/sci/ - Science & Math

>>15141801
>>15141840
I suppose yes to what you say, but I'd probably not phase this in terms of ZFC. Not in set theory, and certainly not in set theory with choice. If you got a set theory with choice, the choice functions in general can't be evaluated either.
You'll have to be more concrete about your talking about "too fine". Most reals in ZFC will be of the uncomputable sort in the above sense, because in fact - worse - by the "tea table argument", most numbers in ZFC are not even definable, i.e. "write-downable".
https://en.wikipedia.org/wiki/Definable_real_number

Aside: The digits of the sequence I stated can't be evaluated already in an arithmetic. If the Church-Turing thesis holds, and we're interested in computability, why import that above ballast.
Let I in the naturals be an index, S in {2, 3} be a sign (2 for minus and 3 for plus), N in the naturals be the numerator and D in the naturals without 0 be the enumerator. Then you can code indexed rationals as naturals e.g. via
2^I · 3^S · 5^N · 7^D
(idx=80, rational=-2/11) = 2^80 · 3^2 · 5^2 · 7^11 \approx e^80
By unique prime decomposition, you can find the index and the rational.
In this way you can define Cauchy sequences in the rational (representatives of Cauchy reals) as subsets of the natural numbers. The second order arithmetic quantifier
[math]\forall S\subset {\mathbb N}.[/math]
thus lets you formulate theorems of the reals.

If ya interested, I took my example directly from a collection of results in computability theory or computable analysis. In particular, if you telly ourself you're just going to restrict your mathematics to computable reals, then it turns out you're forced to give up completeness
https://en.wikipedia.org/wiki/Specker_sequence

>You need a calculator that follows some other theory in order to get a proof of an explicit value for the n'th digit.
The other theory will have the same problem with any of its own enumeration of Turing machines and the corresponding number.

>>12523179
The first chapter of a treatment of QFT, see

https://en.wikipedia.org/wiki/Scalar_field_theory#Quantum_scalar_field_theory

Some sentences in your text don't even end in a dot, so I suppose it's just some online uni exercises sheet

What's an example for when the Newton method (or a variation of it) is used in practice? Bonus points for needing to solve algebraic problems?