Where have all the irrationals gone?

[wij]

What are those numbers near either sides of 1?
0.999... and 1.000..1 do not exist? Because every number approaching 1 is 1?
So, all numbers very close to π is π?
(sin(x+h)-sin(x))/h is very close to cos(x), thus, is EQUAL to cos(x).
e^k= lim(x->∞) (1+k/n)^n ... Woh! rule of exponential arithmetic re-defined!

A[0]=0
A[n]=(A[n-1]+1)/2
What is A[n] when n approaches infinity? Oh! 'when' is invalid! 'infinity' is
a concept, not a number!
But, A[∞] will? get to 1, ℝ is dense, density property is not broken.

A transcendental number is a number that is not algebraic, not the kind of
number +-*/ can yield. But ℝ is closed and complete, Oh! where are all those
irrational numbers gone?

PS. I believe TM (program,algorithm) should/would be the foundation of math.
These are for programmers.

[Ben Bacarisse]

wij <wyni...@gmail.com> writes:

> What are those numbers near either sides of 1?
> 0.999... and 1.000..1 do not exist?

0.999... exists (it's another way to write 1) and infinitely many
numbers of the form 1.000...1 exist.

> Because every number approaching 1 is 1?

No number approaches 1. Sequences (and/or functions) can approach 1.
0.999... denotes a limit whose value is 1. 1.000...1 denotes a set of
numbers all slightly greater than 1.

You need a new notation. You can define 0.999___ in any way you like,
and I think you intended the ... in 1.000...1 so suggest an infinite
sequence of zero digits so you might be better off writing that as
1.(000___)1.

> So, all numbers very close to π is π?

No. Only numbers equal to n are equal to n.

> (sin(x+h)-sin(x))/h is very close to cos(x), thus, is EQUAL to
> cos(x).

No.

> A[0]=0
> A[n]=(A[n-1]+1)/2
> What is A[n] when n approaches infinity? Oh! 'when' is invalid!
> 'infinity' is a concept, not a number!

No, 'when' is fine and infinity is a concept. Why do you worry that
it's a concept? You can make it a number by stating that you are using
a number system that includes infinity.

> But, A[∞] will? get to 1, ℝ is dense, density property is not
> broken.

A[∞] is not (usually) defined, even in number systems that include
infinity because, in the usual models, ∞ - 1 = ∞.

> A transcendental number is a number that is not algebraic, not the kind of
> number +-*/ can yield.

No, they can't be got from a finite number of basic operations on
rationals.

> But ℝ is closed and complete, Oh! where are all those irrational
> numbers gone?

There are there, defined as they always have been. R is closed under
the operation of taking least upper bounds (or taking limits, or
Dedekind cuts or...). This is all basic stuff.

--
Ben.

[Skep Dick]

On Monday, 29 August 2022 at 13:28:34 UTC+2, Ben Bacarisse wrote:
> No number approaches 1. Sequences (and/or functions) can approach 1.

But the Real numbers ARE sequences/functions (of Natural numbers) in the right topoi/model of computation!

https://ncatlab.org/nlab/show/function+realizability

Surely you must mean something else with those words?

[Ben Bacarisse]

Skep Dick <skepd...@gmail.com> writes:

> On Monday, 29 August 2022 at 13:28:34 UTC+2, Ben Bacarisse wrote:
>> No number approaches 1. Sequences (and/or functions) can approach 1.
> But the Real numbers ARE sequences/functions (of Natural numbers) in
> the right topoi/model of computation!

And also in the right model of the reals. Do you think that's what Wij
was talking about? Since he denies that convergent sequences define
numbers, that seems highly unlikely.

You may not believe it, but I try to discuss matters in good faith, and
that almost always involves making an effort to reply based on what the
author is most likely talking about.

--
Ben.

[Skep Dick]

On Monday, 29 August 2022 at 16:24:56 UTC+2, Ben Bacarisse wrote:
> Skep Dick <skepd...@gmail.com> writes:
>
> > On Monday, 29 August 2022 at 13:28:34 UTC+2, Ben Bacarisse wrote:
> >> No number approaches 1. Sequences (and/or functions) can approach 1.
> > But the Real numbers ARE sequences/functions (of Natural numbers) in
> > the right topoi/model of computation!
> And also in the right model of the reals.

No idea what you mean by "right" model, but in the topos I have in mind (e.g NOT the effective topos) all functions are computable.
So you can absolutely represent the reals, but only computable operations ON the reals can be realized.

Standard realizability interpretation stuff.

>Do you think that's what Wij was talking about?

Yes! Very much so. The function realizability topos is a very nice place to do Mathematics. You get to drop the silly Turing Machine restriction where you can't handle IO/interaction with the outside world.

That's a super neat property. Don't you think? It naturally agrees with human intuition about how computation works (we actually interact with, and measure the real world to obtain "real numbers").

https://ncatlab.org/nlab/show/realizability

>Since he denies that convergent sequences define numbers, that seems highly unlikely.

Oh, I absolutely deny that claim also! I have absolutely no idea whether an infinite sequence from Baire space are "convergent" or "divergent". I just know that each sequence represents a Real number.

Of course this may be a semantic gap here. Does an infinite sequence which represents a Real number "define it"? "describe it" ? "Represent it"?

Labels, labels, labels! What do they mean?

> You may not believe it, but I try to discuss matters in good faith, and
> that almost always involves making an effort to reply based on what the
> author is most likely talking about.

You may not believe it, but I also try to discuss matters in good faith, and tha talmost always involves making an effort to reply basd on what the author needs.