>>10890919See also OP's pic to answer your question. The answer is:
No.
The natural numbers N can cardinality |N|. And the continuum (and thus the standard models of the reals) have cardinality of the power set P(N) of naturals. And for any cardinality k, you have that |P(k)| > |k|, i.e. most set theories prove that there are more reals than natural numbers.
You can put all strings in bijection with N, so there's more reals than strings to describe things. Therefore, the bulk of the reals can't be described on an individual level. Set theory just speaks about all the reals and all subsets of the reals as collections, but you can't pin down all reals or all subsets of any infinite set.
Reals like 1 and .
An expression like
is a description of and the partial series gives a way of computing it to an arbitrary precision. Those are all among the strings you enumerated.
However, not all definable numbers are computable, see e.g.
>>10890027And those who aren't even definable* are naturally also not computable.
This thread is largely about whether it's a good thing even to say "those numbers which aren't definable", because they can't be accessed on an individual level anyhow. Set theory or second order logic allows you to reason about collections without the members being accessible - at least in classical nonconstructive logic there are such existence statements.
In constrictive logic of reals you still got uncountability in a formal sense, but it's significantly harder to make existence statements.
>the universe adhere's to set rules>in the platonic realmAlso for you, see
>>10890027>Given two 15-by-15 matrices a,b, let Z(a,b) be true if they can be multiplied in some way, possibly with repetition, (e.g. ab, bab, bbabbba, etc.) so that this product is the zero matrix.It's known that there can't be an algorithm that can solve this
