Galileo's Paradox
in [Math]
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From: MoeBlee on 8 Dec 2006 16:46 Tony Orlow wrote: > You might want to look into Internal Set Theory, a partial > axiomatization of Nonstandard Analysis. Why do you say 'parital'? > Both infinitesimal and infinite > values are "nonstandard", and no reference to "standard" values is > allowed in the definition of any set. Not ANY set. IST includes standard sets too. You do realize that IST is an EXTENSION of ZF, right? MoeBlee
From: Virgil on 8 Dec 2006 18:57 In article <4579d1c7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > When there is a balance between one assumption and another, we are in a > position to justify one or the other. That's logic.... The same way assumption of Santa Claus balances assumption of a tooth fairy? That's TO's logic! > > There are some VERY simple definitions in set theory (either naive or > > ZF, AC or not), and some of them are REALLY as simple as one can > > expect: a set is called "infinite" if there exists a bijection (which > > already has been completely and fully WELL defined) between that set > > and at least one of its proper subsets. Period. That is all there is to > > it. > > Cardinality, yes, is simplistic - no argument. Very simplistic... Math is simple (not always easy, but definitely simple). TO is simplistic. > > > You don't wanna accept this definition? Good, propose > > yours..."potencial infinity", "actual infinity", shminfinity: give us > > DEFINITIONS, axioms to work with...and let's hope that upon checking > > and re-checking, those axioms and definitions aren't shown to be > > inconsistent, which has NOT been proved for ZF, AC or not AC....and > > that they are sufficiently interesting to deal with, of course. > > Okay, a "potential" infinite set is one where each element, like the > naturals, has a specific string associated with it, which has a > left-hand end. If those strings do not have two ends, you have either an uncountably potentiality or a ppotential uncountability. > > After some time interchanging posts, some of these trolls/crankis begin > > to REALLY believe that they have proved inconsistencies, > > contradictions, etc. Just read some of Eckie's posts to see what a diet > > low in potassium can do to human brain. > > No one here claims any such thing. Eckie does! WM does! > One can only claim that certain > logical constructions involved are invalid. In matters mathematical, claims without proofs usually do not count for much. They are called conjectures. > > There is nothing wrong with expecting science to satisfy intuition. In the way that general relativity and quantum field theory do? > > > And one last question from me to you: what do you think of my remark, > > some 5-6 days ago, that as far as I know, NONE of the megacranks is a > > mathematician? Don't you wonder about this? I don't doubt there are > > mathematicians that don't like this ir that part in math, but I bet > > they won't troll about it as you people do, and that's a huge > > difference. > > Tonio > > > > > > > > I chose to work within computer science, after having planned to become > a mathematician, for the obvious reasons.... Couldn't cut it as a mathematician?
From: Virgil on 8 Dec 2006 18:58 In article <4579d233(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > Six wrote: > >> On 6 Dec 2006 07:08:46 -0800, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: > > > >>> Six wrote: > >>>> I am very grateful to you for expanding on this. While I'm almost > >>>> certain I'm missing something, I'm afraid I still don't get it. > >>>> > >>>> How exactly does claiming that a 1:1 C is not necessarily > >>>> indicative of equality of size with infinite sets presuppose an > >>>> inability > >>>> to map (eg) the binary and decimal representations of integers? > >>>> > >>>> There is still a 1:1 C between the two sets. It is still true that > >>>> for any finite sets a 1:1C implies equality of size. Moreover it's still > >>>> reasonable to suppose that a 1:1C implies equality of size in the > >>>> infinite > >>>> case unless there are other, 'functional' reasons to the contrary. > >>>> (Vague, > >>>> I know. Roughly, 1:1 C is a necessary but not sufficient condition for > >>>> equality of size.) > >>>> > >>>> The idea is that the naturals (in any base) form a paradigm or > >>>> norm, a standard against which other sets can be measured. > >>> The set of finite binary strings is a subset of the set of finite > >>> decimal strings. > > > >> I confess I hadn't fully appreciated this simple point, that > >> together with the fact that the strings just are, so to speak, the natural > >> numbers (in a given base). > > > >>> Then b) precludes them being the same size. > >>> > >>> They are also both the same size as the set of natural numbers. > >>> > >>> Thus they are the same size as each other. > >>> > >>> Contradiction. > > > >> One is driven to the conclusion that there is no base-independent > >> size for the natural numbers. > > > > How can the size be base dependent? The natural numbers are not base > > dependent. > > Any natural number can be expressed in any base. There is no natural > > number > > expressible in base 16 that is not expressible in base 10, or base 9, or > > base 2. > > > > I suppose you could claim that there is a set of decimal numbers, and a set > > of base 2 numbers, and a set of hexadecimal numbers, and that they are all > > different, and all have different sizes. But it is a strange notion of > > "different size" given that all the sets represent the same thing. > > > > Stephen > > Measure is something different from the language needed to express it. Which does not exculpate TO from the wrong he has so recently promulgated.
From: MoeBlee on 8 Dec 2006 19:29 Tony Orlow wrote: > You might want to look into Internal Set Theory, a partial > axiomatization of Nonstandard Analysis. Why do you say 'parital'? > Both infinitesimal and infinite > values are "nonstandard", and no reference to "standard" values is > allowed in the definition of any set. Not ANY set. IST includes standard sets too. You do realize that IST is an EXTENSION of ZF, right? MoeBlee
From: cbrown on 8 Dec 2006 21:56
Tony Orlow wrote: > What,in mathematics, has a solution which is neither a real measure, or > the measure of truth of a statement, 0, 1, or somewhere in between? "Find all pairs of distinct naturals (x,y) such that x^y = y^x". The solution to which is the set {(2,4), (4,2)}, which doesn't appear to be a "real measure", nor a "measure of the truth of a statement" (as far as I can understand your meaning of the terms). I assume that you offer more than the trivial observation that all mathematical statements P, including the statement "2^4=4^2", are examples of the (boolean) truth valued statement "it can be proved that P". If so, I would claim that the mathematical question is "Find a proof of P, or proof of not P". And the solution is not "0" or "1"; the solution is either a proof of P, or a proof of not P. "Find all finite groups G having a maximal subgroup S, and having a subgroup T which is isomorphic to S but not maximal". This question was asked in sci.math a few days ago. The groups in question are not "numbers" at all; and a set of them possesses no particularly natural total orderings. They can be partially ordered by "size" (number of elements); but there are, in general, mutliple distinct groups on a set of any given size. You might look at the following threads currently active in sci.math within the last 24 hours (believe it or not, not everyone posts to argue about Cantor/infinity): "some advance algebra q's. please help" "Group on arbitrary ordinal" "Another two universal algebra questions" "? how subspaces are like" "large ordinals, help!" "Continuous injection from a subset of R^n to R" "Smith normal form in a binary field (F2), symbolic" "Infinite width Moebius strip" "notation of fields and rings" "(Universal Algebra) The function determined by a term..." "Intermediate Fields in Galois Theory" "Mapping of integer functions into reals" "Sets of quaternions with no repeats in infinite set of products:" "please help me explain: Counter-Example for Riesz representation Theorem" "how to deduce the algebra structure" "Infinite simple groups and their proper subgroups" "Right cosets intersection" "Valuations in field extensions" "Groups and commutators" "Compactness" .... and so on. There are many mathematical questions of the forms: "does there exist X such that P(X)?", "characterize all X such that P(X)", and "find an X such that P(X)" that do not use a "real measure" to describe X, and are not about "measuring" things with "numbers" in the sense I'm guessing you mean. In particular on sci.math, there's topology, number theory, abstract algebra, linear algebra, and graph theory. These areas exist because of a combination of two factors: they are sometimes useful, and sometimes interesting in their own right to its practitioners. As far as /I'm/ concerened, these are the only actually /interesting/ parts of mathematics. Face it - Calculus is boring! I don't like adding up columns of numbers, so I have a calculator. Nor do I like plodding through many certainly quite well-known transformations; so I look it up in a table if I need it. On the other hand, given my interests, I don't need it that often; so I also don't personally find it very "useful" :). Cheers - Chas |