On 2/2/2013 8:41 PM, Shmuel (Seymour J.) Metz wrote: > In <GbydnfG1Xq3Nu5HMnZ2dnUVZ_jqdnZ2d@giganews.com>, on 02/01/2013 > at 02:32 PM, fom <fomJUNK@nyms.net> said: > >> So, for example, there are "gaps" in the system of rational >> numbers. One can, assuming completed infinities, define infinite >> sets of rational numbers corresponding to the elements of a >> Cauchy sequence. > > What would be the point? You're introducing extra machinery into what > is a very simple construction.
That is not me. I would be happy to understand the real numbers in the simple manner in which it had been taught to me.
The construction of the reals from the natural numbers is a sequence of logical types for which the order relation of the natural numbers grounds the order relation of the derived type.
You can find an excellent construction of the integers in Jacobson:
Lectures in Abstract Algebra Van Nostrand Co., Inc. Princeton NJ, c. 1951
Each integer is an infinite class of pairs.
From this, the similar construction for defining the rationals should be apparent using quotients rather than differences. Some particular may be needed for handling 0.
Once again, each rational is an infinite class of pairs.
From this, one constructs the reals. The following is from Cantor's Grundlagen concerning the logic of definition for a real number:
"I come now to the third definition of real numbers. Here too an *infinite* set of rational numbers of the first power is taken as a basis, but it has properties other than the Weierstrassian definition. [...]
"Every such set (a_v), which also can be characterized by the requirement:
lim_(v=oo) (a_(v+u) - a_v) = 0 (for arbitrary u)
I call a fundamental sequence and correlate with it a number b, *TO BE DEFINED THROUGH IT*, [...]
"Care must be taken on this cardinal point, whose significance can easily be overlooked: in the third definition the number b, say, is *NOT DEFINED AS THE 'LIMIT' OF THE TERMS a_v OF A FUNDAMENTAL SEQUENCE (a_v); FOR THIS WOULD BE A LOGICAL ERROR SIMILAR TO THE ONE DISCUSSED FOR THE FIRST DEFINITION, i.e., WE WOULD BE PRESUMING THE EXISTENCE OF THE LIMIT lim_(v=oo) a_v. BUT THE SITUATION IS RATHER THE REVERSE.
"These definitions agree that an irrational real number is *GIVEN BY A WELL-DEFINED INFINITE SET OF RATIONAL NUMBERS* of the first power. But they differ over the way in which the set is linked with the number it defines, and in the conditions which the set has to fulfill in order to qualify as a foundation for the definition in question.
"In the first definition a set of positive rational numbers a_v is taken as a basis, is designated by (a_v), and satisfies the condition that, whatever and however many of the a_v are summed (so long as the count is finite) this sum always remains less than a specifiable limit. [...]
"One sees here that the creative element which binds the set with the number defined through it lies in the formation of sums; but it must be emphasized that only the summation of an always finite count of rational elements is used *AND THAT THE NUMBER b TO BE DEFINED IS NOT SET AT THE BEGINNING AS EQUAL TO THE SUM Sum_v(a_v) OF THE INFINITE SERIES (a_v); THIS WOULD BE A LOGICAL ERROR,[...]
By this account, the real numbers are a logical type wherein each individual real number is an equivalence class of fundamental sequences.
I have no doubt that I am "wrong" in the eyes of many. But it would be nice to have that error explained.
At what point did we start ignoring the "foundations" of mathematics because the logical rigor was excessive? Believe me. I get the usual axiomatization of the real numbers as presented in any decent calculus text. But strict logical form that places the foundations of mathematics within set theory using only the axioms of set theory obtains each real number through a sequence of definitions describing classes of distinct logical types.
I am fully aware that much research on the real numbers in set theory involves the isomorphic set omega^(omega) which is closely related to the description of real numbers relative to infinite continued fractions (Baire space). But, the sets involved in this isomorphism do not have members identifiable as real numbers relative to construction using continued fractions. Such a definition would also involve equivalence classes of infinite collections of rationals.
Now, I certainly did make the mistake of thinking that the original poster had less knowledge than subsequent posts indicated. But, to the best of my knowledge, my remarks have been accurate and to the point with regard to what modern mathematicians claim to believe when they say the things they say concerning sets -- especially ZFC.
> >> When the limit of the sequence is, itself, a rational number, >> that infinite set becomes a representation of that rational number >> in the complete space whose "numbers" are equivalence classes of >> Cauchy sequences sharing the same limit. > > No; the set of values taken by the sequence is irrelevant. If a Cauchy > sequence in the rationals converges then the *sequence* is a > representative of its limit. >
You are making precisely the logical error spoken of by Cantor.
>> When the limit of a Cauchy sequence does not exist as a rational >> number, that Cauchy sequence becomes a representative of the >> equivalence class of Cauchy sequences that cannot be >> differentiated from that representative using the order relation >> between the rational numbers of the underlying set. > > What are you trying to say? The definition of the equivalence relation > is the same whether the Cauchy sequences converge or not; two > sequences are equivalent if their difference converges to zero. Any > Cauchy sequence is a representative of its equivalence class, by > definition.
This is the point at which the order relation of the underlying set of rationals is used to ground the identity relation for the equivalence classes of objects defined individually as infinite collections of rationals.
> >> These "numbers" have no corresponding rational number as a limit >> and are, therefore, distinguished as a different logical type in >> the *new*, completed space. > > Non sequitor, and false. There is nothing logically special about > equivalence classes of Cauchy sequences that converge.
Sadly I only speak English. But the false part I take issue with. The entire problem since the discovery of incommensurables has been to find a satisfactory logical explanation for them. In Dedekind's definition using cuts, there certainly is a logical difference between cuts containing a maximum value (or minimum depending on which is used) and those that do not.
And Cantor's careful distinctions have been made precisely to accommodate the *DEFINITION* of real numbers, both rational and irrational.
> >> To call a subset of a complete space a dense subset is to say that >> such a logical type construction could be made from that subset >> to recover the original space. > > Completeness only applies to metric spaces, while denseness applies > to subsets of arbitrary topological spaces. You can't in general > reconstruct a topological space from only a dense subset, not even if > the space is compact. > >  Well, slightly more general. >
As I said when I concluded before, there are certainly more knowledgeable topologists than I.
"A uniform space is complete if and only if every Cauchy net in the space converges to a point of the space"
"A uniform space is complete if and only if each family of closed sets which has the finite intersection property and contains small sets has a non-void intersection"
"A pseudo-metrizable uniform space is complete if and only if every Cauchy sequence in the space converges to a point"
"A set A is dense in a topological space X iff the closure of A is X"
All of these statements are from General Topology by Kelley. The first and fourth are definitions. Both contradict what you have just stated. What you are thinking of as a dense subset is a set dense in a subset with the subspace topology.
As I said, I am probably wrong. But I try to be careful.