[Math Noise] (fwd)
Forwarded message:
Yes, the Reals can be constructed from the Rationals. No, the Reals are not a subset of the Rationals.
An arbitrary Real can be constructed from the Rationals. If we accept the proposition, as posed apparently by you and others, of uncountable Reals then your 'assumption' fails, otherwise the 'uncountable' members would be countable.
Er, no. But you can create a number which is not representable as a fraction as a limit point of very many fractions.
A 'limit point' is not the same as 'equal to'. Arbitrarily close is not equivalent, inherent in the definition of a limit point is the concept of 'little o' and 'big o', or worded differently our axiomatic definition of infinity.
If we then consider equivalence classes of those Cauchy sequences which converge to the same limit, and consider an element of the original set to correspond to the class containing the sequence all of whose members are that element, we can consider the classes to form a "completion" of the original set by addition of all its limit points.
We can, but there is no fundamental rule in mathematics that requires me to ignore the small but distinct difference between the element in the original set and the sequences used to approximate it. It is something that must be agreed upon by accepting a particular axiomatic definition of infinity.
Dedekind Cuts are a simple abstraction, often used to construct the Reals from the Rationals in undergraduate calculus courses. Conceptually, one makes a single "cut" in the set of Rationals, dividing it into two parts, all of the members of one part being greater than all of the members of the other. The number of ways of doing this correspond to the Reals.
The number of cuts are 1-to-1 with the Reals, they are not the Reals. There is no way I can make a cut which is 3.1527, only 1-to-1 with the number(s). Important distinction. Jim Choate CyberTects ravage@ssz.com "The best lack all conviction, while the worst are full of passionate intensity." Yeats
Jim Choate <ravage@EINSTEIN.ssz.com> writes:
An arbitrary Real can be constructed from the Rationals. If we accept the proposition, as posed apparently by you and others, of uncountable Reals then your 'assumption' fails, otherwise the 'uncountable' members would be countable.
We can construct the Reals from the Rationals without having to speak of each specific Real while doing so. A formal system, having only a countable number of strings of symbols from its alphabet, can speak of "The Real Numbers" even though it cannot speak of "The Real Number X" for every single X in the Reals.
A 'limit point' is not the same as 'equal to'. Arbitrarily close is not equivalent, inherent in the definition of a limit point is the concept of 'little o' and 'big o', or worded differently our axiomatic definition of infinity.
In standard analysis, the limit of a sequence A[n] is a value x such that given any positive epsilon, no matter how small, we can find a point in the sequence such that all its members after that point are within epsilon of x. Such a limit, if it exists, is unique and exactly defined. "Little o" and "big o" are concepts from complexity theory and I am not precisely sure why you feel they need to be mentioned.
I can, but there is no fundamental rule in mathematics that requires me to ignore the small but distinct difference between the element in the original set and the sequences used to approximate it.
In the general case, the limit point will not be a member of the sequence which approximates it. Although every member of the sequence is a finite distance away from the limit, the limit itself is, as I previously mentioned, exactly known without any ambiguity.
It is something that must be agreed upon by accepting a particular axiomatic definition of infinity.
An "infinity" is simply the property of being able to be put in 1-1 correspondence with a proper subset of oneself.
The number of cuts are 1-to-1 with the Reals, they are not the Reals. There is no way I can make a cut which is 3.1527, only 1-to-1 with the number(s). Important distinction.
Mathematical objects are sets with structure. We generally consider two mathematical objects equivalent if there exists a 1-1 correspondence between the respective sets which is structure preserving. What the actual members of the set are, and how they were constructed, is usually unimportant. For all practical purposes we may refer to any mathematical object isomorphic to the Reals as "The Reals", without any confusion. -- Mike Duvos $ PGP 2.6 Public Key available $ mpd@netcom.com $ via Finger. $
On Sun, 19 Jan 1997 22:34:05 -0600 (CST), Jim Choate wrote: Forwarded message:
Yes, the Reals can be constructed from the Rationals. No, the Reals are not a subset of the Rationals.
An arbitrary Real can be constructed from the Rationals. If we accept the proposition, as posed apparently by you and others, of uncountable Reals then your 'assumption' fails, otherwise the 'uncountable' members would be countable. There are no "uncountable" numbers -- uncountability is a property of the set, not an individual member of the set. The set of reals cannot be placed in one-to-one correspondence with the integers (strictly, positive integers, but it amounts to the same thing).
Dedekind Cuts are a simple abstraction, often used to construct the Reals from the Rationals in undergraduate calculus courses. Conceptually, one makes a single "cut" in the set of Rationals, dividing it into two parts, all of the members of one part being greater than all of the members of the other. The number of ways of doing this correspond to the Reals.
The number of cuts are 1-to-1 with the Reals, they are not the Reals. There is no way I can make a cut which is 3.1527, only 1-to-1 with the number(s). Important distinction. Cut the rational numbers into two sets, A containing all the negative rationals and all those that have squares less than 2, and B containing all the positive rationals that have squares greater than 2. There you have a cut which is 1.41421... (i.e., sqrt(2)). So you can define irrational numbers from the rationals (an irrational number is a cut such that the first set (A) has no largest member and the second set (B) has no smallest member). -- Paul Foley <mycroft@actrix.gen.nz> --- PGPmail preferred PGP key ID 0x1CA3386D available from keyservers fingerprint = 4A 76 83 D8 99 BC ED 33 C5 02 81 C9 BF 7A 91 E8 ---------------------------------------------------------------------- Sometimes a feeling is all we humans have to go on. -- Kirk, "A Taste of Armageddon", stardate 3193.9
Jim Choate allegedly said:
[...]
The number of cuts are 1-to-1 with the Reals, they are not the Reals. There is no way I can make a cut which is 3.1527, only 1-to-1 with the number(s). Important distinction.
They are 1-1 with the Reals, you can define all the operations on them that are defined for reals, for every representable real (like 3.1527) there exists a DC. You have some idea, perhaps that there are "real" reals, as opposed to the various constructs for defining them? Perhaps you could define what a "real" real would be? -- 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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Jim Choate -
Kent Crispin -
mpd＠netcom.com -
Paul Foley