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.

A two sided beginning. No quantity zero math begins it
and then absolute values go beyond it.
Zero and infinitesimal maths are fundamental.

Mitchell Raemsch