Question…
In the current theory of the real numbers does there exist a pair of distinct rational numbers between which there is NOT an infinite number of irrationals? No. How then does a “cut” of Q (the rationals) determine a unique irrational number?
Usual answer: The sqrt2 “cuts” the rationals into two sets: those whose square is greater than 2 and those whose square is less than 2.
That “guess and check” method only works if you have an alternate definition of the irrational….if you already have that “limit” to begin with…that’s not a “construction” by definition. The “sqrt2 example” only works for algebraic numbers. What if the irrational is not algebraic? After all, in the current theory of the reals that set is of measure 0. And so is the whole set of definable transcendental numbers, as in definable via a finite sentence or set of sentences (algorithm).
What about the Cauchy criterion as a foundation for the construction of the irrationals?
Let’s consider the set of irrational numbers as infinite strings of digits in some base or infinite continuing fractions. Then can we apply the Cauchy criterion to define convergence? (For an explanation of the Cauchy criterion see page 368 of “Advanced Calculus: A Differential Forms Approach” by H. M. Edwards…or just google “Cauchy criterion”…you’ll find plenty of sources.)
Quoting Edwards..”a given sequence {qn} is convergent if and only if it is ‘contained in arbitrarily small intervals’.” That amounts to saying that we can ignore any finite prefix of an infinite string…it’s the “tail” that determines the point of convergence. Ignoring the prefix amounts to taking the algebraic quotient of the whole set of Dedekind’s “gaps” by Q …we get cosets all of which contain elements with every possible finite prefix.
The result is a set of equivalence classes called R/Q+. That set has the trivial topology…it’s a single point topologically. If we follow the Cauchy criterion to its “limit” we will have exhausted all finite prefixes and be left with no way to distinguish one “irrational” from another. The cosets in R/Q+ cannot be distinguished from one another….The whole theory collapses along with Q. What it collapses to is an infinitesimal…The infinitesimals that construct the whole set of vectors we can associate with the rationals… https://wordpress.com/post/21stcenturyparadox.com/513.
The idea that “R is a vector space over Q” assumes that those vectors associated with the irrationals are linearly independent. They are not. They are dependent on the construction of Q as a vector space (one such derivation of Q is related to what is shown at the link given above). That construction is by way of exactly two linearly independent vectors. The “gaps” are all the “same” and only get their distinction via an infinitesimal shift of the whole set of vectors Q.
The usual description of elements of R/Q+ as Q+r for some r of R gets it backwards. R is defined by a shift of Q, an infinitesimal shift, and if we “collapse” Q we only get a single element, a “gap”, an infinitesimal. The “location” of that “gap” is totally dependent on its association with Q. The link above shows how a pair of linearly independent vectors over N can construct the whole set of rational numbers as vectors and then use the rationals to give structure to all the “irrational” directions by shifting the whole set of vectors associated with Q infinitesimally. Only the two foundational vectors are linearly independent. The rest are derived from them. To be explained in more detail….
We can think of R as all the directions a vector on the plane can point to from the origin. (See E. Artin Geometric Algebra …where dilatations and translations are described as groups and modeled as exponentials.) What R collapses to in R/Q+ is exactly two cosets…one corresponding to “Left” and one corresponding to “Right”….the two most fundamental directions, the only two directions in one dimension…on a circle we can think of them as clockwise and counterclockwise…and they are related as “complements” in exactly the way fact and counterfactual are….as two complementary (orthogonal) “worlds”.
Just as the roles of fact and counterfactual are applied “after the fact”, and one determines the other, likewise the whole set of “gaps” is determined by an infinitesimal shift of Q in either direction….the foundational infinitesimals in essence “overlap” at infinity (they cannot be differentiated) -> at infinity there’s no way to distinguish one from the other. At infinity “left” and “right” (and “orthogonal”) no longer have meaning…direction is indeterminate….”at infinity” there is no way to orient oneself in space and arbitrarily assigning directions (and hence order) does not change that fact. While the idea of a fixed point is a useful fiction, it is exactly that.
Remember, Cantor started his “search” for certainty with trigonometric functions…functions which depend on orientation or direction for their very definition. The error in the “ultimate” foundations for set theory is the fact that at infinity all basis for orientation is lost….aside from an arbitrary assignment of such via the unary membership relation. The “inclusion” operator is most certainly arbitrary…what is included in what is a two sided question with no predetermined answer. In Huygens terms: “each point on the expanding wavefront is a source”.
Mach was right, Cantor and Hilbert not so much….there is no absolute frame of reference (the current theory of the real numbers is supposed to provide such a “frame”). The only certainty is that fact and counterfactual do not simultaneously exist, by definition…by any actionable definition of “exist”…all fanciful definitions notwithstanding.
21st Century Paradox 