Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: Tony Orlow on 27 Jul 2005 16:43 Robert Kolker said: > Han de Bruijn wrote: > > MoeBlee wrote: > > > >> Meanwhile, I'm still fascinated by your inconsistent theory, posted at > >> your web site, which is: > >> > >> Z, without axiom of infinity, bu with your axiom: Ax x = {x}. > >> > >> Would you say what we are to gain from this inconsistent theory? > > > > > > No. But first we repeat the mantra: > > > > A little bit of Physics would be NO Idleness in Mathematics > > Many parts of mathematics are inspired or motivated physical phenomena. > For example calculus (the theory of real variables) was invented by > Newton specifically to quantify motion of material bodies. > > So physics is a potent motivator for some branches of mathematics and > has been so for 400 years. However, physical applicability is not a > requirement for creating a mathematical theory. Analytic number theory > is not motivated by physical considerations althought it turns out it > has some physical applications. > > In any case, physical applicablility is not the sina qua non for the > validity or soundness of a mathematical theory. Internal consistency is. > > > > > > It is physically correct that every member of a set is at the same time > > a _part_ of the set, > > Sets are not physical. They are abstract. Search the world over and you > will not find a set. In the real world trees exist but forests do not. > Forests exist in our heads. > > Bob Kolker > Forests do not exist? Do people exist, or only cells? Maybe cells don't exist, but only organelles. Do organelles exist? Nope, just molecules. Oh wait, molecules don't exist, only atoms, which are, of course, unsplittable. They're not? Okay they don't exist. Only quarks exist. Quarks are made of what? Okay they don't exist either. Nothing exists, except Bob. -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 16:44 Robert Kolker said: > Han de Bruijn wrote: > > MoeBlee wrote: > > > >> Meanwhile, I'm still fascinated by your inconsistent theory, posted at > >> your web site, which is: > >> > >> Z, without axiom of infinity, bu with your axiom: Ax x = {x}. > >> > >> Would you say what we are to gain from this inconsistent theory? > > > > > > No. But first we repeat the mantra: > > > > A little bit of Physics would be NO Idleness in Mathematics > > Many parts of mathematics are inspired or motivated physical phenomena. > For example calculus (the theory of real variables) was invented by > Newton specifically to quantify motion of material bodies. > > So physics is a potent motivator for some branches of mathematics and > has been so for 400 years. However, physical applicability is not a > requirement for creating a mathematical theory. Analytic number theory > is not motivated by physical considerations althought it turns out it > has some physical applications. > > In any case, physical applicablility is not the sina qua non for the > validity or soundness of a mathematical theory. Internal consistency is. > > > > > > It is physically correct that every member of a set is at the same time > > a _part_ of the set, > > Sets are not physical. They are abstract. Search the world over and you > will not find a set. In the real world trees exist but forests do not. > Forests exist in our heads. > > Bob Kolker > Trees don't exist, only branches. I mean branches don't exist, only leaves. No, leaves don't exist, because they are sets of cells. Do cells exist? -- Smiles, Tony
From: Virgil on 27 Jul 2005 16:42 In article <MPG.1d518d2f358cbb7a989fa8(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > >It seems like axioms are given a status that makes them > > >unquestionable and almost incomprehensible, beyond the application > > >of them. > > > > They are not incomprehensible to the people who work with them. > Right. That's why nobody here seems to have recognized the basis for > the inductive axiom, and misses the fact that it constitutes an > infinite recursion. Sure. You guys really understand your tools. If there were not an axim which allows us to avoid that recursion, we wold be required to execute it, but the axiom says specifically we can have the result without the execution. > Axioms need to be proven too Not by anyone who understands what an axiom system is all about. >> > Nothing in mathematics is excepted without question. Not by >> > mathematicians, anyway. > That's not what they say all the time here. They typically consider > axioms unquestionable truths, or at least act like they are. When working within an axiom system, one is required to consider the axioms true, and the only logical reason to refute an axiom system is internal inconsistency. One may avoid or reject an axiom system one does not care for, and simply not work within that set of axioms, but refutation requires proof of internal inconsistency. TO has consistently failed to show internal inconsistency for ZFC, et al, as his arguments always require assumptions contradicting the axioms.
From: David Kastrup on 27 Jul 2005 16:44 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Define each subset as an infinite binary string, where the first bit > denotes membership of the first natural, the second bit denotes > membership of the second natural, etc. Each subset int he pwoer set > of the naturals is thus represented by a unique infinite binary > string. Each finite binary string is accepted as corresponding to a > finite natural, but the infinite strings, with infinite values, are > rejected on the basis that natural numbers cannot have infinite > values. If they were allowed to have infinite values, the the set of > all infinite binary strings which represents the power set of the > naturals could be put in bijection with the set of natural numbers, > and you would have a bijection between the naturals and their power > set. Wrong, since those infinite binary strings can only account for the set in/exclusion of the finite numbers. Anyway, if you have any mapping n->F(n), where F(n) is a subset of N, then the subset S={k in N: k not in F(k)} is not accounted for by any n. That is because if n happens to be a member of F(n), then it can't be a member of S. And if n happens to be not a member of F(n), then it will be a member of S. For that reason, S is different from _any_ F(n), and thus no mapping F(n) can cover the complete powerset P(N), since S clearly is a member of P(N). -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Virgil on 27 Jul 2005 16:46
In article <MPG.1d518dd16568434b989fa9(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > > Various axioms have their various issues. The most pertinent to this > > > discussion right now, it seems, is Peano's 5th. I don't disagree > > > with the axiom or with the concept of inductive/recursive proof, > > > > There is no such thing as "recursive proof" in this context. > If you don't see it, you're blind. If TO sees what is not there, he is, as we have long suspected, delusional. > > > > > but in order to eb careful that what we are doing is correct, we > > > need to keep in mind the original justifications for axioms when > > > applying them. > > > > Wrong. An axiom needs to stand on its, absolutely. If it requires > > additional considerations, it was ill-chosen. Fortunately, this does > > not appear to be the case with the 5th Peano axiom. > See, Daryl? Unquestioning acceptance without critical examination. Typical. See TO! Unquestioning rejection without critical examination. Typical! |