How do you preform mathematics on unconstriable obejcts?

[Anonymous No.10256717]

makes me ruminante

[Anonymous No.10256718]

unconstructible *

[Anonymous No.10256743]

>>10256717
You don't.

[Anonymous No.10256746]

>>10256743
>computable
>[math]e[/math]
huh?

[Anonymous No.10256754]

>>10256746
You can write a program (i.e. a computable function) that will spit out the digits of [math]e[/math]. There only exist a countable number of computable functions.

[Anonymous No.10256758]

>>10256754
but [math]e[/math] is irrational. a computer can only approximae it to some degree.

[Anonymous No.10256767]

>>10256758
Yes and it can find an arbitrary amount of digits.
There's numbers for which you can't even do that much, which is why we make the distinction.

[Anonymous No.10256769]

>>10256758
In finite time with finite memory it can only compute it to some degree, but computable functions aren't quite that limited. Essentially you write a recursive program where each call spits out a digit and then do a proof by induction that it will spit out the correct digit at each point. Then you know that for any digit of [math]e[/math] that you choose, the function will eventually get to that position and spit it out correctly.

The idea is that the definable numbers, computable numbers, algebraic numbers, rational numbers, etc.. are all countable proper subsets of the reals. The undefinables are pretty much unthinkable numbers and it's probably impossible to give an example of one (in fact, I think do so would break ZFC).

[Anonymous No.10256870]

>>10256717
math is not about "constructible objects" but about eternal truth.
>If nature provides this and I provide that then some stuff will *surely* happen.

[Anonymous No.10256876]

approximate the fuck out of everything
we call it physics

[Anonymous No.10256905]

>>10256717
Let S be any set (like the set of strings on a finite alphabet).
We assume that to every pair (x,y) of elements of S we can assign a "formal Sentence" A(x,y) (whatever that means).
Let T be a subset of the set of formal sentences.
(again T can be anything you wish, like the set of "true sentences", or the set of sentences such that some computer test runs, and halts returning a positive answer).
We say that a subset R of S is "constructible" whenever there is a r in S such that for every x in S, if x belongs to R then A(r,x) belongs to T and conversely if A(r,x) belongs to T then belongs to R.
In a nutshell,informally, R is constructible if it is definable by a formal sentence indexed by some r in S, with T "testing" if the formal sentence is "true".

There is at least one subset of S which is NOT constructible:
The set F of x such that A(x,x) does not belong to T is not constructible. Indeed if it was, we'd have an element m of S defining F in the above sense; but we would also have A(m,m) in T if and only if A(m,m) outside T which is a contradiction.

It means that math will ALWAYS deal with "non constructible objects" no matter what.

[Anonymous No.10257013]

>>10256905
In general whenever you have an uncountable set you can only work with a countable subset of said set.

[Anonymous No.10257023]

>>10256876
Techniques like this exist in areas like topological logic (see Vickers Topology via Logic) but they're considered computer science. Physics doesn't really have enough math pedantry to deal with stuff like this.