/sci/ - Science & Math

can science go harder than this? is it even possible?

>>14508536
>>14508512

Is the speed of light a manifestation of a 4d real number? Is it the pi of the 4d time sphere?

>here's your model of the continuum, bro
>it only comes at the expense of:
> the arithmetic of rational and natural numbers
>the ability to compute with naturals even just those less than 10
>the ability to explain basic math to a child
> all semantics

>You also get as a bonus:
> a whole bunch of natural questions that turn out to be undecidable
> numbers that contain answers to every question ever (which are definitely (assumed to be) real)
>Banach Tarski paradox
>inaccessible dark numbers
>ability to confidently dismiss Zeno's paradoxes by confusing the listener
> a cool theorem in Fourier analysis

How is this shit still taken seriously by mathematicians? Are they retarded?

explain to me what a "real" number is without using any schizo infinities

>pro tip
u cant

explain to me what the "real" numbers are without using schizo infinities

>pro tip : you cant

name a bigger cope than pic related
>pro tip
>you cant

>Reals as Cauchy sequences with N given as a function of epsilon (epsilon rational)
Advantages:
- Can add, subtract two numbers.
- Can multiply and divide two numbers.
- Can form exp(x) and many other operations involving infinite series.
- Possible to check that x!=y when two numbers x,y are NOT equal.
- From a Cauchy sequence x_n can obtain the limit x (provided the sequence has N attached as a function of epsilon)
Disadvantages:
- No way to check whether two numbers are equal (reduces to the Halting problem).

>Reals as Cauchy sequences WITHOUT N given as a function of epsilon (epsilon rational)
Advantages
- Can add, subtract, multiply, divide (provided we know the denominator is nonzero).
Disadvantages:
- Cannot form exp(x) nor most other operations involving infinite series.
- Impossible to detect when x!=y NOR when x=y. I.e. essentially no way to compare two real numbers.
- Cannot get the limit of a Cauchy sequence of reals.
>Reals as Dedekind cuts
Advantages:
- Possible to detect when two numbers x,y are unequal.
Disadvantages:
- No way to detect when two numbers x,y are equal.
- Cannot add, subtract, multiply, divide.
- Cannot take the limit of a Cauchy sequence.
- Cannot form exp(x) nor most other operations involving infinite series.

What am I missing something here? Are Cauchy sequences with N given the clear winner among the implementations of reals?

>>11308564
*blocks your path*

>>10985570
here's some fake maths

pfff
that's piss easy
now define THIS

What's your favorite axiomatization of the real numbers?

The usual way to do it seems to be to view them as a totally ordered field that is Dedekin-Complete, or as Cauchy Sequences of rational numbers.

Personally I like Tarski's axiomatization of the real numbers.

1. Symmetry
If x < y, then not y < x

2. Denseness
If x < z, there exists a y such that x < y and y < z

3. Separation
For all X, Y ⊆ R, if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that
for all x ∈ X and y ∈ Y, if z ≠ x and z ≠ y, then x < z and z < y.

Or in plain language:

"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."

4. Associativity
x + (y + z) = (x + z) + y.

5. Closed under addition
For all x, y, there exists a z such that x + z = y.

6. Exclusion
If x + y < z + w, then x < z or y < w.
Axioms for one (primitives: R, <, +, 1):

7. Non-emptyness
1 ∈ R.

8. Property of 1
1 < 1 + 1.

There we go. With just 8 axioms we defined the real numbers. The standard way of doing it takes 14 different axioms, and they're nested into different groups for fields, order etc. And here, we didn't even have to define multiplication. The fact that two sets of numbers, where all elements in one collection is larger than all in the other, have a number between them also seems more intuitive than the Dedekin-Completeness of saying "all nonempty sets that are bounded have a least upper bound."

What do you think? Is this axiomatization underestimated?

If you have a definition for the natural numbers N, and allow set operations, then I also like that the real numbers can just be seen as the set of all subsets of N.

These do not describe the universe. Infinite precision such as in 'pi' or 'e' does not exist.

I'm okay with mathfags using the reals but at least admit it's just pointless circlejerking that has no actual real world application.