals
> > for the unbounded in both the horizontal and
> > vertical following Zermelo, one universe with
> > everything in it (including itself).
> >
> > Physics is always on the search for a mathematical
> > "theory of everything" (ToE), for mathematics
> > it's the same, a foundation. (A formalist's general
> > mathematical logic.)
> "I think [Z a.J.] asks: what is the ultraclass of ultraclasses? I coin that
> the "group noun game", that either having sets is enough or sets,
> classes, ultraclasses, ..., is never enough, there are not infinitely
> many group nouns, were there a maximal element, might as well call it
> a set. What is the collection of all (proper) ultraclasses?
>
> You might look at one of Russell's various theories of types, in terms
> of those kinds of encapsulations. (Classes are to sets as models are
> to theories.)
>
> I call classes and so on, in a set theory, non-sets, if the set is a
> fundamental collection, defined by its elements, it should be
> sufficient to fulfill its own definition.
>
> Then, there are obviously perceived paradoxes about infinite integers
> and infinite integers, or for example the consideration of less axioms
> than ZF's deciding that ZF's universe is the Russell set, the sputnik
> of quantification of infinity in infinity, (infinite) sets are
> irregular, N E N. (ZF's universe is the Russell set.)"
>
>
>
> Feb.9, 2014:
>
> "Also:
>
> Cantor's nested intervals theorem <-> Finlayson's EF as counterexample
>
> Cantor's antidiagonal argument <-> Finlayson's EF as counterexample
>
> continued fractions <-> Finlayson's EF as counterexample
>
> Cantor's indicator function theorem <-> Finlayson's symmetrical mapping
> as counterexample
>
> Zuhair's binary tree theorem <-> Finlayson's BT = EF as counterexample
>
> Cantor's powerset theorem <-> Finlayson's powerset as order type as
> successor construction, and a dialetheic ur-element
>
> Russell's negated correlates <-> Finlayson's note on statement of
> structurally true languages
>
> irrationals uncountable <-> Finlayson's "A function surjects the
> rationals onto the irrationals"
>
> I give myself a lot of credit.
>
>
>
>
>
> It's not always so simple. Reason might arrive at that
> there are features of the infinite numbers that can't
> be taken up and put down like a hat or gloves or on
> a whim (besides that we can neatly conjecture).
>
> Surely axiomatics offer a neat way to categorize
> modes of thought and given kinds of objects of
> reason (including the absurd) but the reasoner
> doesn't have to accept "axioms" except conscientiously
> hold them as so for a theory. Then here as above
> the numbers as so rich and ubiquitous objects of
> theory, may have conclusions about them that can
> be arrived at quite despite claims they don't
> exist.
>
> Basically this is pointing that having only an
> "ordinary" infinity is as _wrong_ as having
> none. That is _wrong_ in the sense that
> ignorance is wrong, that axiomatics reflect
> proper theories, but there are others.
>
> This is not just that "the numbers exist",
> they're the same to everyone, all of them.
>
> Retro-finitist crankety trolls as computational
> or numerical methods with bounded resources
> and no internal rational agency to posit others
> are simply enough closed systems, and physics is not.
>
> Thanks, though, it's a gentle rejection.
>
>
>
>
>
> "In 1893 Vivanti, the Italian mathematician, wrote
> to Cantor suggesting that his rejection of infinitesimals
> was unjustified. As du Bois-Reymond had shown, "the
> orders of infinity of functions constitute a class of
> one-dimensional magnitudes, which include infinitely
> small and infinitely large elements. Thus there is no
> doubt that your [Cantor's] assertions cannot apply to
> the most general concept of number."
>
> In defending his point of view, Cantor wrote back
> to Vivanti with the vitriolic remark that to the best
> of his knowledge, Johannes Thomae was the first
> to "infect mathematics with the Cholera-Bacillus of
> infinitesimals." Paul du Bois-Reymond, however, was
> soon to follow. Cantor claimed that in systematically
> extending Thomae's ideas, du Bois Reymond found
> "excellent nourishment for the satisfaction of his own
> burning ambition and conceit." Cantor went on to
> discredit du Bois-Reymond's infinitesimals because
> they were self-contradictory, since he rejected without
> compromise the existence of linear numbers which were
> non-zero yet smaller than any arbitrarily small real number.
> Infinitesimals, Cantor replied, were complete nonsense.
> He reserved his strongest words for du Bois-Reymond's
> Infinitäre Pantarchie and his orders of infinitesimals:
> "Can one still call such things numbers? You see, therefore,
> that the `Infinitäre Pantarchie,' of du Bois-Reymond, belongs
> in the wastebasket as nothing but paper numbers!" Cantor
> placed the theory of actual infinitesimals on a par with
> attempts to square the circle, as impossible, sheer folly,
> belonging in the scrap heap rather than in print. Ironically,
> much of Cantor's criticism could have been turned as
> effectively against the transfinite numbers as against
> infinitesimals. And his rejection of infinitesimals was
> certainly fallacious in its reliance upon a petitio principii.
> Having assumed that all numbers must be linear, this was
> equivalent to the Archimedean property, and thus it was
> no wonder that Cantor could "prove" the axiom. The infinitesimals
> were excluded by his original assumptions, and his proof
> of their impossibility was consequently flawed by its own
> circularity." -- Dauben
>
> What comes around goes around.