Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 26 Jul 2005 17:50 Tony Orlow (aeo6) wrote: > In any case, sure, the program will spit out finite numbers, snce it is a > finite machine running in finite time. If the machine had infinite capacity and > infinite funtime, it could conceivably produce infinite results. In order to finish generating the natural numbers in 1 second, it would need to spit out natural number n with a calculation time of 1/2^n. I believe George first noted this fact in a post to Daryl a few years ago. Unfortunately, I've never found a way to configure my java-virtual-machine to accelerate with the number of steps taken, so I've not found much use for his observation. karl m
From: Virgil on 26 Jul 2005 17:56 In article <MPG.1d5069ac3785c055989f9d(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > The big problem in transfinite cardinality is the assumption that a bijection > necessarily indicates equal sizes for infinite sets, as it does for finite > sets. One *defines* a measure on set sizes via injection/bijection and sees where it leads. That TO does not like that neighborhood does not constitute a valid objection. > When the only way to form a bijection is with a mapping function, then > that function needs to be taken into account. AS there are a huge number of possible injections or bijections for all but the smalest of sets, which bijections must be taken into account, and how? > This nonsense about an infinite set of finite whole numbers is pretty > bad too, but probably without any real consequences. The consequences of of requiring infinite naturals in ZFC would be that every statement can be proved both true and false.
From: Virgil on 26 Jul 2005 18:09 In article <MPG.1d506ab0de98a030989f9e(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Either you have an upper bound or you do not, and if there is no > upper bound on the values of the members, then they may be infinite. Or they may not. > how do you have an infinite set of strings with only finite lengths? The usual way, by not having any finite bound on their lengths. > > > > >I am not assuming anything except for this fact. > > > > You are assuming that every set of strings has a natural number L > > such that every string has length L or less. That's false. > I am saying that if L CANNOT be infinite, then S^L CANNOT be > infinite No one is requiring any S^L to be infinite, we merely deny that there is any finite lid on the size the S^L can attain, as S and L are allowed to increase without finite limit. > For finite S, > S^L can ONLY be infinite with infinite L. Why is this so hard to > understand? If S and L are both finite, then S^L is finite, isn't it? But no one is claiming that any of S, L or S^L is infinite, we just say that any of them can be larger that any finite natural you choose to name. that is, there is no finite limit on their sizes. Why TO rails against something that no one is claiming is because he has no valid argument by which to refute what we actually are claiming. That form of argument is called the fallacy of the STRAW MAN. When TO has no counter to an actual argument, he pretends that there was a different argument, which he then attacks. TO has been doing a lot of STRAW MAN arguing recently, to no avail.
From: Daryl McCullough on 26 Jul 2005 17:50 Tony Orlow (aeo6) wrote: >> (3) The answer to "Is there a largest pofnat?" is somehow neither >>'Yes' nor 'No'. >No, the answer is no. But you claimed that the set of all finite naturals is a finite set. Every finite set of naturals has a largest element. -- Daryl McCullough Ithaca, NY
From: stevendaryl301 on 26 Jul 2005 17:50
In article <MPG.1d506b92f3f911f9989f9f(a)newsstand.cit.cornell.edu>, aeo6 says... > >Chris Menzel said: >> On Mon, 25 Jul 2005 11:27:28 -0400, Tony Orlow <aeo6(a)cornell.edu> said: >> > ...You cannot form an infinite number of strings with a finite >> > alphabet, without strings of infinite length. >> >> Oh jeez, you're as clueless about formal languages as you are about set >> theory, not that this is a surprise. Do you know that what you are >> saying here is contradicted in the early chapters of any book on formal >> languages? Really! Go have a look! Here, let me save you some >> trouble: On page 1 of their famous standard text *Introduction to >> Automata Theory, Languages and Computation*, Hopcroft and Ullman define >> a "string" to be *finite* sequence of symbols. (Note: they don't think >> the notion of string can't be generalized to the infinite, it's just >> that in their text they are only interested in the finite ones.) Then, >> on the very next page, they point out that "The set of palindromes >> (strings that read the same forward and backward) over the alphabet >> {0,1} is an infinite language." >yes, I have looked around, and seen the damage done by this madness. When I was >studying that, I shrugged at that nonsense. It's really not important when you >are working with finite computers. It doesn't have any real-world implications, >or at least not regularly. It's still incorrect, and it's still amazing that >this sloppiness is tolerated. >> >> See? Do you realize what a fool you are making of yourself by making >> assertions about set theory, transfinite arithmetic, formal languages, >> computability, etc. that show you don't understand even the most >> elementary concepts, or grasp the most elementary theorems, of set >> theory, transfinite arithmetic, formal languages, computability, etc? >You can call me a fool if it makes you feel better. You might want to read some >Shakespeare and understand what the fool is. >> >> C'mon, stop embarrassing yourself. Go *learn* some mathematics before >> you start spouting off about it. It's painful to watch. >You are enjoying it, more than you should. >> >> Chris Menzel >> >> > |