Galileo's Paradox
in [Math]
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From: Six on 11 Dec 2006 07:09 On Thu, 7 Dec 2006 17:25:40 +0000 (UTC), 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. I agree it would be a strange notion of size. My views have shifted, but I'm afraid you might still find them strange. I do not think we can say those sets are different sizes, but I do not think we can say they are the same size either. Thanks, Six Letters
From: Six on 11 Dec 2006 07:16 On 8 Dec 2006 06:37:10 -0800, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: > >Six wrote: >> On 6 Dec 2006 07:08:46 -0800, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: ><snip> >> >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). > >I'd say the strings are a representation of the set of natural numbers. OK >I'd certainly expect the sets of strings and the set of natural numbers >to be the same size. > >> >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. > >One is driven to the conclusion that EITHER a proper subset can be the >same size as its superset OR that there is no base-independent size for >the natural numbers. Agreed. But following the original argument the latter is what one would be driven to. My view has changed somewhat now. >Frankly the latter makes no sense whatsoever to me for any reasonable >interpretation of "the natural numbers". > >> This does not make the discussion of the relative size of, for >> example, natural numbers and squares meaningless. It's just that a given >> base would have to be understood, and that whatever is said about the >> relative size of the two sets is understood to apply mutatis mutandis to >> any other base. > >So you really believe "the set of natural numbers represented in base >10" is a different set from "the set of natural numbers represented in >base 2"? Hmm. I believe there is a problem with maintaining that they have the same number of members, even though they both represent N. >>But for me at least, it has certainly opened my eyes to the >> implications of the original argument. >> >> Much appreciated, >> >> Six Letters > >Well... you're welcome, I guess. Really, thanks. Six Letters
From: Eckard Blumschein on 11 Dec 2006 07:34 On 12/6/2006 9:08 PM, Virgil wrote: > In article <4576F7E0.3090102(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > And oo is NaN, 2^oo has no meaning. >> >> You are a knowing-all. > > oo is, at best, ambiguous. In what respect? >> >> >> In case of an infinite set, there are not all elements available. >> >> > >> >> > "Available"? >> >> >> >> Yes. You cannot apply the algorithm until you have all numbers. >> > >> > What "algorithm? >> >> For calculating binominal coefficients. > > What does one need that for in accepting the axiom of the power set? Is there evidence for the reals as characterised by DA2 to actually fit ZFC? >> >> >> Fictions are uncountable. >> >> > >> >> > "Uncountable"? > >> > >> > What do you mean by the "uncountability" of one element? How can one >> > element sometimes be countable and sometimes not? >> >> Your question is justified. In order to be part of a counted or at lest >> countable plurality, each element has to be discrete and addressable. So >> the number 1 may belong to a finite set as well as to a cuntable >> infinite set. However 1.000... with perfectly oo much of significant >> nils is not immediately a countable element. > > If it can exist in isolation, as you seem to be assuming by referring to > it, then it can be counted in isolation as one object. This is Bolzano's religious thinking. Pi is not isolated within e.g. the system of decimal numbers but merely like a problem. >> Fortunately, I was never trained in set-quasi-religion. > > No mathematicians have been either, but mathematicians also have not > been trained in your anti-set-actual-religion either. Hilbert: "axiomatic method maintains the advantages of belief in ..."
From: Eckard Blumschein on 11 Dec 2006 07:49 On 12/6/2006 5:48 AM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/4/2006 9:49 PM, David Marcus wrote: >> >> >> According to my reasoning, the power set is based on all elements of a >> >> set. >> > >> > "Based"? >> >> Yes. The power set algorithm does not change what mathematicians still >> used to call cardinality. 2^oo=oo. > > "still used to"? What does that mean? Nobody needs aleph_2 or even higher alephs. Countably infinite replaces aleph_0. Uncountable or continuous replaces aleph_1. > > You still haven't explained what you meant by "based". I refer to a proof by Cantor showing that 2^aleph_0 > aleph_0. >> >> In case of an infinite set, there are nor all elements available. >> > >> > "Available"? >> >> Yes. You cannot apply the algorithm until you have all numbers. > > So, "available" means "have". What does "have" mean? Having a complete numerical address to refer to. >> >> Nonetheless I can do so as if they would exist, and I am calling them a >> >> fiction. >> > >> > "Exist"? "Fiction"? >> >> Exist means, they have their numerical address within a rational order. > > Define "numerical address" and "rational order", please. The words are self-explaining. >> Fiction means, they don't have it but it is reasonable to do so as if. > > So, "fiction" means "not exist, but reasonable to do so as if". What > does "reasonable to do so as if" mean? Exactly what e.g. Leibniz or Vaihinger or to some extent Robinson understood. > >> >> Fictions are uncountable. >> > >> > "Uncountable"? >> >> The continuum cannot really be resolved into countable elements. It can >> just be thought to consist of an actually infinite amount of fictitious >> elements. > > Doesn't seem to answer my question. Let's try again: What does > "uncountable" mean? I do not restrict my knowledge to what you learned. Uncountable is definitely not a property of numbers. Numbers are always countable. Nonetheless a single real "number" is uncountable. >> >> So this power set has no chance but >> >> to be also uncountable. >> >> Try to get the title cardinal Kolker, or at least Bob the Builder and I >> >> will possibly convert. >> > >> > When you tire of religion, you could always learn some math. >> >> My topic is not relegion but outdated quasi-religious mathematics. > > You are the one reading old papers by Cantor instead of reading modern > mathematics texts. Why do you think that what you read has any relevance > to modern, current, up-to-date mathematics? Some basic errors were not yet eradicated.
From: Eckard Blumschein on 11 Dec 2006 08:05
On 12/6/2006 5:17 PM, Tez wrote: > Eckard Blumschein wrote: >> On 12/4/2006 9:49 PM, David Marcus wrote: >> >> >> According to my reasoning, the power set is based on all elements of a >> >> set. >> > >> > "Based"? >> >> Yes. The power set algorithm does not change what mathematicians still >> used to call cardinality. 2^oo=oo. > > "The" power set "algorithm"? Could you write out "the" algorithm here > for us so that you and I can flesh out your claims in detail? If you have N elements, you will get 2^oo combinations (cf. Pasqual's triangle) >> >> In case of an infinite set, there are nor all elements available. >> > >> > "Available"? >> >> Yes. You cannot apply the algorithm until you have all numbers. > > You seem to be looking at this from a computational (but more probably, > a programatic) perspective. No. From that perspective, you are wrong. It > is possible to apply algorithms to infinite data structures. See: > http://en.wikipedia.org/wiki/Haskell_programming_language > http://en.wikipedia.org/wiki/Lazy_evaluation > http://cs.wwc.edu/KU/PR/Haskell.html > > In that last link, look at the section headed "Lazy Evaluation and > Infinite Lists" and examine, for example, the implementation of the > Sieve of Eratosthenes. One can delay operations but not shift them into future. >> >> Nonetheless I can do so as if they would exist, and I am calling them a >> >> fiction. >> > >> > "Exist"? "Fiction"? >> >> Exist means, they have their numerical address within a rational order. >> Fiction means, they don't have it but it is reasonable to do so as if. > > "Address"? Rational "order"? If we're going to play with words, I'm > going to ask: Why don't entities with numerical addresses with an > irrational order "exist"? In mathematics existence means common properties. What about the reals, we have to distinguish genuine and putative reals. Genuine reals are those which were assumed for DA2 and are indeed uncountable since they have an actual infinite number of decimals. Putative reals are defined by Dedekind's cut, nested intervals, Cauchy limits if the tiny word fictitious is missing and the like. |