Obections to Cantor's Theory (Wikipedia article)
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From: Dik T. Winter on 21 Jul 2005 23:01 I mist this gem, so I am going piggyback... > On Thu, 21 Jul 2005 12:21:38 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >stephen(a)nomail.com wrote: > >> Somehow they > >> think an all-encompassing definition of 'infinite' must > >> be provided before someone can say what an infinite set is. > >> I am not sure what they mental hangup is. I wonder > >> how any of them would ever learn a foreign language. > > > >Ha, ha, ha. _This_ anti-Cantorian has learned six languages: Dutch, > >German, French, English, Latin and Greek. We in the Netherlands are > >privileged with our knowledge of foreign languages. Yeah, right. Try to ask directions anywhere on the streets in French and you will draw a blank face (unless you find someone who came from Morocco). Pray step off your high horse. Did you ever hear one of the people from government talk in English? Did you not cringe? > > Yet I find that > >an "all-encompassing definition of 'infinite' must be provided". Why? Do you have an "all-encompassing definition" of the Dutch word "pas"? Offhand I know two meanings, and neither fits the use in the term "bankpas". In mathematics (as in real life) the meaning of a word depends on context. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Poker Joker on 21 Jul 2005 23:56 "Han de Bruijn" <Han.deBruijn(a)DTO.TUDelft.NL> wrote in message news:d4018$42df50e7$82a1e3ad$10536(a)news1.tudelft.nl... >> Nothing of any importance about mathematics would change >> if we substituted different words for the basic concepts. Yes it would. Mathematicians would stop looking so foolish when they say things like: "All natural numbers are finite but there are an infinite number of them." Not only do they seem foolish when they say such things, they also create enough confusion as to create a whole group of people who are known as anti-Cantorians. Furthermore, they appear to be quite ignorant about the problem with the above quote.
From: Virgil on 22 Jul 2005 00:11 In article <Q7_De.154$8g.85(a)tornado.rdc-kc.rr.com>, "Poker Joker" <Poker(a)wi.rr.com> wrote: > "Han de Bruijn" <Han.deBruijn(a)DTO.TUDelft.NL> wrote in message > news:d4018$42df50e7$82a1e3ad$10536(a)news1.tudelft.nl... > > >> Nothing of any importance about mathematics would change > >> if we substituted different words for the basic concepts. > > Yes it would. > > Mathematicians would stop looking so foolish when they say > things like: > > "All natural numbers are finite but there are an infinite number > of them." > > Not only do they seem foolish when they say such things, they > also create enough confusion as to create a whole group of > people who are known as anti-Cantorians. Furthermore, they > appear to be quite ignorant about the problem with the above > quote. All naturals are finite, and since adding 1 to any natural produces a larger natural, there is no end to them. Some people seem think that called this non-ending "non-finite" is non-ignorant.
From: Jesse F. Hughes on 22 Jul 2005 00:22 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > malbrain(a)yahoo.com said: >> Main Entry: fi=B7nite >> Pronunciation: 'fI-"nIt >> Function: adjective >> Etymology: Middle English finit, from Latin finitus, past participle of >> finire >> 1 a : having definite or definable limits <finite number of >> possibilities> b : having a limited nature or existence <finite beings> >> >> This definition from webster should suffice. Binary numbers with ones >> in finite positions have a limited number of possibilities. >> >> karl m >> >> > Thank you Karl. So it means "having a limit", which is pretty much an end. I > have no problem with that definition. It seems that your definition (and also the dictionary definition above) conflate two distinct mathematical notions: finiteness and having a maximal element (or the closely related notion of boundedness). I will give an example, but I don't expect that Tony will get it. Suppose I take the set of natural numbers and adjoin a new element, let's call it "top". Suppose I extend the ordering so that for all n in N, we have n < top. Then my new set looks something like this: 0 < 1 < 2 < .... < n < n+1 < .... < top. The first ellipsis has the numbers between 2 and n while the second ellipsis has *every* natural number greater than n+1 [1]. This is a perfectly well-defined ordered set (in fact, well-ordered). It even has a name: omega + 1. It clearly has a maximal element. Now, Tony is apparently incapable of defining "having an end" in purely mathematical terms, so we are left to guess what he means, but my best guess is "having a maximal element". However, this would give an example of a finite set with an infinite subset. Surely, Tony doesn't want this. But then what the heck *does* "having an end" or "having a limit" mean? Footnotes: [1] If there actually were infinite natural numbers, then they are included in the second ellipsis. -- Jesse F. Hughes "Time and again, history has shown that people who think their beliefs trump reality lose, and lose badly. Luckily, I don't have to listen to you." -- James Harris on reality avoidance
From: Helene.Boucher on 22 Jul 2005 00:46
Virgil wrote: Good a proof reader! > In article <1122004787.370831.6550(a)o13g2000cwo.googlegroups.com>, > Helene.Boucher(a)wanadoo.fr wrote: > > (x)(Sy = x*y + x) > > Shouldn't the above read '(x)(x*Sy = x*y + x)' ? Yes! > > > (x)(!x = 0 => (there exists y)(Sy = x) > > Shouldn't the above read '(x)(x != 0 => (there exists y)(Sy = x)'? Unless I'm missing something, "! x = 0" is the same as "x != 0"... > > > > The last axiom can be proven by induction, so is not included in the > > axioms of PA (at least when PA is defined straightaway, and not from > > Q). For instance, here's a standard defintion of PA (Mendelson), with > > S a total function: > > (x)(y)(Sx = Sy) > > (x)(!Sx = 0) > > (x)(xy)(Sx = Sy => x = y) > > Shouldn't the above read '(x)(y)(Sx = Sy => x = y)' ? Yes! > > (x)(y)(x = Sy => S(x + y)) > > Shouldn't the above read '(x)(y)(x + Sy => S(x + y))' ? Yes! Thank you! I don't know what time it is where you are, but it was 5am in the morning when I wrote that! |