An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 20 Sep 2006 17:11 In article <4511532f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Han de Bruijn wrote: > >> Virgil wrote: > >> > >>> In article <1158489723.269348.27860(a)e3g2000cwe.googlegroups.com>, > >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> > >>>> What's wrong with mathematics ?! > >>> Nothing!! > >> "Mathematics should be a science" is the answer. > > > > Why? > > > > Because we are discovering truths, not making up games, in real math. TO misuses "WE", as he is not discovering anything at all. And to those, like TO, who claim mathematics should be bsed on absolute truth independent of any unproven axioms, that is one of the axioms of their system which they must presume without proof.
From: Virgil on 20 Sep 2006 17:19 In article <451154d5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450d5597(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <450c87cc(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <450c71a1(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>> Aatu Koskensilta wrote: > >>>>>> Given the axioms and rules of inference, the conclusions are provably > >>>>>> true or false. > >>>>>> > >>>>>> Soundness is another issue, regarding the fundamental justification > >>>>>> for > >>>>>> the logical axioms themselves, and whether they are "correct", meaning > >>>>>> "objectively verifiable". > >>>>> If axioms were ever objectively verifiable they would not need to be > >>>>> assumed in the first place, but would be objectively verified. > >>>>> > >>>> In the mathematical world, the greater framework can be considered > >>>> relatively objective. > >>> Greater than what? If one wnats something in one's system, either it is > >>> provable in terms of other things in the system or it must be assumed > >>> without being provable in terms of other things in the sysem, and just > >>> like with having to have undefined terms, at some point you have to have > >>> unproven assumptions. > >>> > >>> In mathematics, when you get to that point, you call those unproven > >>> assumptions axioms. > >>> > >>> TO seems to want to do without any axioms by some sort of daisy chain > >>> circle of proofs lifting the whole mess up by its bootstraps. > >> How on Earth do you read all that from what I said. The "greater > >> framework" is mathematics in general. If a particular axiom or theory > >> contradicts enough other math, then it's trouble. There's no reason that > >> all of mathematics can't be consistent. That's the greater framework. > > > > The axioms system of Euclidean geometry is inconsistent with that of > > various non-Euclidean geometries. and there are a lot of other places > > where one system contradicts another. > > What happened is that the parallel postulate was downgraded to a > postulate, rather than a law. What happened was that it became clear that everything in mathematics, however "obvious" it seemed, was ultimately based on statements which ultimately had to be assumed without proof. Thus the beginning of axiomatics. > > > > What mathematics allows is any system of axioms which does not appear to > > contain any self-contradictions, at least for as long as it maintains > > that appearance. > > If it claims to be a generalization, the it should not contradict those > particulars which it generalizes. > > > > >> I understand that axioms are necessary, but they should not be arbitrary. > > > > Which axioms in which mathematical systems does TO think have not been > > judicially chosen? > > The axiom of choice, for one. That's optional even in your system. AoC has been discussed at great length, and it has even been shown that if ZF is consistent without it, or even including the direct negation of the choice, then ZFC is also consistent. So it appears that one can choose either choice or no-choice or non-choice, as one pleases.
From: Virgil on 20 Sep 2006 17:23 In article <4511550b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450d5757(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > > > >> Without the additional > >> structure which Aatu suggests, bijection may show some kind of > >> equivalence, but it cannot be considered any kind of exact analog for > >> the size of finite sets. You're trying to extract measure from something > >> with no measure in it, like blood from a stone. > > > > On the contrary, Cantor was trying to devise a measure which was > > entirely independent of every property of the members of each set other > > than their being distinguishable from each other. > > > > TO is measuring order relations, not sets. > > I am measuring sets USING order relations in a standard metric space. > Otherwise, there is no measure. In TO's notion of "measure", giving a set a different order relation can change its size. There is a measure on sets which ignores all order relations, and other relations on them. it is called cardinality. Under it, changing the ordering of a set does not change its size.
From: Virgil on 20 Sep 2006 17:25 In article <45115570$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450d5f76(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> Mike, you haven't responded to my use of IFR > > > > An IFR, being dependent on order relations, at best measures order > > relations, not their underlying sets. > > Funny how it DOES measure the sizes of sets perfectly in all finite cases. So does cardinality. By that test they are equally valid. But once one gets into the realm of infinite sets, only cardinality measures the size of sets independently of any order relation imposed on them.
From: Virgil on 20 Sep 2006 17:38
In article <45115f7e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > The axiom of choice, and the axiom of infinity is insufficient. It is > better to start with simpler axioms to build your theorems. With fewer axioms, one can get fewer theorems, so that less of what TO wants will be possible with fewer axioms. > > > Do you understand now that the sum of an infinite series need not > > exist? > > I understand that standard treatment does not assign a value to > divergent series, but I also know it is possible, with a unit infinity, > to represent many of them. As TO can't get a "unit infinity" into even ZF, he will find it even harder with fewer axioms. > > > Do you admit that your attempt at a proof that the naturals can be > > bijected with their powerset was totally bogus? > > No, but I admit that most of the naturals it maps to are not STANDARD > finite naturals. WRONG! Most of the alleged naturals it maps FROM are not natural naturals. What it maps TO is sets of natural naturals. > > > Do you understand that Cardinality doesn't claim to be the only or best > > analogy to "size" for finite sets? > > Mathematicians claim that. If not, then why so much resistance to > attempts to improve upon cardinality? Because those claimed "improvements" are actually detriments. Requiring every set to be tied to some specific ordering, for example, means that the same set with a different ordering imposed may have a different size even though it still has exactly the same members. > It seems someone's toes are > getting stepped on. When people use words like "same size", > "equinumerous", etc, that's what they are implying, and it's only when > backed into a corner that they admit there may be some difference. The difference is not in the sets but in the order relations imposed on them. |