/sci/ - Science & Math

>>10986103
This is obvious, since the descriptions of a fixed alphabet can be enumerated. Say the alpabet we want to use has 256 character
>abc...xyzABC...XYZ+'#-.;....0123456789

and enumerate all books by

>a,b,c,...,7,8,9,aa,ab,ac,...a9,ba,bb,bc,...aaa,aab,...
etc

If you use sets which constructivel can't be enumerated (put in correspondence with the first countably infinite ordinal, say N), then you can't describe (or conceptualize in your head) all its elements as individuals.
Set theory is able to characterize collections as a set, but not seperate out all their contituents.

>>10982527
> the reals, their practicality
What did you mean by this.

It's rare that any engineer even uses formal field extensions - only some physicists do when the work with mathematica. Most of the time they work with 16-digit approximations of real numbers. Would be interested we we lived in a world where you could even calculate with the reals. Try doing Newton method without truncating at each and ever step. You can't even use the method using Q, let alone R.
The reals are "used" in algorithm descriptions to not having to track error terms of trigonometric functions and roots - that's about it.

>>10982557
Axioms make a hoc judgement about boolean sentences. Definitions introduce symbols and their subsitution rules.

If you work within an axiomatic framework, often people introduce predicates that aim at specifying / pinning down a certain set of object (e.g. you formally thin of yourself workin in a set theoretic grounding and talk about groups - then you say what a group is and continue working with sets that, explicitly stated or not, are restricted to those fulfilling the group axioms). In that case, it looks like an axiom, but in reality you're just introducing a conditional for all statements that follow.

>>9521323
I see no scenario in which you'd have come from learning about existing AI and not think "ah, it's just math". The surprising part is maybe that it's the systems you will probably have looked at are just regression challenges, and for that you only need to know methods to find extrema, i.e. optimization algorithms, in large spaces.