[Math Noise] (fwd)
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Only countably many real numbers, or members of any uncountable set, are denumerable. It is the property of being uncountable, rather than of being real or complex, which is important here.
In short you are saying there are Reals which can not be expressed in the format: AmEm + Am-1Em-1 + ... + A0E0 . B0E-1 + B1E-2 + ... + BnE-n+1 where m and n -> infinity. Another way of saying this is that there are Reals for which membership in a set of Reals, because they are uncountable and therefore unrepresentable, is not possible. From a set perspective this means, R = [[-infinity, ..., m] [n, ..., 0] [0, ..., p] [q, ..., +infinity]] such that there are uncountable numbers between m,n or p,q; etc. Clearly in contradiction with the base axioms of mathematics as described by Euclid in defining a line.
In general, only countably many members of any uncountable set can be precisely specified within any formal system, given names comprised of strings of symbols, or other similar things.
And I contend that ANY number which is Real can be expressed by the decimal expansion above. Which clearly qualifies as a formal system. There are three ways of looking at the total of mathematics, symbolic set theory (Bourbaki School) geometric (Euclidian) While it is true the 3 are equivalent, the choice of approach does have a relevance on how difficult, if even tractible, the proof is for any particular concept.
It is often convenient, such as when drawing contour maps, to consider the complex numbers to be in 1-1 correspondence with the points of the plane. However, I wouldn't necessarily consider regions of the complex plane to have "area" in the Euclidian sense.
The Euclidian plane IS a complex plane. The -j or -i used in Complex symboligy simply means 'rotate the second axis of measure 90 degree counter-clockwise (by agreement) to the axis of the first measure'. If it requires 2 or more numbers (integer or real) taken as a set to represent the quantity it is a Complex. Regions of a Complex plane are directly comparable (1-to-1) with the concept of 'area' on the Euclidian plane.
We can't physically draw a line segment to arbitrary high precision.
Irrelevant. If the hole in my approach is that I can't draw a line of arbitrary precision in practice then your own 'uncountable numbers' argument falls for the same reason because if it is truly uncountable you can't point to it on a number line and say "there is an uncountable". This discussion is one of principles, not one of practice.
We can conceive of the notion of line segments being in 1-1 correspondence with the reals, but we can specify at most countably many "finitely denumerable" line segments if we wish to discuss their lengths individually.
To say there are Reals for which there is no linear representation is the same as saying there are lengths which can't be measured. Now since a line is nothing but a set of points, which don't have size, but only position this obviously holds no water, unless you are saying it is not possible to place two points arbitrarily close together, which would imply that points have some sort of width, clearly against the definition of a point. This all goes back to what I said in a earlier post, the problem comes from our axiomatic (ie taken on faith, unprovable) use of infinity. Without a clear and precise dilineation of those axioms prior to the proof such conclusions are worthless. Several of you have said "infinity is not a number", this is an axiom. Change the axioms and the whole structure changes. I am simply saying that perhaps we should look at the "infinity is not a number" axiom, much as geometers look at Euclids Fifth Postulate. There is nothing inherent in nature that prefers one axiomatic expression of infinity over the other. By changing our axiomatic definition of infinity we reduce the sets we have to work with from [Integer, Irrational, Real, Complex] to [Integer, Real, Complex]. Now, whether it is worth the trouble is at this time unanswerable because nobody has ever done the research. Jim Choate CyberTects ravage@ssz.com "The laws of mathematics, as far as they refer to reality, are not certain, and as far as they are certain, do not refer to reality." Albert Einstein
Jim Choate wrote:
Forwarded message:
Only countably many real numbers, or members of any uncountable set, are denumerable. It is the property of being uncountable, rather than of being real or complex, which is important here.
In short you are saying there are Reals which can not be expressed in the format:
AmEm + Am-1Em-1 + ... + A0E0 . B0E-1 + B1E-2 + ... + BnE-n+1
All reals are equivalent to sequences of digits, but there are reals such that there is no algorithm to generate their digits. It happens because there are "more" real numbers than algorithms.
In general, only countably many members of any uncountable set can be precisely specified within any formal system, given names comprised of strings of symbols, or other similar things.
And I contend that ANY number which is Real can be expressed by the decimal expansion above. Which clearly qualifies as a formal system.
I suggest the following mental exercise. FORGET FOR A MOMENT ABOUT REAL NUMBERS. Let's deal with mummies: DEFINITION: I define a mummy as possibly infinite sequence of characters, separated by one dot, such that only characters abcdefghij are allowed. Also, mummies that are represented by finite sequences of characters are by this definition equivalent to mummies that end with an infinite sequence of letters "a". END DEFINITION. Examples: dce.abdefhaabdaaa ae.cacacacacacaca... and so on. Obviously, some of the mummies, such as c.cccccc... (with an ininite sequence of "c") CAN be generated by algorithms. The interesting fact, that i will prove below, is that some of them cannot be generated by any algorithm. THEOREM: The set of mummies is more than countable PROOF: if it is countable, we can construct a mummy that is not counted. it is easy. THEOREM: there are mummies such that there is no algorithm that can print them. PROOF: the set of mummies is more than countable, the set of algorithms is countable, therefore there is no way to construct a one-to-one correspondence between mummies and algorithms. Do you agree? Now let's back to the original problem of real numbers: the only difference between mummies and real numbers is that digits 0123456789 are replaced by characters abcdefghij. Not a whole lot of difference, so everything that applies to mummies applies to real numbers. - Igor.
Jim Choate <ravage@EINSTEIN.ssz.com> writes:
In short you are saying there are Reals which can not be expressed in the format:
AmEm + Am-1Em-1 + ... + A0E0 . B0E-1 + B1E-2 + ... + BnE-n+1
where m and n -> infinity. Another way of saying this is that there are Reals for which membership in a set of Reals, because they are uncountable and therefore unrepresentable, is not possible. From a set perspective this means,
Again, we may speak of a representation for the Reals within a formal system, although we may not speak of "The Representation of X" for every single Real X. We cannot assign to every Real a finite representation, but we can talk about the infinite representation all Reals have within a formal system without running out of space. The inability of a formal system to talk about each Real individually, or equivalently, that there are Real numbers which are not finitely denumerable, does not mean that there are Reals which do not have an representation as a non-ending sequence of symbols.
R = [[-infinity, ..., m] [n, ..., 0] [0, ..., p] [q, ..., +infinity]]
such that there are uncountable numbers between m,n or p,q; etc.
Infinity does not have a predecessor, so it makes no sense to count back from it a finite number of steps.
Clearly in contradiction with the base axioms of mathematics as described by Euclid in defining a line.
This point completely escapes me.
And I contend that ANY number which is Real can be expressed by the decimal expansion above. Which clearly qualifies as a formal system.
Useful formal systems employ finite strings from some alphabet. The set of all possible such strings is countable. The set of all sequences from the same alphabet is uncountable, but not particularly useful for theorem-proving, at least in a finite amount of time.
To say there are Reals for which there is no linear representation is the same as saying there are lengths which can't be measured. Now since a line is nothing but a set of points, which don't have size, but only position this obviously holds no water, unless you are saying it is not possible to place two points arbitrarily close together, which would imply that points have some sort of width, clearly against the definition of a point.
No. Points do not have width.
This all goes back to what I said in a earlier post, the problem comes from our axiomatic (ie taken on faith, unprovable) use of infinity. Without a clear and precise dilineation of those axioms prior to the proof such conclusions are worthless.
In Axiomatic Set Theory, it is necessary to postulate (either implicitly or explicitly) the existance of one infinite set. This is an act of faith. Whether any infinities really exist is a matter for philosophy.
Several of you have said "infinity is not a number", this is an axiom. Change the axioms and the whole structure changes. I am simply saying that perhaps we should look at the "infinity is not a number" axiom, much as geometers look at Euclids Fifth Postulate. There is nothing inherent in nature that prefers one axiomatic expression of infinity over the other.
If one constructs the Ordinals, which are isomorphism classes of well-ordered sets, and the Cardinals, which are equivalence classes of equipotent sets, one will automatically end up with all sorts of transfinite numbers. We normally don't include infinities when we build the rationals, the reals, or the complex numbers, unless we need them for a particular application, such as in using the extended real number line in defining measures. -- Mike Duvos $ PGP 2.6 Public Key available $ mpd@netcom.com $ via Finger. $
Jim Choate allegedly said:
Forwarded message:
Only countably many real numbers, or members of any uncountable set, are denumerable. It is the property of being uncountable, rather than of being real or complex, which is important here.
In short you are saying there are Reals which can not be expressed in the format:
AmEm + Am-1Em-1 + ... + A0E0 . B0E-1 + B1E-2 + ... + BnE-n+1
No, that's not what he is saying. What you have written does not represent a *specific* number. He is saying that IF you have a particular scheme for representing *specific* numbers, you can only represent countably many -- for any given scheme, there are numbers you can't represent. To put it another way a scheme that says "you can represent numbers as half infinite strings of digits with a single period somewhere" doesn't actually *specify* any numbers. A scheme that says "start with the number 1 and increment it 400 times" actually specifies a number.
And I contend that ANY number which is Real can be expressed by the decimal expansion above. Which clearly qualifies as a formal system.
To be a formal system of the type required, you would also have to specify deterministic rules that could generate the "Ai" values. The key distinction is between "expressed by" and "generated by". -- Kent Crispin "No reason to get excited", kent@songbird.com,kc@llnl.gov the thief he kindly spoke... PGP fingerprint: 5A 16 DA 04 31 33 40 1E 87 DA 29 02 97 A3 46 2F
participants (4)
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ichudov＠algebra.com -
Jim Choate -
Kent Crispin -
mpd＠netcom.com