An uncountable countable set
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From: MoeBlee on 25 Oct 2006 20:47 Lester Zick wrote: > Yeah, Moe, look you do a lot of talking to justify one primitive. I'm not justifying a primitive. > My > primitive is "contradiction" or "differences" and I can justfy that as > true in one sentence. You say all your contentions "revert" (whatever > that may mean) to one "undefined non logical primitive" by which I > assume you mean one outright non demonstrable assumption No, I don't mean that. And I've already explained the difference. And again, since you are unfamiliar with even the most basic notions such as a primitive symbol, I'm pretty much wasting my time every time I write up explanations for you to receive in the lap of your own ignorance. > >No, primitives are individual symbols. > > So they're just symbolic assumptions of truth. No, they are very much NOT that. > >Yes, the term 'cardinality' is not in the above definition. > > So now if you don't say "cardinality" the concept isn't there? It just doesn't even work in the way you pose the question. I just can't explain how to do certain tasks on a computer to someone who won't even push the "on" button on the machine. I just can't explain how to regard mathematical defintions using mathematical logic to someone who just won't even read page one of a textbook. > In other words if I can't intuit your faith based belief in standard > set analytical techniques you need to get back to the practice of your > religion and leave me to my own devices? No, my point is that I can't explain page 100 to you if you won't read page 1. Nor can I keep very interested in a conversation with someone such as you asks me questions on the subject but then replies to my answers with essentially, "Who asked you? Who cares?" MoeBlee
From: Ross A. Finlayson on 25 Oct 2006 21:13 MoeBlee wrote: > MoeBlee wrote: > > Ross A. Finlayson wrote: > > > Eh, ZF is consistent with itself. That's because anything can be > > > proven in an inconsistent theory, except where it would conflict with A > > > theory of course, because there are only and all true statements in A > > > theory. > > > > What benefit, what satisfactions do you derive from typing nonsense > > such as above? > > > > (The only correct thing you said above is that an inconsistent theory > > proves every statement (that is, every statement in the language of the > > theory)). > > Correction: As to "ZF is consistent with itself", I do not mean to > claim that ZF is inconsistent. > > MoeBlee Cardinals are ordinals and in a set theory: sets. Generally it's known that you get cardinals from the previous using the powerset operation, and ordinals from the previous using the successor operation. So, consider the formation of the ordinals where the powerset of the previous ordinal is the successor. The powerset is order type is successor with ubiquitous ordinals, and naturals. You might say "the von Neumann ordinals are simpler than that" and also they contain less information. For example in comparing ordinals as integers sets are defined by their elements. Infinite sets are equivalent. That says there exists bijective mappings between any two infinite sets, with composition. If you're familiar with Cantor's results you've seen and probably accepted proofs, or as I call them results, about functions between the naturals and reals or naturals and powerset of reals. Basically that's about three things: the antidiagonal argument and nested intervals about well-ordering the reals and mapping them to the naturals, and the powerset result and coded binary powerset result about inversion, and the set of natural numbers being itself. The reals are ordering sensitive in that the way you order them in mapping to other naturally ordered sets affects the ability of them to be described. The real numbers are consistently not a set in ZF, not thus a set theory. There's only one theory with no axioms. This ball and vase problem, marbles in a jar, you have a discrete process occurring at each t = 1/ 2^n for integer n > 0, or n >= 0, whatever. Then, there's some n = oo which is when the process completes, because, for no other value of n would this "completion" occur. So, the numbers go right to N, infinity, or, it's ridiculous to suggest the rate out overcomes the greater rate in. That might violate the problem. Ross
From: MoeBlee on 25 Oct 2006 21:39 Ross A. Finlayson wrote: > Cardinals are ordinals and in a set theory: sets. > > Generally it's known that you get cardinals from the previous using the > powerset operation, and ordinals from the previous using the successor > operation. > > So, consider the formation of the ordinals where the powerset of the > previous ordinal is the successor. > > The powerset is order type is successor with ubiquitous ordinals, and > naturals. > > You might say "the von Neumann ordinals are simpler than that" and also > they contain less information. > > For example in comparing ordinals as integers sets are defined by their > elements. > > Infinite sets are equivalent. That says there exists bijective > mappings between any two infinite sets, with composition. If you're > familiar with Cantor's results you've seen and probably accepted > proofs, or as I call them results, about functions between the naturals > and reals or naturals and powerset of reals. > > Basically that's about three things: the antidiagonal argument and > nested intervals about well-ordering the reals and mapping them to the > naturals, and the powerset result and coded binary powerset result > about inversion, and the set of natural numbers being itself. > > The reals are ordering sensitive in that the way you order them in > mapping to other naturally ordered sets affects the ability of them to > be described. The real numbers are consistently not a set in ZF, not > thus a set theory. > > There's only one theory with no axioms. There are a few correct statements in there, but most of the above is just plainly incorrectl mixed with gibberish. I'm just cursious to know what satisfcation you derive from typing up and posting such nonsense, post after post after post, year after year after year. MoeBlee
From: Dik T. Winter on 25 Oct 2006 21:50 In article <1161825225.141821.66550(a)e3g2000cwe.googlegroups.com> "Ross A. Finlayson" <raf(a)tiki-lounge.com> writes: .... > Cardinals are ordinals and in a set theory: sets. > > Generally it's known that you get cardinals from the previous using the > powerset operation, Through what powerset operation do you get the cardinal 3? > and ordinals from the previous using the successor > operation. Through what successor operation do you get the ordinal omega? > So, consider the formation of the ordinals where the powerset of the > previous ordinal is the successor. In that way you do not construct the ordinals, but the Finlayson numbers. They go like 0, 1, 2, 4, 8, 16, 32, ... > You might say "the von Neumann ordinals are simpler than that" and also > they contain less information. Well, at least they contain ordinal numbers like 3, 5, 6, 7 etc. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Ross A. Finlayson on 25 Oct 2006 22:11
MoeBlee wrote: > Ross A. Finlayson wrote: > > Cardinals are ordinals and in a set theory: sets. > > > > Generally it's known that you get cardinals from the previous using the > > powerset operation, and ordinals from the previous using the successor > > operation. > > > > So, consider the formation of the ordinals where the powerset of the > > previous ordinal is the successor. > > > > The powerset is order type is successor with ubiquitous ordinals, and > > naturals. > > > > You might say "the von Neumann ordinals are simpler than that" and also > > they contain less information. > > > > For example in comparing ordinals as integers sets are defined by their > > elements. > > > > Infinite sets are equivalent. That says there exists bijective > > mappings between any two infinite sets, with composition. If you're > > familiar with Cantor's results you've seen and probably accepted > > proofs, or as I call them results, about functions between the naturals > > and reals or naturals and powerset of reals. > > > > Basically that's about three things: the antidiagonal argument and > > nested intervals about well-ordering the reals and mapping them to the > > naturals, and the powerset result and coded binary powerset result > > about inversion, and the set of natural numbers being itself. > > > > The reals are ordering sensitive in that the way you order them in > > mapping to other naturally ordered sets affects the ability of them to > > be described. The real numbers are consistently not a set in ZF, not > > thus a set theory. > > > > There's only one theory with no axioms. > > There are a few correct statements in there, but most of the above is > just plainly incorrectl mixed with gibberish. > > I'm just cursious to know what satisfcation you derive from typing up > and posting such nonsense, post after post after post, year after year > after year. > > MoeBlee I understand it. Please identify something you see as incorrect or don't understand. There's no universe in ZF. So, quantify over sets. Basically, I plan to cut and paste the posts into other documents, in the evolving discussion. Ross |