The field of “useful reals” between rational and real numbers (2019)

Post by [deleted] » Wed, Mar 25, 2020, 1:01 AM UTC

effie wrote: post #41

klodolph wrote: post #11

Earlier quoted context omitted.

It’s larger than the rational numbers in the sense that it is a strict superset. Cardinality is what a lot of people reach for when they are talking about “larger” or “smaller”, but there are lots of other useful concepts which we can translate to “larger” and “smaller”. So when someone says “larger” or “smaller”, your first step might be to try and translate that relationship into a more precise mathematical concept…

> “closeness” does not necessarily exist in a vector space. The asker will give a definition. For example, two vectors are close if sqrt of dot product of difference of the two vectors is smaller than some number delta.

Post by lonelappde » Wed, Mar 25, 2020, 1:25 AM UTC

pdonis wrote: post #21

ur-whale wrote: post #17

Earlier quoted context omitted.

> Guess it's not. Isn't there a theorem that speaks of the existence or non-existence of a set whose cardinality is strictly larger than Q and strictly smaller than R. And a conjecture that says this theorem might well be unprovable?

A "theorem" is a proved proposition, so there is no such thing as a theorem that might well be unprovable. The proposition you refer to (which is not a theorem since no proof is known, and in fact it has been shown that this proposition is logically independent of the usual foundations of set theory, so it cannot be proved in that framework) is called the Continuum Hypothesis: https://en.wikipedia.org/wiki/Continuum_…

Post by btilly » Wed, Mar 25, 2020, 1:35 AM UTC

NelsonMinar wrote: post #70

Another related topic of interest is constructivism in mathematics. Unfortunately the wikipedia article is pretty abstruse, anyone have a more down to earth one? https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_... (Note this is different from constructible numbers, which the author mentions. That has to do with classical geometry.)

Post by lonelappde » Wed, Mar 25, 2020, 1:44 AM UTC

thaumasiotes wrote: post #46

OscarCunningham wrote: post #38

> A “useful real” is just a real number that can be precisely described (not just approximated!) by some symbolic notation. Obviously, this definition is loose and depends greatly on your choice of symbols and their definitions. In fact, the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn'…

> the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn't be precisely described Is this true without the Axiom of Choice? Don't you need a choice function to order the numbers before you can diagonalize them?

Post by naniwaduni » Wed, Mar 25, 2020, 1:48 AM UTC

perl4ever wrote: post #9

xtacy wrote: post #6

Earlier quoted context omitted.

Yep, that's right. Its cardinality is the same as rationals, since it's countable.

How disappointing. Unlike most of the time, I read the article first and now that I'm here, that was the question I had - clearly it's smaller than reals, but how and why is this field larger than rational numbers? Guess it's not.