Galileo's Paradox
in [Math]
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From: Tony Orlow on 14 Dec 2006 13:02 Eckard Blumschein wrote: > On 12/11/2006 7:10 PM, Tony Orlow wrote: >> Eckard Blumschein wrote: > >>> More is just inappropriate as to describe something infinite. There are >>> not more naturals than rationals. There are not equaly many of them, >>> there are not less naturals than rationals. >>> >>> >> So, you wouldn't agree that every natural is a rational, > > Every natural is a rational, yes. > >> and that there are "more" rationals besides those? > > No. In this even the dullest Cantorians agree with me. > They say: There is a bijection between both. > I add: Galilei: incomparable > > Bijections aside, if you have the set of naturals, and you wish to turn it into the set of rationals, you do not remove any elements, but between every two elements you had originally, you have to add infinitely many "more", to get the desired set. The bijection approach alone is hundreds of years old. Galileo was brilliant, but he was not the "last prophet", any more than Georg Cantor. There is no such animal. It's time to work measure into count, and distinguish what are obviously different size sets.
From: Tony Orlow on 14 Dec 2006 13:05 Eckard Blumschein wrote: > On 12/6/2006 8:18 PM, David Marcus wrote: >> Eckard Blumschein wrote: > >>> He assumes that his list of all reals is complete and shows that this is >>> not the case. From this contradiction he was forced to conclude that the >>> reals are uncountable >>> but he intentionally misinterpreted the outcome by >>> claiming there are more real than rational numbers. >> Please define the word "more" in this context. What exactly are you >> saying is not correct (or misinterpreted)? > > The word more is nearly synonymous to the > relation. > There is no number x of all real "numbers". > There is also no number y of all rational numbers. > There is not even a number z of all natural numbers. > > So it may be intuitively obvious but it is not justified to guess x>y>z. > > > Sorry, Eckard. That's a copout. "They don't exist, so stop drawing conclusions about them." You have no basis for saying there is no reasonable way of quantifying these sets, just because you can't do it. I'm not advocating the transfinitological approach, but it's not wrong in ALL respects.
From: Tony Orlow on 14 Dec 2006 13:07 Eckard Blumschein wrote: > On 12/11/2006 9:56 PM, Virgil wrote: > >> Cantor was able to show >>> himself that all natural and rational numbers do not have a different >>> "size". Infinity is not a number. >> >> No mathmatician says "infinity" in that context. They will call some >> sets "infinite", but that is not the same thing. > > Yes. Infinity is a fiction. It cannot be achieved by conting. Infinite > is almost the opposite. > >>> It has been understood like something >>> which cannot be enlarged and not exhausted either: >> Infinite sets can be enlarged in the sense that for each >> of them there are proper supersets, at least in ZFC and NBG. > > So infinite sets are potentially infinite. Quite reasonble. > >> They can be exhausted by taking away all their members. > > One more indication for my insight that ZFC does not really cover pi. > > Unfortunately, I will be hindered for a while to continue our discussion. > Who could grasp my insights and suggestions? > Robert Kolker is out of anything. > TO does not even understand Cantor. I do, but I disagree. You are the one who claimed that the evens being half as numerous as the naturals was "Cantorian nonsense", when it is directly contradictory to transfinite set theory. So, I would imagine it's YOU that has no idea what he's talking about. You agree with more with Cantor than I do. These kind of personal attacks are beneath you. > Virgil will not be able to surrender. > Lester Zick? ??? > Maybe someone else. > > EoD. I regret it. >
From: Tony Orlow on 14 Dec 2006 13:11 MoeBlee wrote: > Tony Orlow wrote: >> You cannot have a smallest infinity, and also not have it. > > You are actually TRYING NOT to listen. > > 'Smallest infinity' means two DIFFERENT things in two DIFFERENT > contexts. I've explained too many times already now but you just keep > going right past the point to make your same error over and over and > over. > > MoeBlee > The infinite numbers in set theory have no relation to the infinite numbers in NSA. They exist within two different theories. In pressing this point, I see set theorists backing away from the notion that omega is even a "number" It's an "ordinal" with no relation to "real" numbers. I'm satisfied with that admission. Omega is really not a number. There is no establish language for it that supplies anything like a calculation. Real numbers? They exist for me. Sorry, Eckard, they're real. "Limit ordinals" and "transfinite cardinals"? Those are fictions, concoctions with no relation to reality. Fun? Sure. Meaningful? Not. ToeKnee
From: Tony Orlow on 14 Dec 2006 13:17
MoeBlee wrote: > Tony Orlow wrote: >> Can you or anyone please cite where Robinson mentions omega in >> Nonstandard Analysis? I'm not saying it isn't there, but I haven't seen >> it. Granted, I'm not very far through it, but so far I see no need for it. > > Page 52. Not only does he mention omega, but he explicitly uses it in a > formula of ordinal arithmetic to state the order type of the > nonstandard *N. > > Now will you PLEASE stop claiming that Robinson's nonstandard analysis > is incompatible with having a least infinite ordinal? > > Maybe, just MAYBE, you'll take ONE MINUTE to think toward understanding > what I've said to you so many times already about your confusing two > different senses of 'infinite'. > > MoeBlee > Yes, I stand corrected. He mentions it in the context of countable neighborhoods surrounding each infinite value, in each direction. That's exactly like m T-riffic countable neighborhoods around each declared infinite limit point (digit), with rational "repeating" digit strings connecting them. He doesn't refer to omega as being "infinite" in the sense that he's using for his theory, however, does he? He's referring to a countable set in the way most easily understood by his readers. ToeKnee |