4

From this statement

As there is no surjection from $\mathbb{N}$ onto $\mathcal{P}(\mathbb{N})$, thus there must exist an undecidable language.

I would like to understand why similar reasoning does not work with a finite set $B$ which also has no surjection onto $\mathcal{P}(B)$! (with $|B|=K$ and $K \in \mathbb{N}$)

Why is there a minimum need for the infinite set?

EDIT Note:

Although I chose an answer, many answers and all comments are important.

Hernan_eche
  • 723
  • 1
  • 7
  • 23

2 Answers2

6

Let us first recapitulate in which context the cited statement makes sense.

  1. Let us restrict ourselves to the domain of (decision) functions in $\mathbb{N} \to \{0,1\}$ (a subset of all functions).
  2. Every such function corresponds to one element of $\mathcal{P}(\mathbb{N})$ (see characteristic function).
  3. There are countably many Turing machines (simple encoding).

By 1. and 2., there are uncountably many functions. Therefore, there are functions that have no corresponding Turing machine, that is they are not computable. There are simply too many functions; this is what is meant by "there is no surjection".

Now, there are only countably many finite sets in $\mathcal{P}(\mathbb{N})$(extension of Cantor's pairing function). Therefore, the same contradiction can not be derived when only considering finite sets.

If you add some infinite sets to your base set so it becomes uncountable, there is no reason to believe some of the finite sets were undecidable; you only know that there are some undecidable sets. In fact, all finite sets are decidable, so the culprits are always infinite sets.

Community
  • 1
Raphael
  • 73,212
  • 30
  • 182
  • 400
6

If you take any finite set $A$ of TMs, there is a language not decided by any TM in $A$ and the finite powerset would suffice for that. But this is not what we want. We want to show that there is an undecidable language, i.e. a language that no TM can decide it. The cardinality difference between a finite set and its power set would not show that. You need the cardinality difference with the set of all TMs which is countable to say there is a language which is not decided by any machine in the set.

Kaveh
  • 22,661
  • 4
  • 53
  • 113