Is there a connection between the topology of $\mathbb{R}/\mathbb{Q}$ and the logic fundamental to Smooth Infinitesimal Analysis?

Question

As I read through the discussion related to this question (Visualizing quotient groups: $\mathbb{R/Q}$) posted 11 years ago I am reminded of a line on page 20 in a book by J. L. Bell, A Primer of Infinitesimal Analysis: "Thus we have automatically allowed for the possibility (which will turn out to be a reality in S) that two locations, although not in fact coincident, are nonetheless sufficiently indistinguishable that it cannot be decided whether one is to the right or left of the other."

In particular, in the most popular answer to the above mentioned MathStack question I find this quote, "Thus if such a preimage is nonempty, it contains an open interval,..." As far as I can see this is essentially what the foundational axiom of "Microstraightness" in SIA means.

Is there a connection between the fact that the underlying logic of SIA is intuitionist logic and the fact that this quote accurately describes what we 'see' when examining the quotient $\mathbb{R}/\mathbb{Q}$ from the point of view of point-set topology?

Is it reasonable to consider $\mathbb{R}/\mathbb{Q}$ as a one point compactification of $\mathbb{R}$?...as the topological equivalent of identifying the endpoints of the interval $[0, 1]$ (thereby enabling us to 'shrink' $\mathbb{R}$ to a single point?)...making any 'order' impossible ('order' assumes a minimum of two)?

Do $\mathbb{R}/\mathbb{Q}$, R/Q+ and R/Q,+ mean different things? — S.C., May 19 '24 at 06:57
R/Q,+ (and R/Q in the original quoted question) refer to a group over addition …the cosets of Q in R are the elements of the group, the group operation is addition — 21stCenturyParadox, May 20 '24 at 07:29
The algebraic quotient R/Q where R is considered as a group over addition and Q a normal subgroup — 21stCenturyParadox, May 20 '24 at 07:56

score 1 · Answer 1 · answered May 20 '24 at 08:14

1

You may be confusing $\mathbb R/\mathbb Z$ and $\mathbb R/\mathbb Q$. The former can indeed be viewed as the interval $[0.1]$ with endpoints identified, but the latter is much more complicated. There is no connection between the topology of $\mathbb R/\mathbb Q$ and the phenomena in Smooth Infinitesimal Analysis (SIA). As you pointed out, the latter are heavily dependent on intuitionistic logic, which is unrelated to the analysis of the topology of $\mathbb R/\mathbb Q$ in the question you linked.

answered May 20 '24 at 08:14

Mikhail Katz

47,573

I am considering this quote from the Wiki entry on the Real projective line: "The real projective line can be modeled by the projectively extended real line, which consists of the real line together with a point at infinity; i.e., the one-point compactification of R"...which is modeled as a circle...given that form would there not be a map to the circle arrived at by identifying the endpoints of the interval [0, 1]? Is Q/Z (Q restricted to [0, 1]) not a normal subgroup of R/Z? – 21stCenturyParadox May 21 '24 at 10:38
...it would seem that what can be said of R/Q can also be said of R[0, 1]/Q[0, 1] (that is considering R/Q only over the interval [0, 1] as in the construction of the Vitali set). – 21stCenturyParadox May 21 '24 at 10:48

Is there a connection between the topology of $\mathbb{R}/\mathbb{Q}$ and the logic fundamental to Smooth Infinitesimal Analysis?

1 Answers1