Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: Virgil on 26 Jul 2005 15:29 In article <MPG.1d50146f85fc1fb3989f87(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Right. The Peano axioms don't fully address this issue, hence the > confusion. However, you cannot argue that any particular S^L can be > infinite with finite S and L, can you? Which S and which L are the largest possible ones such that one cannot have more than S^L finite natural numbers? Or are larger finite S and larger finite L ALWAYS possible, so that there is no finite limit on =the size of S^L? TO wants to put a lid on the allowable size for "finite" naturals, but will not say how that lid is to be determined. We, on the other hand, say that for every Peano natural (i.e., finite natural) there is a larger Peano natural obtainable by adding 1. Thus there is no end to our sequence of Peano naturals.
From: Daryl McCullough on 26 Jul 2005 15:13 Tony Orlow (aeo6) says: >Except for the fact that somehow you got an infinite set in >a finite number of steps, producing 1 element at a time. >How does that work? No, if you produce one element at a time, then there is never a time in which you will have produced an infinite set. Once again, you're having quantifier problems. Let enum(e,s,t) mean "enumeration e produces element s at or before step t". Then 1. S is finite <-> exists enumeration e, exists step t, forall s in S, enum(e,s,t) 2. S is countable <-> exists enumeration e, forall s in S, exists step t, enum(e,s,t) Note the difference between 1. and 2. The difference is in the order of quantifiers. That difference is important. The definition of finite says that there is some *maximum* number of steps t, independent of the element s, at which you know that you will have enumerated s. The definition of countable says that for each s, there is a corresponding t (depending on s) at which you will have enumerated s. -- Daryl McCullough Ithaca, NY
From: Virgil on 26 Jul 2005 15:30 In article <MPG.1d50159c1a3d29a2989f88(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Han de Bruijn said: > > Tony Orlow (aeo6) wrote: > > > > > In my book, there are 2^N > > > log2's of natural numbers. That doesn't make it uncountable. It just > > > makes it a > > > bigger set. > > > > Please tell us what the title of your book is. > > > > Han de Bruijn > > > > > Sorry, Han. That's an English colloquialism. It's as if everyone has a book > they are writing, or taking notes in, in their head. I simply meant, the way > I > see it, it is perfectly easy to enumerate the members of a powerset of a set > that is countable. > > But, I think I will title my book: "0 1 oo: The Square Circle". ;) Written by the Square Head?
From: Virgil on 26 Jul 2005 15:32 In article <MPG.1d5015c77abb6b38989f89(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > > > Tony Orlow (aeo6) wrote: > > > > > >> In my book, there are 2^N log2's of natural numbers. That doesn't > > >> make it uncountable. It just makes it a bigger set. > > > > > > Please tell us what the title of your book is. > > > > "Increasing your member size beyond natural". > > > > Just guessing. > > > > > I don't need help with my member size, thanks. ;) Perhaps it is using all the blood that should be going to his brain. That would explain a lot.
From: Chris Menzel on 26 Jul 2005 15:15
On Tue, 26 Jul 2005 13:12:33 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > Han de Bruijn said: >> Peter Webb wrote: >> >> > You seem to think that somehow mathematics is a physical science, >> > and the axioms are like physical laws, which can be true or false. >> > You think that you can observe that zero does not have a suucessor >> > just as you observe that every action has an equal and opposite >> > reaction. >> >> That's true. But mathematical axioms start to behave _as if_ they >> were physical laws, as soon as they become being _applied_ to i.e. >> physics. >> >> > The axioms of set theory (ZFC) make a really, really interesting >> > game. >> >> Set theory doesn't deserve such a predominant place in mathematics. >> After the discovery of Russell's paradox et all, everybody should >> have become most reluctant. >> >> Han de Bruijn >> > Not to mention the Banach-Tarski spheres. Doesn't that derivation > constitute a disproof by contradiction? Isn't the result absolutely > nonsensical? And yet, it is accepted, somehow, as truth that one can > chop a ball into a finite number of pieces and reassemble them into > two solid balls, each the same size as the original, despite all > evidence and logic to the contrary. It's a clear sign of something > wrong in the system, when it produces results like that. The T-B "paradox" arises from thinking that (finite, noncontinuous) physical objects in physical space should have all the mathematical properties of (uncountably large, infinitely dense, and continuous) mathematical objects in abstract Euclidean space. There are *enough* structural similarities between physical and mathematical objects that mathematics turns out often to be very useful in characterizing the behavior of physical objects. But to suppose mathematical and physical objects are structurally identical in every respect -- even those that rely on infinite size and complexity -- is spectacularly naive. But granted, it is so much easier to use one's Powerful (albeit completely uneducated) Intuitions as the guide to Truth than to have to learn all about all those obviously silly mathematical notions like uncountability, continuity, and measurability. Chris Menzel |