Hilbert's Paradox and Cantor's Diagonal Proof



30082012, 07:17 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
I'm struggling here.
Why isn't bunging another number in at the dots here: { 0, 1, 2, 3, 4, ....} the same as bunging another number in at the dots here: { 0, ... 0.1, ... 0.2, ... 0.5, ... 1} ? (please be gentle with me, I haven't touched maths for 30 years!) 

30082012, 07:21 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29082012 10:15 PM)Starcrash Wrote:(29082012 08:36 PM)Chas Wrote: Cantor's Diagonal Method clearly proves that there are infinities of at least two sizes. There are more real numbers than rational numbers, for instance. If we're talking math, you need to be precise. I full well know what prove means, and Cantor's diagonal method does prove that the cardinality of the real numbers is greater than the cardinality of the natural numbers. You belief about 'using up' the members of an infinite set is meaningless in mathematics. Induction gets us all the way there. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

30082012, 07:23 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29082012 08:33 PM)Bucky Ball Wrote:(29082012 08:28 PM)Chas Wrote: The law of the excluded middle is foundational to the definition of binary logic. It states that a proposition is either true or its negation is true. There are logics that don't include it. See W. V. O. Quine. Have at at it, Bucky! I don't see how you'll get there, but I admire your willingness to take the journey. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

30082012, 07:26 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(29082012 02:33 PM)guitar_nut Wrote: So I've been reading a lot of arguments for and against infinity, specifically the two in the subject. A few theists, most famously Craig, have used similar arguments to show the universe has a definite beginning. Here's a reference if you're interested: dood, you had the same inquiry i did 3 decades back but i used it differently. I applied to the tangible, mathematically. I then left each variable open to time on either end "Before and after" more to define the process, versus the constants. It may seem confusing but i am objective by nature, so i used it. If you all want the scary realization (IMO): If the system (existence itself) is closed, then existence created itself, and we are defining it to do so. In a sense, we are 'it' defining itsef, for its beginning to exist. There's a minf duck. You want esoteric, i be your huckleberry! 

30082012, 07:26 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 07:17 AM)DLJ Wrote: I'm struggling here. The Wikipedia article on Cantor's Diagonal Method is pretty good. The basic idea is that we can use up all of the natural numbers {1, 2, 3, ...} as labels on real numbers, but when we're done, we have (an infinite number of) unlabeled real numbers left over, but we're all out of natural numbers. Clear? Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

30082012, 07:42 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 07:26 AM)Chas Wrote:(30082012 07:17 AM)DLJ Wrote: I'm struggling here. Nope. I have trouble with the "use up" bit and the "when we're done" bit. How can we use up an infinite number (of natural numbers)? 

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30082012, 07:45 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 07:26 AM)Bishadi Wrote:(29082012 02:33 PM)guitar_nut Wrote: So I've been reading a lot of arguments for and against infinity, specifically the two in the subject. A few theists, most famously Craig, have used similar arguments to show the universe has a definite beginning. Here's a reference if you're interested: This is what I think I'm sort of thinking about, in terms of Godel's incompleteness. "If the system (existence itself) is closed, then existence created itself, and we are defining it to do so. In a sense, we are 'it' defining itsef, for its beginning to exist." There's something very wrong about that. I'm not quite sure how to express it yet. Insufferable knowitall.
Those who were seen dancing were thought to be insane by those who could not hear the music  Friedrich Nietzsche 

30082012, 08:09 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 07:45 AM)Bucky Ball Wrote:(30082012 07:26 AM)Bishadi Wrote: "If the system (existence itself) is closed, then existence created itself, and we are defining it to do so. In a sense, we are 'it' defining itsef, for its beginning to exist." It makes you a part of it! talk about a 'spooky action at a distance'. did ya just have a daja vu? 

30082012, 08:29 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 07:42 AM)DLJ Wrote:(30082012 07:26 AM)Chas Wrote: The Wikipedia article on Cantor's Diagonal Method is pretty good. It's called induction. Skepticism is not a position; it is an approach to claims. Science is not a subject, but a method. 

30082012, 08:32 AM




RE: Hilbert's Paradox and Cantor's Diagonal Proof
(30082012 08:29 AM)Chas Wrote:(30082012 07:42 AM)DLJ Wrote: Nope. I have trouble with the "use up" bit and the "when we're done" bit. How can we use up an infinite number (of natural numbers)? Hahaha! That's just typical! "Induction" was what we started doing when I dropped out of the education system. I knew it would come back to haunt me. 

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