The Definition of Points
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From: Tony Orlow on 19 Apr 2007 10:07 Virgil wrote: > In article <46266273(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <46251314(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>> >>>> But, I have a question. What, exactly, is the difference between >>>> "equality" and "equivalence"? >>> "Equality" usually means being the same in every detectable respect. >>> "equivalent" usually means the same in one respect, or in a limited >>> number of respects, while still possibly distinguishable in other >>> respects. >>> >>> >>> For example, parity of a natural number is an equivalence. >>> >>> Naturals having the same parity (evenness or oddness) are equivalent >>> with respect to parity even though not equal as natural numbers. >>> >>> 1 and 3 are equivalent with respect to parity but are not equal. >> Right. I stand corrected. Thanks you. >> >> But, the question remains in this regard. How do we know we have >> actually looked at all possible respects in which could compare? When we >> know of some way in which two things differ, but under a limited set of >> criteria they are not distinguishable, I can see that they are >> "equivalent" in that "respect". But, how can we be sure that we have >> actually taken into account all respects in which two objects may be >> compared? > > By considering context! > > "George Washington" and "the first two term President of the United > States" are the same in some respects and different in others, so one > must consider the context in which they are being compared to determine > whether they are at best equivalent or actually equal in all respects > relevant to that context. > > There are contexts in which '1.000...' and '0.999...' are considered > equal, and others in which they are considered no more than equivalent, > and in some contexts perhaps not even that. Then, are you admitting that the cranks that complain that 1 and 0.999... are not the same may actually have a valid point?
From: Tony Orlow on 19 Apr 2007 10:13 Virgil wrote: > In article <46266bb2(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: >>> On 18 Apr, 06:17, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Mike Kelly wrote: >>>> >>>> < snippery > >>>> >>>>> You've lost me again. A bad analogy is like a diagonal frog. >>>>> -- >>>>> mike. >>>> Transfinite cardinality makes very nice equivalence classes based almost >>>> solely on 'e', >>> Why "almost solely"? >>> >> The successor relation is a relation not related to 'e' except in a >> particular model of the naturals upo which the rest is built. > > Can TO define or construct anything like successorship in any set > theory which doe not depend, even indirectly, on 'e'? Well, sure. Consider '+' to be predefined geometrically, or define it according to its algebraic properties. Then, we can properly define the naturals. 1. 0eN 2. 1eN 3. A xeN -> x+1eN >>>> but in my opinion doesn't produce believable results. >>> Which "results" of cardinality are not believable? You don't think the >>> evens and the naturals and the rationals are mutually bijectible? Or >>> you do think the reals and the naturals are bijectible? Or did you >>> mean you have a problem not with what cardinality actually is but with >>> what you hallucinate cardinality to be? >>> >> As long as transfinite "equivalence" classes are referred to as such, >> and not said to denote "equal" size or numerosity, then I have no problem. > > If we choose to define "size" and "equinumerosity' in terms of > injections and bijections, then TO's problems are purely his own, and > need not affect anyone else. My problem with it is that it violates certain notions which are central for finite sets, but discarded for infinite sets. 1+omega=omega>omega+1 violates the principle tat a+b=b+a. I don't like that. >>>> I'm working on a better theory, bit by bit. > > Of which TO so far has zip. Not complete, but not "zip". >>> You have never managed to articulate a problem with what cardinality >>> actually says (all it says is which sets are bijectbile). You've never >>> managed to explain the motivation behind your "better theory". I don't >>> even know whether your ideas are supposed to be formulated in ZF or in >>> something else, possibly a new foundation of your own devising(or, >>> perhaps, never formalised at all?). FWIW I don't think it would be >>> very hard to define, say, "density in the naturals" in ZF. I just >>> don't see the motivation behind doing so. >>> >> The idea is to create a method that works for comparing countable and >> uncountable sets alike, and properly defining the relationship between >> the two. > > That implies, contrary to fact, that has already been done is improper. > It seems improper to me. An axiom set may be internally consistent, and yet have no bearing on anything externally plausible. >> I suppose it's more the way it's spoken about. Leave out "size" and >> "equinumerous" and talk about "equivalence classes" and not "equality", >> and I have no problem with the consistency of standard theory. > > Since we define size and equinumerosity in terms of injections and > bijections, TO will just have to learn to deal with his problems on his > own. That's what I'm doing. Thanks for the advice.
From: Tony Orlow on 19 Apr 2007 10:18 Virgil wrote: > In article <46266dd9(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: >>> On Apr 17, 10:02 am, Tony Orlow <t...(a)lightlink.com> wrote: >>> >>>> I am saying it's obvious that any countable set has a well ordering. It >>>> is not obvious for uncountable sets. >>> It's not only obvious, but it's trivial to prove that every countable >>> set has a well ordering. >>> >>> So, just to be sure you understand, what is at stake with an axiom of >>> countable choice is not at all that every countable set has a well >>> ordering (since we don't need any choice principle to prove that) but >>> rather that for any countable set there is a function that chooses >>> exactly one member from every nonempty subset of that countable set >>> (and actually, we need concern ourselves only with denumerable sets, >>> since we don't need a choice principle to prove that for every finite >>> set there is a function that chooses exactly one member from every >>> nonempty subset of the finite set). >>> >>> MoeBlee >>> >> It's not trivial to prove any such thing for an uncountable set. That >> requires full Choice, does it not? In fact, it was Virgil, I believe, >> who said that was the MOTIVATION behind Zermelo's formulation of Choice, >> so that ALL sets could be considered well orderable. However, while I >> see that a choice function produces a well ordering for any countable >> set, I don't see that it does for an uncountable set. > > TO's selective blindness strikes again. > > A function which assigns to every nonempty subset of a given set a > particular element in the set, which is what the AOC states exists,IS a > well ordering of that set that set. One merely identifies as the first > element of each set the value of that function. Can one partition an uncountable set into a countable set of countable partitions? Only if one assumes Choice, right? >> Maybe I am confused about something. > > > To is confused about almost everything. > >> Can a well ordering include an >> uncountable number of limit elements? > > > Why not? All any well ordering of any set requires is that every > non-empty subset of that set have a first member under that ordering. I thought it required that there be no infinite descending chains within the well order. Does that not include an infinite descending chain of limit elements? > >> I just don't see that a well order >> can be accomplished in principle on an uncountable set, whether one >> declares an axiom to that effect of not. > > > There is a critical difference in mathematics between proving something > exists and instantiating it. But, you don't PROVE it exists with Choice. You ASSUME it exists axiomatically without any real justification that I can tell. > > Given the AOC, one can trivially prove that for every set there exists a > well ordering. Actually constructing an explicit well ordering may be a > whole different ball of wax. It's plainly impossible for an uncountable set. If you disagree with that statement, by all means prove me wrong with a counterexample. If infinite descending chains of limit values are allowed in a well order, then the H-riffics can easily be well ordered.
From: Tony Orlow on 19 Apr 2007 10:22 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>> Mike Kelly wrote: >>>>> The point that it DOESN'T MATTER whther you take cardinality to mean >>>>> "size". It's ludicrous to respond to that point with "but I don't take >>>>> cardinality to mean 'size'"! >>>>> >>>>> -- >>>>> mike. >>>>> >>>> You may laugh as you like, but numbers represent measure, and measure is >>>> built on "size" or "count". >>> What "measure", "size" or "count" does the imaginary number i represent? Is i a number? >>> The word "number" is used to describe things that do not represent any sort of "size". >>> >>> Stephen > >> Start with zero: E 0 >> Define the naturals: Ex -> Ex+1 >> Define the integers: Ex -> Ex-1 >> Define imaginary integers: Ex -> sqrt(x) > >> i=sqrt(0-(0+1)), so it's built from 0 and 1, using three operators. It's >> compounded from the naturals. > > That does not answer the question of what "measure", "size" or "count" i represents. > And it is wrong on other levels as well. You just pulled "sqrt" out of the > air. You did not define it. Claiming that it is a primitive operator seems > a bit like cheating. And if I understand your odd notation, the sqrt(2) > is an imaginary integer according to you? And sqrt(4) is also an imaginary integer? No, but sqrt on the negatives produces imaginary numbers. Besides, sqrt can be defined, like + or -, geometrically, through construction. > > You also have to be careful about about claiming that i=sqrt(-1). It is much safer > to say that i*i=-1. If you do not see the difference, maybe you should explore the > implications of i=sqrt(-1). > > So what is wrong with > Start with zero: E 0 > Define the naturals: Ex -> Ex+1 > Define omega: Ax > I did that using only one operator. > > Ax? You mean, Ax x<w? That's fine, but it doesn't mean that w-1<w is incorrect. >> A nice picture of i is the length of the leg of a triangle with a >> hypotenuse of 1 and a leg of sqrt(2), if that makes any sense. It's kind >> of like the difference between a duck. :) > > That does not make any sense. There is no point in giving a nonsensical > answer, unless you are aiming to emulate Lester. > > Stephen > > It's not nonsensical, and may even apply to uses of imaginary numbers in practice, but you can ignore it as I knew you would. That's okay. Tony
From: David R Tribble on 19 Apr 2007 10:36
Alan Smaill wrote: >> and Zick was the one who claimed that he would use l'Hospital to work >> out the right answer for 0/0. such a japester, eh? > David R Tribble wrote: >> Geez. How many posts before someone points >> out that it's l'Hôpital's rule? L'Hospital is where >> you take someone after they get punched in the >> nose by a mathematician after saying "l'Hospital's rule". > Brian Chandler wrote: > Well, given that "ô" is only a French spelling for "o-with-the- > following-s-omitted", seems to me that l'Hospital is a pretty > reasonable asciification. Well, I guess we all learn something new every day. I learned it as "l'Hopital" way back when, but I don't recall if the "o" was diacriticized in my textbook or not. I'll have to check. |