An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 7 Sep 2006 16:42 In article <4500215f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <44fd9eba(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Dik T. Winter wrote: > >>>>> In article <44ef3da9(a)news2.lightlink.com> Tony Orlow > >>>>> <tony(a)lightlink.com> > >>>>> writes: > >>>>> Your axiom uses things that are not defined. What is the *meaning* of > >>>>> "x<z"? > >>>> Geometrically it means that x is left of z on the number line. > >>> And for someone standing on the other side of the number line would x be > >>> on the right of z? > >>> > >>> And does the line stay horizontal as one moves around earth? Which way > >>> is larger if the line ever goes vertical. And how does the "larger" work > >>> at antipodes? > >>> > >> Silly questions. > >> > >>> > >>>> It means > >>>> A y (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y). That says about all it > >>>> needs to, wouldn't you say? > >>> Not hardly. > >>> A y (y = x) v (y = z) v (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y) > >>> is a bit better but still insufficient. > >> True, I should have specified y<>x and y<>z. I guess it's usually done > >> using <= for this reason, eh? > >> > >>>>> > > That is not a definition, because it makes no sense. "The set of > >>>>> > > naturals > >>>>> > > is as large as every natural"? > >>>>> > > >>>>> > It is not larger than all naturals > >>>>> > >>>>> That is something completely different again. > >>>> It's not LARGER than every finite. > >>> Which natural(s) is it "not larger" than", in the sense of not being a > >>> proper superset of that natural or having that natural as a member? > >> ....11111 binary (all bit positions finite) > > > > That isn't a natural number, Tony. > > > > Are you sure? Pay close attention. Let S be the set of all natural number representable as a finite string of binary digits. Is the first natural a member of S? Yes! If any natural is in S is its successor in S? Yes! Does induction require that every natural be a member of S? Yes! So that in ZF, ZFC or NBG, does any natural require an infinite string of binary digits? NO! Is there any extant axiom system in which any natural requires more that a finite number of binary digits for its representation? NO! Is TO wrong to claim otherwise? YES!
From: Virgil on 7 Sep 2006 16:53 In article <45002415(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Which axiom of ZFC set theory do you find objectionable? > > I object to the claim that the Axiom of infinity defines an infinite > set The AoI defines sets with certain properties, we chose to define "infinite" of sets in a way which applies to those sets. If TO thinks that the word "infinite" has some meaning which cannot be overridden, he is not very familiar with mathematical use of common words in technical senses. "Group", "ring", "field", Associative", commutative", "distributive", among many others. Once a mathematical definition is made in some mathematical context, that meaning supercedes all other meanings that the word might have in other contexts so long as one remains within that context. Mathematics has worked that way since long before TO was a twinkle in his parents eyes, and will be that way long after TO is dust. So if TO is upset by this, he should avoid mathematics entirely.
From: Virgil on 7 Sep 2006 17:31 In article <4500259b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Dik T. Winter wrote: > >>> In article <44fdcaf1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > >>> writes: > >>> > Dik T. Winter wrote: > >>> > > In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com> > >>> > > mueckenh(a)rz.fh-augsburg.de writes: > >>> ... > >>> > > > A number has only one of the following properties: It is larger > >>> > > > than or > >>> > > > smaller than or equal to any natural number. > >>> > > > >>> > > So omega and aleph-0 are numbers. They satisfy the definition. > >>> > > Thanks. > >>> > > >>> > So, they are numbers which are larger than any finite number? Why then > >>> > do we not consider an inductive proof of the form E y e N A x>y P(x) > >>> > not > >>> > to prove P(aleph_0) or P(omega)? > >>> > >>> That would be a new axiom, and you may consider it, but it leads to > >>> contradictions when you retain the current definitions of aleph_0 and > >>> omega. > >> Yes, I am well aware of that. That's why I have chosen to reject those > >> concepts in favor of finding something better, based on this > >> infinite-case induction. Noting personal. > >> > >> :) > > > > Better in what sense? What mathematics are you hoping to be able to do > > with your new foundation that cannot be done with ZFC? > > > > It accounts for changes of a single element between infinite sets and > therefore provides for a full spectrum of ordered infinite sets, thus > making the Continuum Hypothesis null and void. It relates the continuum > to the hypernaturals formulaically, and provides for an integration of > sets and measure in the infinite case. It rids us of anomalies like > omega-1=omega. > > If you remove an element, the proper subset should ALWAYS be smaller by > 1. If one orders by subset, a proper subset is always smaller than its superset, but proper subset ordering is not a total order, and cannot be extended to a total order on a set and its subest with more than one element in the "largest" set. > That is the case for me. For a theory to claim a proper subset is the > same "size" as the proper superset is an immediate deal-breaker for me. There is no total ordering on sets allowing that restriction, so TO is SOL.
From: Virgil on 7 Sep 2006 17:40 In article <450042de(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Yes, I was saying that there may be no such thing as a urelement, if > every object is considered to be the set of all of its attributes. That is self-contradictory if one is to have an empty set, as the empty set has attributes and thus cannot be empty. > However, what we have here are two different kinds of sets: sets of > objects, and sets of attributes. Sensible people get along with only one set theory, not two. > If a urobject, which has no object > elements, represents a set of attributes, then this indicates a need to > distinguish between sets of objects and sets of attributes. I suppose > this is the basis for type theory, which I'll try to read up on. But > Virgil says it's an anachronism. We'll see. That is not at all what the Russell theory of types was about. > > > > > So your passage above is yet another example of what people mean by > > your blathering. You just blather whatever mathematical terminology > > comes into your head, even if it makes no sense. > > Blah blah blah. AS much sense as TO ever makes. > > > > Because it is frustrating talking with you, as I prefer not to consider > > you stupid, but your willful ignorance (not just in refusing to a read > > a single book on the subject but also in ignoring so many crucial > > points made in posts made to you) and arrogance cause you to say so > > many stupid things. > > My arrogance is driven by conviction and success in formulating an > alternative. "Conviction" I take leave to doubt, but for TO to claim "success" is clearly false. > Agreed. {x | x is red} = {x | is_red() e x} > > That is, the property is not a member of the set of all things HAVING > that property, but it is a member of each of those things in the set > which HAS that property as a member of its deifnition. GIGO.
From: Virgil on 7 Sep 2006 17:51
In article <450046d9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > It's trivial given the axiom that there are Big'un reals in every unit > interval. Denseness is defined for any real interval givent he axiom of > internal infinity: x<z -> Ey x<y<z Given enough axioms anything and it negation are provable. Denseness already has a definition, but is not an axiom, as there are sets which are not dense. There are even sets which are nor ordered. > > > > > Assuming that you could prove such a thing, you then have > > to realize that such a system is incompatible with standard > > arithmetic and set theory. You see why, don't you? > > > > Yes, of course it's incompatible. That's because set theory ignores measure. Set theory is the foundation on which eventually a measure on the reals is based, but as usual, TO has cart-before-horse-itis. > > If all infinite values are greater than finite values, and I prove P(n) > for all n>m (finite m), then I have proven it fall all infinite n, since > they are greater than m. That would have to be claimed as an axiom, as it doe not follow from anything. And I strongly suspect that once it is claimed on top of the axioms of ZF, that one can prove that 2 - 1 and all sorts of curious things. > > What does "in the infinite case" mean? > > > > When n is an actually infinite value, such as the number of points in a > unit interval (Big'un). Not defined in ZF, and TO has no system in which it is defined. > > > > It would help if you could define what it means for n=oo in the > > first place. But be warned that since that has no meaning in > > standard arithmetic, whatever system you come up with in which > > it does have some kind of meaning is a system that cannot be > > compatible with standard arithmetic. > > > > Yeah, I know. I've known that for 25 years. That doesn't stop me from > considering alternatives to transfinitology any more than being > surrounded by McDonald's' and Pizza Huts prevents me from cooking real food. But getting anybody sane to taste your food might be a bit harder. |