Galileo's Paradox
in [Math]
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From: stephen on 11 Dec 2006 12:47 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> David Marcus wrote: >>>> Tony Orlow wrote: >>>>> David Marcus wrote: >>>>>> Tony Orlow wrote: >>>>>>> David Marcus wrote: >>>>>>>> Tony Orlow wrote: >>>>>>>>> Why do you have a problem with the mere suggestion of an infinite value? >>>>>>>> I don't have a problem with infinite values. However, you have to do >>>>>>>> more than merely say "positive infinite n". Assuming your positive >>>>>>>> infinite things aren't the same as something that we already know about >>>>>>>> (in which case, you should just say so), you either need to give a >>>>>>>> construction of these positive infinite things or you need to specify >>>>>>>> their properties. You haven't done either. For example, how many of >>>>>>>> these things are there? How do they relate to each other? How do they >>>>>>>> interact with the natural numbers? Are the operations of addition and >>>>>>>> multiplication defined for them? >>>>>>> Well, I have been through much of that regarding such specific language >>>>>>> approaches as the T-riffic digital numbers, but that's not necessary for >>>>>>> this purpose. It suffices to say that, if a statement is proved true for >>>>>>> all n greater than some finite k, that that also includes any postulated >>>>>>> infinite values of n, since they are greater than any finite k. I don't >>>>>>> need to construct these numbers. Consider them axiomatically declared. >>>>>> Then list the axioms for them. >>>>> (sigh) >>>>> >>>>> infinite(x) <-> A yeR x>y >>>> Is that your only axiom? If so, then state your first theorem about them >>>> and give the proof. >>>> >> >>> That's the only one necessary for what defining a positive infinite n. A >>> whole array of theorems pop forth from infinite-case induction and IFR, >>> such as that the size of the even naturals is half that of the naturals. >>> That's a no-brainer. Go back to where I first answered your question at >>> length, and read again, at length. It's not transfinitology, but it's >>> also not nonsense. >> >>> Allow me to add another: >> >>> |{ x| yeR and 0<(x-y)<=1}|=Big'un. >> >>> That's the unit infinity. >> >> What is the definition of | | ? >> >> Stephen > "Size" of the set. Or, is "|...|" reserved only for "cardinality"? I > think I can adopt the previous symbol for absolute value, as set > theorists have, without stepping on too many toes. > Tony No, | | is not reserved for cardinality, but you have to define | | before you use it. It you say it means "size", you have to define "size". Is "size" a primitive? If not, then it needs to be defined. If it is a primitive, you need some axioms that describe how size behaves. Is Big'un a primitive? Or was the above meant to be a definition of Big'un? Are there other rules governing Big'un? You still do not understand the root of the criticisms levelled at you. It is not that you are trying to define a system that is different, it is that you refuse to provide real definitions and axioms. Nobody is going to be interested in a system that only exists in your head. Stephen
From: Tony Orlow on 11 Dec 2006 12:55 stephen(a)nomail.com wrote: > Six wrote: >> On Thu, 7 Dec 2006 17:25:40 +0000 (UTC), stephen(a)nomail.com wrote: > >>> Six wrote: >>>> On 6 Dec 2006 07:08:46 -0800, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: >>>>> Six wrote: >>>>>> I am very grateful to you for expanding on this. While I'm almost >>>>>> certain I'm missing something, I'm afraid I still don't get it. >>>>>> >>>>>> How exactly does claiming that a 1:1 C is not necessarily >>>>>> indicative of equality of size with infinite sets presuppose an inability >>>>>> to map (eg) the binary and decimal representations of integers? >>>>>> >>>>>> There is still a 1:1 C between the two sets. It is still true that >>>>>> for any finite sets a 1:1C implies equality of size. Moreover it's still >>>>>> reasonable to suppose that a 1:1C implies equality of size in the infinite >>>>>> case unless there are other, 'functional' reasons to the contrary. (Vague, >>>>>> I know. Roughly, 1:1 C is a necessary but not sufficient condition for >>>>>> equality of size.) >>>>>> >>>>>> The idea is that the naturals (in any base) form a paradigm or >>>>>> norm, a standard against which other sets can be measured. >>>>> The set of finite binary strings is a subset of the set of finite >>>>> decimal strings. >>>> I confess I hadn't fully appreciated this simple point, that >>>> together with the fact that the strings just are, so to speak, the natural >>>> numbers (in a given base). >>>>> Then b) precludes them being the same size. >>>>> >>>>> They are also both the same size as the set of natural numbers. >>>>> >>>>> Thus they are the same size as each other. >>>>> >>>>> Contradiction. >>>> One is driven to the conclusion that there is no base-independent >>>> size for the natural numbers. >>> How can the size be base dependent? The natural numbers are not base dependent. >>> Any natural number can be expressed in any base. There is no natural number >>> expressible in base 16 that is not expressible in base 10, or base 9, or base 2. >>> >>> I suppose you could claim that there is a set of decimal numbers, and a set >>> of base 2 numbers, and a set of hexadecimal numbers, and that they are all >>> different, and all have different sizes. But it is a strange notion of >>> "different size" given that all the sets represent the same thing. > >> I agree it would be a strange notion of size. My views have >> shifted, but I'm afraid you might still find them strange. I do not think >> we can say those sets are different sizes, but I do not think we can say >> they are the same size either. > >> Thanks, Six Letters > > Well if they do not have the same size, and they do not have different > sizes, then the most sensible conclusion is that they simply > do not have any size at all. That is not at all unreasonable. > If your definition of "size" requires the size to be a natural > number, then the set of natural numbers does not have a size. > This of course gets back to the main point, which is you have > to decide what "size" means in this context. If you do not have > a definition of "size", it is rather pointless to talk about it. > > Stephen > I'd say that, if you have two different methods of measuring size, and one equates two sets while the other distinguishes between them, then it is reasonable to say that, according to the second measure, there is a difference between the sizes of the two sets. The main problem that may arise within any given method of measure is a situation where |A|<|B| and |A|>|B|. If a given measure produces such a result, then it doesn't work. Otherwise, it is a valid measure, and any difference it detects is something that exists, even if another measure does not detect it. Tony
From: Bob Kolker on 11 Dec 2006 12:59 Eckard Blumschein wrote: > > > Concerning its putative fundamentals obviously with no avail. On the > contrary. Mathematics has burgeoning in the 20-th century. Almost all of Hilbert's famous challange problems have been solved. Bob Kolker
From: Tony Orlow on 11 Dec 2006 13:06 Eckard Blumschein wrote: > On 11/28/2006 9:30 PM, Tony Orlow wrote: > >>> The relations smaller, equally large, and larger are invalid for >>> infinite quantities. >>> >> Galileo's conclusions notwithstanding, there are certainly relationships >> among many countably and uncountably infinite sets which indicate >> unequal relative measures. I certainly consider 1 inch to be twice as >> infinitely many points > > Twice as infinitely many is Cantorian nonsense. Cantor was able to show > himself that all natural and rational numbers do not have a different > "size". Infinity is not a number. It has been understood like something > which cannot be enlarged and not exhausted either: > > oo * 2 = oo. > THAT'S the Cantorian nonsense. What I said is not Cantorian, nor in line with transfinitology, as is your statement. My statement is a statement of the result I and others intuit, that there are twice as many reals in twice as large an interval, like there are twice as many naturals as even naturals. That can be accomodated mathematically, but not by set theory as it stands.
From: Tony Orlow on 11 Dec 2006 13:10
Eckard Blumschein wrote: > On 12/6/2006 8:02 PM, David Marcus wrote: >> Eckard Blumschein wrote: >>> On 12/5/2006 7:06 PM, Tony Orlow wrote: >>>> Eckard Blumschein wrote: >>>>> On 12/1/2006 9:59 PM, Virgil wrote: >>>>> >>>>>> Depends on one's standard of "size". >>>>>> >>>>>> Two solids of the same surface area can have differing volumes because >>>>>> different qualities of the sets of points that form them are being >>>>>> measured. >>>>> Both surface and volume are considered like continua in physics as long >>>>> as the physical atoms do not matter. >>>>> Sets of points (i.e. mathematical atoms) are arbitrarily attributed. >>>>> There is no universal rule for how fine-grained the mesh has to be. >>>>> Therefore one cannot ascribe more or less points to these quantities. >>>>> >>>>> Look at the subject: Galileo's paradox: The relations smaller, equally >>>>> large, and larger are pointless in case of infinite quantities. >>>> Now, just a minute, Eckard. You're contradicting yourself, if these >>>> objects are infinite sets of points. They have different measure. >>> I did not consider measures. Let's get concrete. Are there more naturals >>> than odd naturals? This question could easily be answered if there was a >>> measure of size. >> It can easily be answered once you say what you mean by "more". Why do >> you think that common English words have unambiguous mathematical >> meanings? > > 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, and that there are "more" rationals besides those? |